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Many problems in two-dimensional topology (cf. [[Topology of manifolds|Topology of manifolds]]) arise from, or have to do with, attempts to lift algebraic operations performed on the chain complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l1201701.png" /> of a universal covering complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l1201702.png" /> to geometric operations on the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l1201703.png" /> (here and below, "complex"  means a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l1201705.png" />-complex, i.e. a polyhedron with a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l1201706.png" />-structure, see [[#References|[a4]]] for a precise definition; for simplicity, one may think of a polyhedron): The chain complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l1201707.png" /> encodes the relators of the presentation (cf. [[Presentation|Presentation]]) associated to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l1201708.png" /> only up to commutators between relators.
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A first classical example for this phenomenon occurs in the proof of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017010.png" />-cobordism theorem (see [[#References|[a7]]], which thus only works for manifolds of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017011.png" />. In this context, J. Andrews and M. Curtis (see [[#References|[a8]]]) asked whether the unique <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017012.png" />-dimensional thickening of a compact connected <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017013.png" />-dimensional complex (in short, a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017015.png" />-complex) in a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017016.png" />-dimensional piecewise-linear manifold (a PL-manifold) is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017017.png" />-dimensional ball.
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Many problems in two-dimensional topology (cf. [[Topology of manifolds|Topology of manifolds]]) arise from, or have to do with, attempts to lift algebraic operations performed on the chain complex $\underline{\underline{C}} ( \tilde { K } )$ of a universal covering complex $\tilde { K } ^ { 2 }$ to geometric operations on the complex $K ^ { 2 }$ (here and below,  "complex"  means a $P L C W$-complex, i.e. a polyhedron with a $C W$-structure, see [[#References|[a4]]] for a precise definition; for simplicity, one may think of a polyhedron): The chain complex $\underline{\underline{C}} ( \tilde { K } )$ encodes the relators of the presentation (cf. [[Presentation|Presentation]]) associated to $K ^ { 2 }$ only up to commutators between relators.
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A first classical example for this phenomenon occurs in the proof of the $s$-cobordism theorem (see [[#References|[a7]]], which thus only works for manifolds of dimension $\geq 6$. In this context, J. Andrews and M. Curtis (see [[#References|[a8]]]) asked whether the unique $5$-dimensional thickening of a compact connected $2$-dimensional complex (in short, a $2$-complex) in a $5$-dimensional piecewise-linear manifold (a PL-manifold) is a $5$-dimensional ball.
  
 
They show that this is implied by the Andrews–Curtis conjecture.
 
They show that this is implied by the Andrews–Curtis conjecture.
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This conjecture reads:
 
This conjecture reads:
  
AC) any contractible finite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017018.png" />-complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017019.png" />-deforms to a point, i.e. there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017020.png" />-dimensional complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017021.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017022.png" /> collapses to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017023.png" /> and to a point: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017024.png" />. (Cf. [[#References|[a4]]] for the precise notion of a collapse, which is a deformation retraction through  "free faces" .)
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AC) any contractible finite $2$-complex $3$-deforms to a point, i.e. there exists a $3$-dimensional complex $L^3$ such that $L^3$ collapses to $K ^ { 2 }$ and to a point: $K ^ { 2 } \swarrow L ^ { 3 } \searrow \operatorname{pt}$. (Cf. [[#References|[a4]]] for the precise notion of a collapse, which is a deformation retraction through  "free faces" .)
  
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/l120170a.gif" />
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<img src="https://www.encyclopediaofmath.org/legacyimages/common_img/l120170a.gif" style="border:1px solid;"/>
  
 
Figure: l120170a
 
Figure: l120170a
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A sequence of  "elementary"  collapses yielding a collapse
 
A sequence of  "elementary"  collapses yielding a collapse
  
To a contractible finite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017025.png" />-complex there corresponds a balanced presentation (cf. [[Presentation|Presentation]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017026.png" /> of the trivial group. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017027.png" />-deformations can be translated into a sequence of Andrews–Curtis moves on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017028.png" />:
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To a contractible finite $2$-complex there corresponds a balanced presentation (cf. [[Presentation|Presentation]]) $\mathcal{P} = \langle x _ { 1 } , \dots , x _ { n } | R _ { 1 } , \dots , R _ { n } \rangle$ of the trivial group. $3$-deformations can be translated into a sequence of Andrews–Curtis moves on $\mathcal{P}$:
  
1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017029.png" />;
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1) $R _ { i } \rightarrow R _ { i } ^ { - 1 }$;
  
2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017031.png" />;
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2) $R _ { i } \rightarrow R _ { i } R _ { j }$, $i \neq j$;
  
3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017033.png" /> any word;
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3) $R _ { i } \rightarrow w R _ { i } w ^ { - 1 }$, $w$ any word;
  
4) add a generator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017034.png" /> and a relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017035.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017036.png" /> any word in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017037.png" />.
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4) add a generator $x_{n+1}$ and a relation $wx_{n+1}$, $w$ any word in $x _ { 1 } , \ldots , x _ { n }$.
  
 
Hence, an equivalent statement of the Andrews–Curtis conjecture is: Any balanced presentation of the trivial group can be transformed into the empty presentation by Andrews–Curtis moves.
 
Hence, an equivalent statement of the Andrews–Curtis conjecture is: Any balanced presentation of the trivial group can be transformed into the empty presentation by Andrews–Curtis moves.
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Here are some prominent potential counterexamples to AC):
 
Here are some prominent potential counterexamples to AC):
  
1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017038.png" /> (E.S. Rapaport, see [[#References|[a42]]]);
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1) $\langle a , b , c | c ^ { - 1 } b c = b ^ { 2 } , a ^ { - 1 } c a = c ^ { 2 } , b ^ { - 1 } a b = a ^ { 2 } \rangle$ (E.S. Rapaport, see [[#References|[a42]]]);
  
2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017039.png" /> (R.H. Crowell and R.H. Fox, see [[#References|[a10]]], and [[#References|[a11]]] for a generalization to an infinite series);
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2) $\langle a , b | b a ^ { 2 } b ^ { - 1 } = a ^ { 3 } , a b ^ { 2 } a ^ { - 1 } = b ^ { 3 } \rangle$ (R.H. Crowell and R.H. Fox, see [[#References|[a10]]], and [[#References|[a11]]] for a generalization to an infinite series);
  
3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017040.png" /> (S. Akbulut and R. Kirby, see [[#References|[a12]]]). This example corresponds to a homotopy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017041.png" />-sphere which is shown to be standard by a judicious addition of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017042.png" />-, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017043.png" />-handle pair, see [[#References|[a13]]], and [[#References|[a6]]];
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3) $\langle a , b | a b a = b a b , a ^ { 4 } = b ^ { 5 } \rangle$ (S. Akbulut and R. Kirby, see [[#References|[a12]]]). This example corresponds to a homotopy $4$-sphere which is shown to be standard by a judicious addition of a $2$-, $3$-handle pair, see [[#References|[a13]]], and [[#References|[a6]]];
  
4) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017044.png" /> (C.McA. Gordon).
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4) $\langle a , b | a = [ a ^ { p } , b ^ { q } ] , b = [ a ^ { r } , b ^ { s } ] \rangle$ (C.McA. Gordon).
  
An analogue of the conjecture is true in all dimensions different from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017045.png" />; in fact, the following generalization of it to non-trivial groups and keeping a subcomplex fixed holds (see [[#References|[a14]]] for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017046.png" /> and [[#References|[a15]]] for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017047.png" />; cf. also [[Homotopy type|Homotopy type]]): Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017048.png" />; and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017049.png" /> be a simple-homotopy equivalence of connected, finite complexes, inducing the identity on the common subcomplex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017050.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017051.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017052.png" /> is homotopic rel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017053.png" /> to a deformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017054.png" /> which leaves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017055.png" /> fixed throughout. A deformation is a composition of expansions and collapses; if the maximal cell dimension involved is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017056.png" />, this will be denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017057.png" />, see [[#References|[a7]]].
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An analogue of the conjecture is true in all dimensions different from $2$; in fact, the following generalization of it to non-trivial groups and keeping a subcomplex fixed holds (see [[#References|[a14]]] for $n \geq 3$ and [[#References|[a15]]] for $n = 1$; cf. also [[Homotopy type|Homotopy type]]): Let $n \neq 2$; and let $f : K _ { 0 } \rightarrow K _ { 1 }$ be a simple-homotopy equivalence of connected, finite complexes, inducing the identity on the common subcomplex $L$, $n = \operatorname { max } ( \operatorname { dim } ( K _ { 0 } - L ) , \operatorname { dim } ( K _ { 1 } - L ) )$. Then $f$ is homotopic rel $L$ to a deformation $K _ { 0 } ^ { n + 1 } \searrow  K _ { 1 }$ which leaves $L$ fixed throughout. A deformation is a composition of expansions and collapses; if the maximal cell dimension involved is $n$, this will be denoted by $K N L$, see [[#References|[a7]]].
  
The corresponding statement for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017058.png" /> is called the relative generalized Andrews–Curtis conjecture ( "generalized"  because the fundamental group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017059.png" /> may be non-trivial;  "relative"  because of the fixed subcomplex). The subcase <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017060.png" />, i.e. the expectation that a simple-homotopy equivalence between finite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017061.png" />-dimensional complexes can always be replaced by a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017062.png" />-deformation, is called the generalized Andrews–Curtis conjecture, henceforth abbreviated AC'); see [[#References|[a4]]].
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The corresponding statement for $n = 2$ is called the relative generalized Andrews–Curtis conjecture ( "generalized"  because the fundamental group of $K_i$ may be non-trivial;  "relative"  because of the fixed subcomplex). The subcase $L = \phi$, i.e. the expectation that a simple-homotopy equivalence between finite $2$-dimensional complexes can always be replaced by a $3$-deformation, is called the generalized Andrews–Curtis conjecture, henceforth abbreviated AC'); see [[#References|[a4]]].
  
Suppose <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017063.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017064.png" /> are presentations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017065.png" /> such that
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Suppose $\mathcal{P} = \langle a _ { 1 } , \dots , a _ { g } | R _ { 1 } , \dots , R _ { n } \rangle$ and $\mathcal{Q} = \langle a _ { 1 } , \dots , a _ { g } | S _ { 1 } , \dots , S _ { n } \rangle$ are presentations of $\pi$ such that
  
D) each difference <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017066.png" /> is a consequence of commutators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017067.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017068.png" />) of relators, then the corresponding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017069.png" />-dimensional complexes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017070.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017071.png" /> are simple-homotopy equivalent. Furthermore, up to Andrews–Curtis moves the converse is true, see [[#References|[a16]]].
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D) each difference $R _ { i } S _ { i } ^ { - 1 }$ is a consequence of commutators $[ R _ { j } , R _ { k } ]$ ($1 \leq j , k \leq n$) of relators, then the corresponding $2$-dimensional complexes $K ^ { 2 }$ and $L^{2}$ are simple-homotopy equivalent. Furthermore, up to Andrews–Curtis moves the converse is true, see [[#References|[a16]]].
  
Thus, in terms of presentations, AC') states that under the assumption D), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017072.png" /> can actually be made to coincide with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017073.png" /> by Andrews–Curtis moves, for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017074.png" />. Even though AC') is expected to be false, D) implies that the difference <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017075.png" /> between the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017076.png" />th relators by Andrews–Curtis moves can be pushed to become a product of arbitrarily high commutators of relators, see [[#References|[a17]]]. Furthermore, taking the one-point union not only with a finite number of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017077.png" />-spheres, but also with certain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017078.png" />-complexes of minimal Euler characteristic, eliminates any potential difference between simple-homotopy and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017079.png" />-deformations: A simple homotopy equivalence between finite connected <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017080.png" />-complexes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017081.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017082.png" /> gives rise to a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017083.png" />-deformation between the one point union of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017084.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017085.png" />) with a sufficiently large number of standard complexes of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017086.png" />, see [[#References|[a16]]]. For a detailed discussion on the status of the conjectures AC), AC') and relAC'), see [[#References|[a4]]], Chap. XII.
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Thus, in terms of presentations, AC') states that under the assumption D), $R_i$ can actually be made to coincide with $S _ { i }$ by Andrews–Curtis moves, for all $i$. Even though AC') is expected to be false, D) implies that the difference $R _ { i } S _ { i } ^ { - 1 }$ between the $i$th relators by Andrews–Curtis moves can be pushed to become a product of arbitrarily high commutators of relators, see [[#References|[a17]]]. Furthermore, taking the one-point union not only with a finite number of $2$-spheres, but also with certain $2$-complexes of minimal Euler characteristic, eliminates any potential difference between simple-homotopy and $3$-deformations: A simple homotopy equivalence between finite connected $2$-complexes $K ^ { 2 }$, $L^{2}$ gives rise to a $3$-deformation between the one point union of $K ^ { 2 }$ (respectively, $L^{2}$) with a sufficiently large number of standard complexes of ${\bf Z} _ { 2 } \times {\bf Z} _ { 4 }$, see [[#References|[a16]]]. For a detailed discussion on the status of the conjectures AC), AC') and relAC'), see [[#References|[a4]]], Chap. XII.
  
There is a close relation between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017087.png" />-complexes and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017088.png" />-manifolds. (cf. [[Three-dimensional manifold|Three-dimensional manifold]]): Every compact connected <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017089.png" />-dimensional manifold with non-empty boundary collapses to a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017090.png" />-dimensional complex, called a spine (see [[#References|[a4]]], Chap. I, §2.2), and thus determines a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017091.png" />-deformation class of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017092.png" />-complexes. A counterexample to AC) which is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017093.png" />-manifold with spine <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017094.png" /> would disprove the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017096.png" />-dimensional Poincaré conjecture (cf. [[Three-dimensional manifold|Three-dimensional manifold]])
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There is a close relation between $2$-complexes and $3$-manifolds. (cf. [[Three-dimensional manifold|Three-dimensional manifold]]): Every compact connected $3$-dimensional manifold with non-empty boundary collapses to a $2$-dimensional complex, called a spine (see [[#References|[a4]]], Chap. I, §2.2), and thus determines a $3$-deformation class of $2$-complexes. A counterexample to AC) which is a $3$-manifold with spine $K ^ { 2 }$ would disprove the $3$-dimensional Poincaré conjecture (cf. [[Three-dimensional manifold|Three-dimensional manifold]])
  
 
==Zeeman conjecture.==
 
==Zeeman conjecture.==
This prominent conjecture on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017097.png" />-complexes actually implies the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017098.png" />-dimensional Poincaré conjecture. The Zeeman conjecture states that (see [[#References|[a23]]]):
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This prominent conjecture on $2$-complexes actually implies the $3$-dimensional Poincaré conjecture. The Zeeman conjecture states that (see [[#References|[a23]]]):
  
Z) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l12017099.png" /> is a compact contractible <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170100.png" />-dimensional complex, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170101.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170102.png" /> is an interval. Note that Z) also implies AC), as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170103.png" /> would be a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170104.png" />-deformation. Examples which fulfil <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170105.png" /> are the dunce hat, Bing's house and the house with one room, see [[#References|[a4]]]. However, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170106.png" /> is not even established (as of 1999) for most of the standard <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170107.png" />-complexes of presentations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170108.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170109.png" />, even though these are Andrews–Curtis equivalent to the empty presentation.
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Z) if $K ^ { 2 }$ is a compact contractible $2$-dimensional complex, then $K ^ { 2 } \times I \searrow \operatorname{pt}$, where $I$ is an interval. Note that Z) also implies AC), as $K ^ { 2 } \nearrow K ^ { 2 }\times I \searrow \operatorname {pt}$ would be a $3$-deformation. Examples which fulfil $K ^ { 2 } \times I \searrow \operatorname{pt}$ are the dunce hat, Bing's house and the house with one room, see [[#References|[a4]]]. However, $K ^ { 2 } \times I \searrow \operatorname{pt}$ is not even established (as of 1999) for most of the standard $2$-complexes of presentations $\langle a , b | a ^ { p } b ^ { q } , a ^ { r } b ^ { s } \rangle$ where $p s - q r = \pm 1$, even though these are Andrews–Curtis equivalent to the empty presentation.
  
 
As for AC), there is a straightforward generalization to non-trivial groups; the generalized Zeeman conjecture:
 
As for AC), there is a straightforward generalization to non-trivial groups; the generalized Zeeman conjecture:
  
Z') <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170110.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170111.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170112.png" />. Of course, Z') implies both Z) and AC'). It is open (as of 1999) whether AC') implies Z'), but given a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170113.png" />-deformation between finite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170114.png" />-complexes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170115.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170116.png" /> can be expanded by a sequence of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170117.png" />-expansions to a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170118.png" />-complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170119.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170120.png" />, see [[#References|[a18]]].
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Z') $K ^ { 2 } / \searrow L ^ { 2 }$ implies $K ^ { 2 } \times I \searrow L ^ { 2 }$ or $L ^ { 2 } \times I \searrow  K ^ { 2 }$. Of course, Z') implies both Z) and AC'). It is open (as of 1999) whether AC') implies Z'), but given a $3$-deformation between finite $2$-complexes $K ^ { 2 }/ \stackrel { 3 } { \searrow  } L ^ { 2 }$, then $K ^ { 2 }$ can be expanded by a sequence of $2$-expansions to a $2$-complex $K ^ { \prime 2 } \searrow K ^ { 2 }$ such that $K ^ { \prime 2 } \times I \searrow \operatorname{pt}$, see [[#References|[a18]]].
  
In the special case of expansion of a single <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170121.png" />-ball, followed by a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170122.png" />-collapse, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170123.png" />, it is true that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170124.png" />, see [[#References|[a19]]], [[#References|[a20]]], [[#References|[a21]]]. This can be viewed as a first step in proving Z') modulo AC'), as every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170125.png" />-deformation between finite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170126.png" />-complexes can be replaced by one where each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170127.png" />-ball is transient, i.e. is collapsed (in general from a different free face) immediately after its expansion, see [[#References|[a22]]]. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170129.png" />, this method is called collapsing by adding a cell and works for all above-mentioned examples for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170130.png" />.
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In the special case of expansion of a single $3$-ball, followed by a $3$-collapse, $K ^ { 2 } \nearrow K ^ { 2 } \cup _ { B ^ { 2 } } B ^ { 3 } \searrow L ^ { 2 }$, it is true that $K ^ { 2 } \times I \searrow L ^ { 2 }$, see [[#References|[a19]]], [[#References|[a20]]], [[#References|[a21]]]. This can be viewed as a first step in proving Z') modulo AC'), as every $3$-deformation between finite $2$-complexes can be replaced by one where each $3$-ball is transient, i.e. is collapsed (in general from a different free face) immediately after its expansion, see [[#References|[a22]]]. For $L ^ { 2 } = \operatorname {pt}$, this method is called collapsing by adding a cell and works for all above-mentioned examples for $K ^ { 2 } \times I \searrow \operatorname{pt}$.
  
A second general method for collapsing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170131.png" /> was proposed by A. Zimmermann (see [[#References|[a24]]]) and is called prismatic collapsing. At first one gets rid of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170132.png" />-dimensional part of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170133.png" /> as follows: For each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170134.png" />-cell <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170135.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170136.png" /> one collapses <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170137.png" /> to the union of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170138.png" /> and a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170139.png" />-cell <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170140.png" /> such that the direct product projection maps <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170141.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170142.png" /> homeomorphically. Then one looks for a collapse of the resulting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170143.png" />-complex.
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A second general method for collapsing $K ^ { 2 } \times I$ was proposed by A. Zimmermann (see [[#References|[a24]]]) and is called prismatic collapsing. At first one gets rid of the $3$-dimensional part of $K ^ { 2 } \times I$ as follows: For each $2$-cell $C ^ { 2 }$ of $K ^ { 2 }$ one collapses $C ^ { 2 } \times I$ to the union of $\partial C ^ { 2 } \times I$ and a $2$-cell $C ^ { * } \subset C ^ { 2 } \times I$ such that the direct product projection maps $\operatorname { lnt }  C ^ { * }$ onto $\operatorname { lnt }  C ^ { 2 }$ homeomorphically. Then one looks for a collapse of the resulting $2$-complex.
  
One may say that prismatic collapsing is a very rough method, but exactly this roughness allows one to give an algebraic criterion for the prismatic collapsibility of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170144.png" />: Attaching mappings for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170145.png" />-cells of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170146.png" /> have to determine a basis-up-to-conjugation in the free fundamental group of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170147.png" />-dimensional skeleton (see [[#References|[a7]]]) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170148.png" />.
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One may say that prismatic collapsing is a very rough method, but exactly this roughness allows one to give an algebraic criterion for the prismatic collapsibility of $K ^ { 2 } \times I$: Attaching mappings for $2$-cells of $K ^ { 2 }$ have to determine a basis-up-to-conjugation in the free fundamental group of the $1$-dimensional skeleton (see [[#References|[a7]]]) of $K ^ { 2 }$.
  
Z) becomes true if one admits multiplication of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170149.png" /> by the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170150.png" />-fold product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170151.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170152.png" />: For each contractible <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170153.png" /> there exists an integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170154.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170155.png" />, see [[#References|[a19]]], [[#References|[a20]]]. In fact, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170156.png" /> suffices for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170157.png" />, see [[#References|[a25]]]. It is surprising that there is such a large gap between the presently (1999) known (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170158.png" />) and Zeeman's conjectured (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170159.png" />) values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170160.png" />.
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Z) becomes true if one admits multiplication of $K ^ { 2 }$ by the $n$-fold product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170151.png"/> of $I$: For each contractible $K ^ { 2 }$ there exists an integer $n$ such that $K ^ { 2 } \times I ^ { n } \searrow \operatorname{pt}$, see [[#References|[a19]]], [[#References|[a20]]]. In fact, $n = 6$ suffices for all $K ^ { 2 }$, see [[#References|[a25]]]. It is surprising that there is such a large gap between the presently (1999) known ($n = 6$) and Zeeman's conjectured ($n = 1$) values of $n$.
  
On the other hand, a generalization of Z) to higher-dimensional complexes is false, since for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170161.png" /> there exists a contractible complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170162.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170163.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170164.png" /> is not collapsible, see [[#References|[a26]]]. The proof of non-collapsibility is based on a very specific (one may say  "bad" ) local structure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170165.png" />. So, the idea to investigate Z) for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170166.png" />-dimensional polyhedra with a  "nice"  local structure (such polyhedra are called special) seems to be very promising.
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On the other hand, a generalization of Z) to higher-dimensional complexes is false, since for any $n &gt; 2$ there exists a contractible complex $K ^ { n }$ of dimension $n$ such that $K ^ { n } \times 1$ is not collapsible, see [[#References|[a26]]]. The proof of non-collapsibility is based on a very specific (one may say  "bad" ) local structure of $K ^ { n }$. So, the idea to investigate Z) for $2$-dimensional polyhedra with a  "nice"  local structure (such polyhedra are called special) seems to be very promising.
  
In fact, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170167.png" /> is a special spine of a homotopy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170168.png" />-ball <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170169.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170170.png" /> collapses onto a homeomorphic copy of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170171.png" />, see [[#References|[a27]]]. It follows that Z) is true for all special spines of a genuine <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170172.png" />-ball and that for special spines of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170173.png" />-manifolds, Z) is equivalent to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170174.png" />-dimensional Poincaré conjecture. Surprisingly, for special polyhedra that cannot be embedded in a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170175.png" />-manifold, Z) turns out to be equivalent to AC) (see [[#References|[a28]]]), so that for special polyhedra, Z) is equivalent to the union of AC) and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170176.png" />-dimensional Poincaré conjecture.
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In fact, if $K ^ { 2 }$ is a special spine of a homotopy $3$-ball $M ^ { 3 }$, then $K ^ { 2 } \times I$ collapses onto a homeomorphic copy of $M ^ { 3 }$, see [[#References|[a27]]]. It follows that Z) is true for all special spines of a genuine $3$-ball and that for special spines of $3$-manifolds, Z) is equivalent to the $3$-dimensional Poincaré conjecture. Surprisingly, for special polyhedra that cannot be embedded in a $3$-manifold, Z) turns out to be equivalent to AC) (see [[#References|[a28]]]), so that for special polyhedra, Z) is equivalent to the union of AC) and the $3$-dimensional Poincaré conjecture.
  
==<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170177.png" />-question.==
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==$\operatorname{Wh} ^ { * }$-question.==
Another situation where dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170178.png" /> presents a severe difficulty in passing from chain complexes to geometry concerns the Whitehead group and the Whitehead torsion of a pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170179.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170180.png" /> is a strong deformation retraction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170181.png" /> (cf. [[Whitehead group|Whitehead group]], [[Whitehead torsion|Whitehead torsion]]). All elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170182.png" /> can be realized by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170183.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170184.png" /> be the set of those torsion values that can be realized by a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170185.png" />-dimensional extension, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170186.png" />. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170188.png" />-question is whether <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170189.png" /> can happen; see [[#References|[a4]]]. If so, another related question is whether <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170190.png" /> is a subgroup.
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Another situation where dimension $2$ presents a severe difficulty in passing from chain complexes to geometry concerns the Whitehead group and the Whitehead torsion of a pair $( K , L )$, where $L$ is a strong deformation retraction of $K$ (cf. [[Whitehead group|Whitehead group]], [[Whitehead torsion|Whitehead torsion]]). All elements of $\operatorname{Wh} ( \pi )$ can be realized by $\operatorname { dim } K = 3$. Let $\operatorname{Wh} ^ { * } ( \pi ) \subseteq \operatorname{Wh} ( \pi )$ be the set of those torsion values that can be realized by a $2$-dimensional extension, i.e. $\operatorname { dim } ( K - L ) \leq 2$. The $\operatorname{Wh} ^ { * }$-question is whether $\operatorname{Wh} ^ { * } ( \pi ) \neq \{ 0 \}$ can happen; see [[#References|[a4]]]. If so, another related question is whether $\operatorname{ Wh} ^ { * } ( \pi )$ is a subgroup.
  
A famous result of O.S. Rothaus is that there exist examples <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170191.png" /> for dihedral groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170192.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170193.png" />; see [[#References|[a29]]]. This result was the basis for work by M.M. Cohen [[#References|[a26]]] on the generalization of Z) to higher dimensions.
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A famous result of O.S. Rothaus is that there exist examples $\tau \in \operatorname{Wh} ( \pi )$ for dihedral groups $\pi$ with $\tau \notin \operatorname{Wh} ^ { * } ( \pi )$; see [[#References|[a29]]]. This result was the basis for work by M.M. Cohen [[#References|[a26]]] on the generalization of Z) to higher dimensions.
  
 
==Whitehead's asphericity question.==
 
==Whitehead's asphericity question.==
A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170194.png" />-complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170195.png" /> is called aspherical if its second [[Homotopy group|homotopy group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170197.png" /> is trivial (or equivalently, if all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170198.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170199.png" /> are trivial). J.H.C. Whitehead asked, (see [[#References|[a30]]]), whether subcomplexes of aspherical <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170200.png" />-complexes are themselves aspherical. An affirmative answer to this question is called the Whitehead conjecture:
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A $2$-complex $K$ is called aspherical if its second [[Homotopy group|homotopy group]] $\pi_2 ( K )$ is trivial (or equivalently, if all $\pi _ { n } ( K )$ for $n \geq 2$ are trivial). J.H.C. Whitehead asked, (see [[#References|[a30]]]), whether subcomplexes of aspherical $2$-complexes are themselves aspherical. An affirmative answer to this question is called the Whitehead conjecture:
  
WH) A subcomplex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170201.png" /> of an aspherical <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170202.png" />-complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170203.png" /> is aspherical.
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WH) A subcomplex $K$ of an aspherical $2$-complex $L$ is aspherical.
  
 
A lot of work has already been done in trying to solve this conjecture and there are about six false results in the literature which would imply WH).
 
A lot of work has already been done in trying to solve this conjecture and there are about six false results in the literature which would imply WH).
  
WH) is known to be true if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170204.png" /> has at most one <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170205.png" />-cell and also in the case where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170206.png" /> is either finite, Abelian or free, see [[#References|[a31]]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170207.png" /> is a subcomplex of an aspherical <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170208.png" />-complex, then one can show that the second homology of the covering <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170209.png" /> corresponding to the commutator subgroup is trivial. In fact, J.F. Adams has shown [[#References|[a32]]] that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170210.png" /> has an acyclic regular covering <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170211.png" /> (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170212.png" />). A counterexample to WH) can thus be covered by an acyclic complex, but not by a contractible one.
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WH) is known to be true if $K$ has at most one $2$-cell and also in the case where $\pi_1 ( L )$ is either finite, Abelian or free, see [[#References|[a31]]]. If $K$ is a subcomplex of an aspherical $2$-complex, then one can show that the second homology of the covering $\overline { K } \rightarrow K$ corresponding to the commutator subgroup is trivial. In fact, J.F. Adams has shown [[#References|[a32]]] that $K$ has an acyclic regular covering $K ^ { * } \rightarrow \overline { K } \rightarrow K$ (i.e. $H _ { 2 } ( K ^ { * } ) = H _ { 1 } ( K ^ { * } ) = 0$). A counterexample to WH) can thus be covered by an acyclic complex, but not by a contractible one.
  
In any counterexample <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170213.png" /> to WH), the kernel of the inclusion induced mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170214.png" /> has a non-trivial, finitely generated, perfect subgroup, [[#References|[a33]]].
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In any counterexample $K \subset L$ to WH), the kernel of the inclusion induced mapping $\pi _ { 1 } ( K ) \rightarrow \pi _ { 1 } ( L )$ has a non-trivial, finitely generated, perfect subgroup, [[#References|[a33]]].
  
J. Howie has shown [[#References|[a34]]] that if WH) is false, then there exists a counterexample <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170215.png" /> satisfying either
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J. Howie has shown [[#References|[a34]]] that if WH) is false, then there exists a counterexample $K \subset L$ satisfying either
  
a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170216.png" /> is finite and contractible, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170217.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170218.png" />-cell <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170219.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170220.png" />; or
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a) $L$ is finite and contractible, and $K = L - e$ for some $2$-cell $e$ of $L$; or
  
b) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170221.png" /> is the union of an infinite ascending chain of finite non-aspherical subcomplexes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170222.png" /> such that each inclusion mapping is nullhomotopic.
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b) $L$ is the union of an infinite ascending chain of finite non-aspherical subcomplexes $K = K _ { 0 } \subset K _ { 1 } \subset \ldots$ such that each inclusion mapping is nullhomotopic.
  
 
This result has been sharpened by E. Luft, who showed that if WH) is false, then there must even exist an infinite counterexample of type b).
 
This result has been sharpened by E. Luft, who showed that if WH) is false, then there must even exist an infinite counterexample of type b).
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170223.png" /> be a finite presentation where each relator is of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170224.png" />. Such a presentation may be represented by a graph <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170225.png" /> in the following way: For each generator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170226.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170227.png" />, define a vertex labelled <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170228.png" /> and for each relator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170229.png" /> define an edge oriented from the vertex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170230.png" /> to the vertex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170231.png" /> labelled by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170232.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170233.png" /> is a tree, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170234.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170235.png" /> or the standard-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170236.png" />-complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170237.png" /> modelled on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170238.png" /> is called a labelled oriented tree.
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Let $\mathcal{P} = \langle x _ { 1 } , \dots , x _ { g } | R _ { 1 } , \dots , R _ { n } \rangle$ be a finite presentation where each relator is of the form $x _ { i } = x _ { j } x _ { k } x _ { j } ^ { - 1 }$. Such a presentation may be represented by a graph $T _ { \mathcal{P} }$ in the following way: For each generator $x_{i}$ of $\mathcal{P}$, define a vertex labelled $i$ and for each relator $x _ { i } = x _ { j } x _ { k } x _ { j } ^ { - 1 }$ define an edge oriented from the vertex $i$ to the vertex $k$ labelled by $j$. If $T _ { \mathcal{P} }$ is a tree, then $\mathcal{P}$ or $T _ { \mathcal{P} }$ or the standard-$2$-complex $K _ { \mathcal{P} }$ modelled on $\mathcal{P}$ is called a labelled oriented tree.
  
Now Howie showed [[#References|[a34]]] that if the Andrews–Curtis conjecture is true and all labelled oriented trees are aspherical, then there are no counterexamples of type a) to WH). Conversely, if there are no counterexamples of type a) to WH), then all labelled oriented trees are aspherical, which is easy to see since adding an extra relator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170239.png" /> to a labelled oriented tree yields a balanced presentation of the trivial group and hence a contractible complex.
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Now Howie showed [[#References|[a34]]] that if the Andrews–Curtis conjecture is true and all labelled oriented trees are aspherical, then there are no counterexamples of type a) to WH). Conversely, if there are no counterexamples of type a) to WH), then all labelled oriented trees are aspherical, which is easy to see since adding an extra relator $x _ { 1 } = 1$ to a labelled oriented tree yields a balanced presentation of the trivial group and hence a contractible complex.
  
So the finite case of WH) can be reduced to the study of the asphericity of labelled oriented trees. Every knot group has a labelled oriented tree presentation (the Wirtinger presentation, see, e.g., [[#References|[a6]]]) and by a theorem of C.D. Papakyriakopoulos, [[#References|[a36]]], it is known that these labelled oriented trees are aspherical. Every labelled oriented tree satisfying the small cancellation conditions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170240.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170241.png" /> or a more refined curvature condition such as the weight or cycle test, [[#References|[a37]]], is aspherical. Apart from that, there are not many classes of aspherical labelled oriented trees known: Howie, [[#References|[a35]]], shows the asphericity of labelled oriented trees of diameter at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170242.png" /> and G. Huck and S. Rosebrock have two other classes of aspherical labelled oriented trees satisfying certain conditions on the relators.
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So the finite case of WH) can be reduced to the study of the asphericity of labelled oriented trees. Every knot group has a labelled oriented tree presentation (the Wirtinger presentation, see, e.g., [[#References|[a6]]]) and by a theorem of C.D. Papakyriakopoulos, [[#References|[a36]]], it is known that these labelled oriented trees are aspherical. Every labelled oriented tree satisfying the small cancellation conditions $C ( 4 )$, $T ( 4 )$ or a more refined curvature condition such as the weight or cycle test, [[#References|[a37]]], is aspherical. Apart from that, there are not many classes of aspherical labelled oriented trees known: Howie, [[#References|[a35]]], shows the asphericity of labelled oriented trees of diameter at most $3$ and G. Huck and S. Rosebrock have two other classes of aspherical labelled oriented trees satisfying certain conditions on the relators.
  
 
An overview on WH), where further aspects of this conjecture are treated, can be found in [[#References|[a4]]], Chap. X.
 
An overview on WH), where further aspects of this conjecture are treated, can be found in [[#References|[a4]]], Chap. X.
  
 
==Wall's domination problem.==
 
==Wall's domination problem.==
Given a [[CW-complex|CW-complex]], it is natural to ask whether it can be replaced by a simpler one having the same homotopy type. Questions of this kind were first considered by J.H.C. Whitehead, who posed in particular the question: When is a CW-complex homotopy equivalent to a finite dimensional one? In [[#References|[a38]]], C.T.C. Wall answered this by giving an algebraic characterization of finiteness. He also showed that a finite complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170243.png" /> dominated by a finite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170244.png" />-complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170245.png" /> has the homotopy type of a finite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170246.png" />-complex if and only if a certain algebraic obstruction vanishes. (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170247.png" /> is dominated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170248.png" /> if the  "homotopy of X survives passing through Y" , i.e. if there are mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170249.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170250.png" /> such that the composition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170251.png" /> is homotopic to the identity). Whether  "max3,n"  can simply be replaced by  "n"  is still (1999) unanswered, due to difficulties when attempting to geometrically realize an algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170252.png" />-complex.
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Given a [[CW-complex|CW-complex]], it is natural to ask whether it can be replaced by a simpler one having the same homotopy type. Questions of this kind were first considered by J.H.C. Whitehead, who posed in particular the question: When is a CW-complex homotopy equivalent to a finite dimensional one? In [[#References|[a38]]], C.T.C. Wall answered this by giving an algebraic characterization of finiteness. He also showed that a finite complex $X$ dominated by a finite $n$-complex $Y$ has the homotopy type of a finite $\operatorname{max}( 3 , n )$-complex if and only if a certain algebraic obstruction vanishes. ($X$ is dominated by $Y$ if the  "homotopy of X survives passing through Y" , i.e. if there are mappings $f : X \rightarrow Y$, $g : Y \rightarrow X$ such that the composition $X \stackrel { f } { \rightarrow } Y \stackrel { g } { \rightarrow } X$ is homotopic to the identity). Whether  "max3,n"  can simply be replaced by  "n"  is still (1999) unanswered, due to difficulties when attempting to geometrically realize an algebraic $2$-complex.
  
In order to explain this in more detail, assume <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170253.png" /> is a chain complex of free <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170254.png" />-modules,
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In order to explain this in more detail, assume $B$ is a chain complex of free ${\bf Z} G$-modules,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170255.png" /></td> </tr></table>
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\begin{equation*} B _ { 2 } \stackrel { d } { \rightarrow } B _ { 1 } \stackrel { d _ { 1 } } { \rightarrow } B _ { 0 } \rightarrow 0, \end{equation*}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170256.png" /> is freely generated by a single element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170257.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170258.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170259.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170260.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170261.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170262.png" /> for some group element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170263.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170264.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170265.png" />. Wall asked if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170266.png" /> is necessarily the cellular chain complex of the [[Universal covering|universal covering]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170267.png" /> of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170268.png" />-complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170269.png" /> with [[Fundamental group|fundamental group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170270.png" />. An affirmative answer would resolve the difficulties in dimension two mentioned above.
+
where $B_0$ is freely generated by a single element $e_0$, $B _ { 1 }$ by $\{ e _ { 1 } ^ { i } \}$, $B _ { 2 }$ by $\{ e _ { 2 } ^ { j } \}$, $d _ { 1 } ( e _ { 1 } ^ { i } ) = g _ { i } e _ { 0 } - e _ { 0 }$ for some group element $g_i$, and $H _ { 1 } ( B ) = 0$, $H _ { 0 } ( B ) = \mathbf{Z}$. Wall asked if $B$ is necessarily the cellular chain complex of the [[Universal covering|universal covering]] $\widetilde { K }$ of a $2$-complex $K$ with [[Fundamental group|fundamental group]] $G$. An affirmative answer would resolve the difficulties in dimension two mentioned above.
  
This topological set-up can also be rephrased in terms of combinatorial group theory. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170271.png" /> be the free group generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170272.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170273.png" /> be the kernel of the homomorphism from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170274.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170275.png" />, sending <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170276.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170277.png" />. The image of the second boundary mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170278.png" /> can be shown to be isomorphic to the relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170279.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170280.png" />. Wall's question of geometric realizability now translates to asking whether the relation module generators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170281.png" /> lift to give a set of normal generators for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170282.png" />. This was answered negatively by M. Dunwoody (see [[#References|[a39]]]).
+
This topological set-up can also be rephrased in terms of combinatorial group theory. Let $F$ be the free group generated by $\{ x _ { i } \}$ and let $N$ be the kernel of the homomorphism from $F$ to $G$, sending $x_{i}$ to $g_i$. The image of the second boundary mapping $d _ { 2 }$ can be shown to be isomorphic to the relation ${\bf Z} G$-module $N / [ N , N ]$. Wall's question of geometric realizability now translates to asking whether the relation module generators $d _ { 2 } ( e _ { 2 } ^ { j } )$ lift to give a set of normal generators for $N$. This was answered negatively by M. Dunwoody (see [[#References|[a39]]]).
  
 
==Relation gap question.==
 
==Relation gap question.==
M. Dyer showed that a more serious failure of this lifting problem, the relation gap question, would actually show that there does exist a finite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170283.png" />-complex dominated by a finite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170284.png" />-complex, with vanishing obstruction, that is not homotopically equivalent to a finite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170285.png" />-complex. Here, a finite presentation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170286.png" /> of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170287.png" /> is said to have a relation gap if no normal generating set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170288.png" /> gives a minimal generating set for the relation module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170289.png" />. There have been many attempts to construct a relation gap in finitely presented groups (see [[#References|[a4]]], p. 50). The existence of an infinite relation gap for a certain finitely-generated infinitely-related group was established in the influential paper of M. Bestvina and N. Brady [[#References|[a1]]].
+
M. Dyer showed that a more serious failure of this lifting problem, the relation gap question, would actually show that there does exist a finite $3$-complex dominated by a finite $2$-complex, with vanishing obstruction, that is not homotopically equivalent to a finite $2$-complex. Here, a finite presentation $F / N$ of a group $G$ is said to have a relation gap if no normal generating set of $N$ gives a minimal generating set for the relation module $N / [ N , N ]$. There have been many attempts to construct a relation gap in finitely presented groups (see [[#References|[a4]]], p. 50). The existence of an infinite relation gap for a certain finitely-generated infinitely-related group was established in the influential paper of M. Bestvina and N. Brady [[#References|[a1]]].
  
 
==Eilenberg–Ganea conjecture.==
 
==Eilenberg–Ganea conjecture.==
Another problem revolving around geometric realizability, connected to the relation gap problem and the Whitehead conjecture, is the Eilenberg–Ganea conjecture. A group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170290.png" /> is of cohomological dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170291.png" /> if there exists a projective resolution of length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170292.png" />
+
Another problem revolving around geometric realizability, connected to the relation gap problem and the Whitehead conjecture, is the Eilenberg–Ganea conjecture. A group $G$ is of cohomological dimension $n$ if there exists a projective resolution of length $n$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170293.png" /></td> </tr></table>
+
\begin{equation*} 0 \rightarrow P _ { n } \rightarrow \ldots \rightarrow P _ { 0 } \rightarrow \mathbf{Z} \rightarrow 0 \end{equation*}
  
but no shorter one (see [[#References|[a2]]] for a good reference on these matters). It was shown by S. Eilenberg, T. Ganea and J. Stallings ([[#References|[a40]]], [[#References|[a41]]]) that a group of cohomological dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170294.png" /> admits an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170295.png" />-dimensional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170296.png" /> complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170297.png" />. In particular, there is a geometric resolution of length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170298.png" /> arising as the augmented cellular chain complex of the universal covering of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170299.png" />.
+
but no shorter one (see [[#References|[a2]]] for a good reference on these matters). It was shown by S. Eilenberg, T. Ganea and J. Stallings ([[#References|[a40]]], [[#References|[a41]]]) that a group of cohomological dimension $n \neq 2$ admits an $n$-dimensional $K ( G , 1 )$ complex $K$. In particular, there is a geometric resolution of length $n$ arising as the augmented cellular chain complex of the universal covering of $K$.
  
The Eilenberg–Ganea conjecture states that this is true in dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170300.png" /> as well. This conjecture is widely believed to be wrong; promising potential counterexamples have been exhibited by Bestvina and also by Bestvina and Brady [[#References|[a1]]]. If the group in question does not have a relation gap, then J.A. Hillman showed that a weaker version of the conjecture is true, see [[#References|[a3]]]. In particular, if the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170301.png" /> does not have a relation gap and acts freely and co-compactly on an acyclic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170302.png" />-complex, then it also admits a co-compact free action on a contractible <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170303.png" />-complex.
+
The Eilenberg–Ganea conjecture states that this is true in dimension $2$ as well. This conjecture is widely believed to be wrong; promising potential counterexamples have been exhibited by Bestvina and also by Bestvina and Brady [[#References|[a1]]]. If the group in question does not have a relation gap, then J.A. Hillman showed that a weaker version of the conjecture is true, see [[#References|[a3]]]. In particular, if the group $G$ does not have a relation gap and acts freely and co-compactly on an acyclic $2$-complex, then it also admits a co-compact free action on a contractible $2$-complex.
  
 
A perhaps unsuspected connection between the Eilenberg–Ganea and the Whitehead conjecture was found by Bestvina and Brady in [[#References|[a1]]]: at least one of the conjectures must be wrong!
 
A perhaps unsuspected connection between the Eilenberg–Ganea and the Whitehead conjecture was found by Bestvina and Brady in [[#References|[a1]]]: at least one of the conjectures must be wrong!
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Bestvina,  N. Brady,  "Morse theory and finiteness properties of groups"  ''Invent. Math.'' , '''129'''  (1997)  pp. 445–470</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  K. Brown,  "Cohomology of groups" , ''GTM'' , '''87''' , Springer  (1982)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J.A. Hillman,  "2-knots and their groups" , ''Austral. Math. Soc. Lecture Notes 5'' , Cambridge Univ. Press  (1989)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  C. Hog-Angeloni,  W. Metzler,  A. Sieradski,  "Two-dimensional homotopy and combinatorial group theory" , ''London Math. Soc.'' , '''197''' , Cambridge Univ. Press  (1993)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  R. Kirby,  "Problems in low-dimensional topology"  W.H. Kazez (ed.) , ''Geometric Topology (1993 Georgia Internat. Topology Conf.)'' , '''2''' , Amer. Math. Soc. &amp;Internat. Press  (1993)  pp. 35–473</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  D. Rolfsen,  "Knots and links" , Publish or Perish  (1976)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  C.P. Rourke,  B.J. Sanderson,  "Introduction to piecewise linear topology" , Springer  (1972)</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  J.J. Andrews,  M.L. Curtis,  "Free groups and handlebodies"  ''Proc. Amer. Math. Soc.'' , '''16'''  (1965)  pp. 192–195</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  H. Tietze,  "Ueber die topologische Invarianten mehrdimensionaler Mannigfaltigkeiten"  ''Monatschr. Math. Phys.'' , '''19'''  (1908)  pp. 1–118</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  R.H. Crowell,  R.H. fox,  "Introduction to knot theory" , Ginn  (1963)</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  C.F. Miller,  P.E. Schupp,  ''Letter to M.M. Cohen'' , '''Oct.'''  (1979)</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  S. Akbulut,  R. Kirby,  "A potential smooth counterexample in dimension 4 to the Poincaré conjecture, the Schoenflies conjecture and the Andrews–Curtis conjecture"  ''Topology'' , '''24'''  (1985)  pp. 375–390</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  R.E. Gompf,  "Killing the Akbulut–Kirby sphere with relevance to the Andrews–Curtis and Schoenflies problems"  ''Topology'' , '''30'''  (1991)  pp. 97–115</TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top">  C.T.C. Wall,  "Formal deformations"  ''Proc. London Math. Soc.'' , '''16'''  (1966)  pp. 342–354</TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top">  W. Metzler,  "Aequivalenzaklassen von Gruppenbeschreibungen, Identitäten und einfacher Homotopietyp in niederen Dimensionen" , ''Lecture Notes London Math. Soc.'' , '''36''' , Cambridge Univ. Press  (1979)  pp. 291–326</TD></TR><TR><TD valign="top">[a16]</TD> <TD valign="top">  C. Hog-Angeloni,  W. Metzler,  "Stabilization by free products giving rise to Andrews–Curtis equivalences"  ''Note di Mat.'' , '''10''' :  Suppl. 2  (1990)  pp. 305–314</TD></TR><TR><TD valign="top">[a17]</TD> <TD valign="top">  C. Hog-Angeloni,  W. Metzler,  "Andrews–Curtis Operationen mit höhere Kommutatoren der Relatorengruppe"  ''J. Pure Appl. Algebra'' , '''75'''  (1991)  pp. 37–45</TD></TR><TR><TD valign="top">[a18]</TD> <TD valign="top">  R. Kreher,  W. Metzler,  "Simpliziale Transformationen von Polyedren und die Zeeman-Vermutung"  ''Topology'' , '''22'''  (1983)  pp. 19–26</TD></TR><TR><TD valign="top">[a19]</TD> <TD valign="top">  P. Dierker,  "Notes on collapsing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170304.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170305.png" /> is a contractible polyhedron" ''Proc. Amer. Math. Soc.'' , '''19''' (1968)  pp. 425–428</TD></TR><TR><TD valign="top">[a20]</TD> <TD valign="top">  W.B.R. Lickorish,  "On collapsing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170306.png" />" , ''Topology of Manifolds'' , Markham  (1970)  pp. 157–160</TD></TR><TR><TD valign="top">[a21]</TD> <TD valign="top">  D. Gillman,  "Bing's house and the Zeeman conjecture"  ''Topology Appl.'' , '''24'''  (1986)  pp. 147–151</TD></TR><TR><TD valign="top">[a22]</TD> <TD valign="top">  P. Wright,  "Group presentations and formal deformations"  ''Trans. Amer. Math. Soc.'' , '''208'''  (1975)  pp. 161–169</TD></TR><TR><TD valign="top">[a23]</TD> <TD valign="top">  E.C. Zeeman,  "On the dunce hat"  ''Topology'' , '''2'''  (1964) pp. 341–358</TD></TR><TR><TD valign="top">[a24]</TD> <TD valign="top">  A. Zimmermann,  "Eine spezielle Klasse kollabierbarere Komplexe <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170307.png" />"  ''Thesis Frankfurt am Main''  (1978)</TD></TR><TR><TD valign="top">[a25]</TD> <TD valign="top">  M.M. Cohen,  "Dimension estimates in collapsing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170308.png" />"  ''Topology'' , '''14'''  (1975)  pp. 253–256</TD></TR><TR><TD valign="top">[a26]</TD> <TD valign="top">  M.M. Cohen,  "Whitehead torsion, group extensions and Zeeman's conjecture in high dimensions"  ''Topology'' , '''16'''  (1977)  pp. 79–88</TD></TR><TR><TD valign="top">[a27]</TD> <TD valign="top">  D. Gillman,  D. Rolfsen,  "The Zeeman conjecture for standard spines is equivalent to the Poincaré conjecture"  ''Topology'' , '''22'''  (1983)  pp. 315–323</TD></TR><TR><TD valign="top">[a28]</TD> <TD valign="top">  S.V. Matveev,  "Zeeman conjecture for unthickenable special polyhedra is equivalent to the Andrews–Curtis conjecture"  ''Sib. Mat. Zh.'' , '''28''' :  6 (1987)  pp. 66–80  (In Russian)</TD></TR><TR><TD valign="top">[a29]</TD> <TD valign="top">  O.S. Rothaus,  "On the nontriviality of some group extensions given by generators and relations"  ''Ann. of Math.'' , '''106'''  (1977)  pp. 599–612</TD></TR><TR><TD valign="top">[a30]</TD> <TD valign="top">  J.H.C. Whitehead,  "On adding relations to homotopy groups"  ''Ann. of Math.'' , '''42'''  (1941)  pp. 409–428</TD></TR><TR><TD valign="top">[a31]</TD> <TD valign="top">  W.H. Cockroft,  "On two-dimensional aspherical complexes groups"  ''Proc. London Math. Soc.'' , '''4'''  (1954)  pp. 375–384</TD></TR><TR><TD valign="top">[a32]</TD> <TD valign="top">  J.F. Adams,  "A new proof of a theorem of W.H. Cockroft"  ''J. London Math. Soc.'' , '''30'''  (1955)  pp. 482–482</TD></TR><TR><TD valign="top">[a33]</TD> <TD valign="top">  J. Howie,  "Aspherical and acyclic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120170/l120170309.png" />-complexes"  ''J. London Math. Soc.'' , '''20'''  (1979)  pp. 549–558</TD></TR><TR><TD valign="top">[a34]</TD> <TD valign="top">  J. Howie,  "Some remarks on a problem of J.H.C. Whitehead"  ''Topology'' , '''22'''  (1983)  pp. 475–485</TD></TR><TR><TD valign="top">[a35]</TD> <TD valign="top">  J. Howie,  "On the Asphericity of ribbon disc complements"  ''Trans. Amer. Math. Soc.'' , '''289'''  (1985)  pp. 419–430</TD></TR><TR><TD valign="top">[a36]</TD> <TD valign="top">  C.D. Papakyriakopoulos,  "On Dehn's lemma and the asphericity of knots"  ''Ann. of Math.'' , '''66'''  (1957)  pp. 1–26</TD></TR><TR><TD valign="top">[a37]</TD> <TD valign="top">  G. Huck,  S. Rosenbrock,  "Eine verallgemeinerter Gewichtstest mit Anwendungen auf Baumpräsentationen"  ''Math. Z.'' , '''211'''  (1992)  pp. 351–367</TD></TR><TR><TD valign="top">[a38]</TD> <TD valign="top">  C.T.C. Wall,  "Finiteness conditions for CW-complexes"  ''Ann. of Math.'' , '''81'''  (1965)  pp. 56–69</TD></TR><TR><TD valign="top">[a39]</TD> <TD valign="top">  J. Dunwoody,  "Relation modules"  ''Bull. London Math. Soc.'' , '''4'''  (1972)  pp. 151–155</TD></TR><TR><TD valign="top">[a40]</TD> <TD valign="top">  S. Eilenberg,  T. Ganea,  "On the Lyusternik–Schnirelman category of abstract groups"  ''Ann. of Math.'' , '''46'''  (1945)  pp. 480–509</TD></TR><TR><TD valign="top">[a41]</TD> <TD valign="top">  J.R. Stallings,  "On torsion-free groups with infinitely many ends"  ''Ann. of Math.'' , '''88'''  (1968)  pp. 312–334</TD></TR><TR><TD valign="top">[a42]</TD> <TD valign="top">  E.S. Rapaport,  "Groups of order 1, some properties of presentations"  ''Acta Math.'' , '''121'''  (1968)  pp. 127–150</TD></TR></table>
+
<table><tr><td valign="top">[a1]</td> <td valign="top">  M. Bestvina,  N. Brady,  "Morse theory and finiteness properties of groups"  ''Invent. Math.'' , '''129'''  (1997)  pp. 445–470</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  K. Brown,  "Cohomology of groups" , ''GTM'' , '''87''' , Springer  (1982)</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  J.A. Hillman,  "2-knots and their groups" , ''Austral. Math. Soc. Lecture Notes 5'' , Cambridge Univ. Press  (1989)</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  C. Hog-Angeloni,  W. Metzler,  A. Sieradski,  "Two-dimensional homotopy and combinatorial group theory" , ''London Math. Soc.'' , '''197''' , Cambridge Univ. Press  (1993)</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  R. Kirby,  "Problems in low-dimensional topology"  W.H. Kazez (ed.) , ''Geometric Topology (1993 Georgia Internat. Topology Conf.)'' , '''2''' , Amer. Math. Soc. &amp;Internat. Press  (1993)  pp. 35–473</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  D. Rolfsen,  "Knots and links" , Publish or Perish  (1976)</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  C.P. Rourke,  B.J. Sanderson,  "Introduction to piecewise linear topology" , Springer  (1972)</td></tr><tr><td valign="top">[a8]</td> <td valign="top">  J.J. Andrews,  M.L. Curtis,  "Free groups and handlebodies"  ''Proc. Amer. Math. Soc.'' , '''16'''  (1965)  pp. 192–195</td></tr><tr><td valign="top">[a9]</td> <td valign="top">  H. Tietze,  "Ueber die topologische Invarianten mehrdimensionaler Mannigfaltigkeiten"  ''Monatschr. Math. Phys.'' , '''19'''  (1908)  pp. 1–118</td></tr><tr><td valign="top">[a10]</td> <td valign="top">  R.H. Crowell,  R.H. fox,  "Introduction to knot theory" , Ginn  (1963)</td></tr><tr><td valign="top">[a11]</td> <td valign="top">  C.F. Miller,  P.E. Schupp,  ''Letter to M.M. Cohen'' , '''Oct.'''  (1979)</td></tr><tr><td valign="top">[a12]</td> <td valign="top">  S. Akbulut,  R. Kirby,  "A potential smooth counterexample in dimension 4 to the Poincaré conjecture, the Schoenflies conjecture and the Andrews–Curtis conjecture"  ''Topology'' , '''24'''  (1985)  pp. 375–390</td></tr><tr><td valign="top">[a13]</td> <td valign="top">  R.E. Gompf,  "Killing the Akbulut–Kirby sphere with relevance to the Andrews–Curtis and Schoenflies problems"  ''Topology'' , '''30'''  (1991)  pp. 97–115</td></tr><tr><td valign="top">[a14]</td> <td valign="top">  C.T.C. Wall,  "Formal deformations"  ''Proc. London Math. Soc.'' , '''16'''  (1966)  pp. 342–354</td></tr><tr><td valign="top">[a15]</td> <td valign="top">  W. Metzler,  "Aequivalenzaklassen von Gruppenbeschreibungen, Identitäten und einfacher Homotopietyp in niederen Dimensionen" , ''Lecture Notes London Math. Soc.'' , '''36''' , Cambridge Univ. Press  (1979)  pp. 291–326</td></tr><tr><td valign="top">[a16]</td> <td valign="top">  C. Hog-Angeloni,  W. Metzler,  "Stabilization by free products giving rise to Andrews–Curtis equivalences"  ''Note di Mat.'' , '''10''' :  Suppl. 2  (1990)  pp. 305–314</td></tr><tr><td valign="top">[a17]</td> <td valign="top">  C. Hog-Angeloni,  W. Metzler,  "Andrews–Curtis Operationen mit höhere Kommutatoren der Relatorengruppe"  ''J. Pure Appl. Algebra'' , '''75'''  (1991)  pp. 37–45</td></tr><tr><td valign="top">[a18]</td> <td valign="top">  R. Kreher,  W. Metzler,  "Simpliziale Transformationen von Polyedren und die Zeeman-Vermutung"  ''Topology'' , '''22'''  (1983)  pp. 19–26</td></tr><tr><td valign="top">[a19]</td> <td valign="top">  P. Dierker,  "Notes on collapsing $K \times I$ where $K$ is a contractible polyhedron" ''Proc. Amer. Math. Soc.'' , '''19'''  (1968)  pp. 425–428</td></tr><tr><td valign="top">[a20]</td> <td valign="top">  W.B.R. Lickorish,  "On collapsing $X ^ { 2 \times } I$" , ''Topology of Manifolds'' , Markham (1970)  pp. 157–160</td></tr><tr><td valign="top">[a21]</td> <td valign="top">  D. Gillman,  "Bing's house and the Zeeman conjecture" ''Topology Appl.'' , '''24'''  (1986)  pp. 147–151</td></tr><tr><td valign="top">[a22]</td> <td valign="top">  P. Wright,  "Group presentations and formal deformations"  ''Trans. Amer. Math. Soc.'' , '''208'''  (1975)  pp. 161–169</td></tr><tr><td valign="top">[a23]</td> <td valign="top">  E.C. Zeeman,  "On the dunce hat"  ''Topology'' , '''2'''  (1964)  pp. 341–358</td></tr><tr><td valign="top">[a24]</td> <td valign="top">  A. Zimmermann,  "Eine spezielle Klasse kollabierbarere Komplexe $K ^ { 2 \times  }I$"  ''Thesis Frankfurt am Main''  (1978)</td></tr><tr><td valign="top">[a25]</td> <td valign="top">  M.M. Cohen,  "Dimension estimates in collapsing $X \times I ^ { 2 }$" ''Topology'' , '''14'''  (1975) pp. 253–256</td></tr><tr><td valign="top">[a26]</td> <td valign="top">  M.M. Cohen,  "Whitehead torsion, group extensions and Zeeman's conjecture in high dimensions"  ''Topology'' , '''16'''  (1977)  pp. 79–88</td></tr><tr><td valign="top">[a27]</td> <td valign="top">  D. Gillman,  D. Rolfsen,  "The Zeeman conjecture for standard spines is equivalent to the Poincaré conjecture"  ''Topology'' , '''22'''  (1983)  pp. 315–323</td></tr><tr><td valign="top">[a28]</td> <td valign="top">  S.V. Matveev,  "Zeeman conjecture for unthickenable special polyhedra is equivalent to the Andrews–Curtis conjecture"  ''Sib. Mat. Zh.'' , '''28''' :  6 (1987)  pp. 66–80  (In Russian)</td></tr><tr><td valign="top">[a29]</td> <td valign="top">  O.S. Rothaus,  "On the nontriviality of some group extensions given by generators and relations"  ''Ann. of Math.'' , '''106'''  (1977)  pp. 599–612</td></tr><tr><td valign="top">[a30]</td> <td valign="top">  J.H.C. Whitehead,  "On adding relations to homotopy groups"  ''Ann. of Math.'' , '''42'''  (1941)  pp. 409–428</td></tr><tr><td valign="top">[a31]</td> <td valign="top">  W.H. Cockroft,  "On two-dimensional aspherical complexes groups"  ''Proc. London Math. Soc.'' , '''4'''  (1954)  pp. 375–384</td></tr><tr><td valign="top">[a32]</td> <td valign="top">  J.F. Adams,  "A new proof of a theorem of W.H. Cockroft"  ''J. London Math. Soc.'' , '''30'''  (1955)  pp. 482–482</td></tr><tr><td valign="top">[a33]</td> <td valign="top">  J. Howie,  "Aspherical and acyclic $2$-complexes"  ''J. London Math. Soc.'' , '''20'''  (1979)  pp. 549–558</td></tr><tr><td valign="top">[a34]</td> <td valign="top">  J. Howie,  "Some remarks on a problem of J.H.C. Whitehead"  ''Topology'' , '''22'''  (1983)  pp. 475–485</td></tr><tr><td valign="top">[a35]</td> <td valign="top">  J. Howie,  "On the Asphericity of ribbon disc complements"  ''Trans. Amer. Math. Soc.'' , '''289'''  (1985)  pp. 419–430</td></tr><tr><td valign="top">[a36]</td> <td valign="top">  C.D. Papakyriakopoulos,  "On Dehn's lemma and the asphericity of knots"  ''Ann. of Math.'' , '''66'''  (1957)  pp. 1–26</td></tr><tr><td valign="top">[a37]</td> <td valign="top">  G. Huck,  S. Rosenbrock,  "Eine verallgemeinerter Gewichtstest mit Anwendungen auf Baumpräsentationen"  ''Math. Z.'' , '''211'''  (1992)  pp. 351–367</td></tr><tr><td valign="top">[a38]</td> <td valign="top">  C.T.C. Wall,  "Finiteness conditions for CW-complexes"  ''Ann. of Math.'' , '''81'''  (1965)  pp. 56–69</td></tr><tr><td valign="top">[a39]</td> <td valign="top">  J. Dunwoody,  "Relation modules"  ''Bull. London Math. Soc.'' , '''4'''  (1972)  pp. 151–155</td></tr><tr><td valign="top">[a40]</td> <td valign="top">  S. Eilenberg,  T. Ganea,  "On the Lyusternik–Schnirelman category of abstract groups"  ''Ann. of Math.'' , '''46'''  (1945)  pp. 480–509</td></tr><tr><td valign="top">[a41]</td> <td valign="top">  J.R. Stallings,  "On torsion-free groups with infinitely many ends"  ''Ann. of Math.'' , '''88'''  (1968)  pp. 312–334</td></tr><tr><td valign="top">[a42]</td> <td valign="top">  E.S. Rapaport,  "Groups of order 1, some properties of presentations"  ''Acta Math.'' , '''121'''  (1968)  pp. 127–150</td></tr></table>

Revision as of 15:30, 1 July 2020

Many problems in two-dimensional topology (cf. Topology of manifolds) arise from, or have to do with, attempts to lift algebraic operations performed on the chain complex $\underline{\underline{C}} ( \tilde { K } )$ of a universal covering complex $\tilde { K } ^ { 2 }$ to geometric operations on the complex $K ^ { 2 }$ (here and below, "complex" means a $P L C W$-complex, i.e. a polyhedron with a $C W$-structure, see [a4] for a precise definition; for simplicity, one may think of a polyhedron): The chain complex $\underline{\underline{C}} ( \tilde { K } )$ encodes the relators of the presentation (cf. Presentation) associated to $K ^ { 2 }$ only up to commutators between relators.

A first classical example for this phenomenon occurs in the proof of the $s$-cobordism theorem (see [a7], which thus only works for manifolds of dimension $\geq 6$. In this context, J. Andrews and M. Curtis (see [a8]) asked whether the unique $5$-dimensional thickening of a compact connected $2$-dimensional complex (in short, a $2$-complex) in a $5$-dimensional piecewise-linear manifold (a PL-manifold) is a $5$-dimensional ball.

They show that this is implied by the Andrews–Curtis conjecture.

Andrews–Curtis conjecture.

This conjecture reads:

AC) any contractible finite $2$-complex $3$-deforms to a point, i.e. there exists a $3$-dimensional complex $L^3$ such that $L^3$ collapses to $K ^ { 2 }$ and to a point: $K ^ { 2 } \swarrow L ^ { 3 } \searrow \operatorname{pt}$. (Cf. [a4] for the precise notion of a collapse, which is a deformation retraction through "free faces" .)

Figure: l120170a

A sequence of "elementary" collapses yielding a collapse

To a contractible finite $2$-complex there corresponds a balanced presentation (cf. Presentation) $\mathcal{P} = \langle x _ { 1 } , \dots , x _ { n } | R _ { 1 } , \dots , R _ { n } \rangle$ of the trivial group. $3$-deformations can be translated into a sequence of Andrews–Curtis moves on $\mathcal{P}$:

1) $R _ { i } \rightarrow R _ { i } ^ { - 1 }$;

2) $R _ { i } \rightarrow R _ { i } R _ { j }$, $i \neq j$;

3) $R _ { i } \rightarrow w R _ { i } w ^ { - 1 }$, $w$ any word;

4) add a generator $x_{n+1}$ and a relation $wx_{n+1}$, $w$ any word in $x _ { 1 } , \ldots , x _ { n }$.

Hence, an equivalent statement of the Andrews–Curtis conjecture is: Any balanced presentation of the trivial group can be transformed into the empty presentation by Andrews–Curtis moves.

Note that redundant relations cannot be added, since by Tietze's theorem (see [a9]) any two presentations of a group become equivalent under insertion and deletion of redundant relations and Andrews–Curtis moves.

Here are some prominent potential counterexamples to AC):

1) $\langle a , b , c | c ^ { - 1 } b c = b ^ { 2 } , a ^ { - 1 } c a = c ^ { 2 } , b ^ { - 1 } a b = a ^ { 2 } \rangle$ (E.S. Rapaport, see [a42]);

2) $\langle a , b | b a ^ { 2 } b ^ { - 1 } = a ^ { 3 } , a b ^ { 2 } a ^ { - 1 } = b ^ { 3 } \rangle$ (R.H. Crowell and R.H. Fox, see [a10], and [a11] for a generalization to an infinite series);

3) $\langle a , b | a b a = b a b , a ^ { 4 } = b ^ { 5 } \rangle$ (S. Akbulut and R. Kirby, see [a12]). This example corresponds to a homotopy $4$-sphere which is shown to be standard by a judicious addition of a $2$-, $3$-handle pair, see [a13], and [a6];

4) $\langle a , b | a = [ a ^ { p } , b ^ { q } ] , b = [ a ^ { r } , b ^ { s } ] \rangle$ (C.McA. Gordon).

An analogue of the conjecture is true in all dimensions different from $2$; in fact, the following generalization of it to non-trivial groups and keeping a subcomplex fixed holds (see [a14] for $n \geq 3$ and [a15] for $n = 1$; cf. also Homotopy type): Let $n \neq 2$; and let $f : K _ { 0 } \rightarrow K _ { 1 }$ be a simple-homotopy equivalence of connected, finite complexes, inducing the identity on the common subcomplex $L$, $n = \operatorname { max } ( \operatorname { dim } ( K _ { 0 } - L ) , \operatorname { dim } ( K _ { 1 } - L ) )$. Then $f$ is homotopic rel $L$ to a deformation $K _ { 0 } ^ { n + 1 } \searrow K _ { 1 }$ which leaves $L$ fixed throughout. A deformation is a composition of expansions and collapses; if the maximal cell dimension involved is $n$, this will be denoted by $K N L$, see [a7].

The corresponding statement for $n = 2$ is called the relative generalized Andrews–Curtis conjecture ( "generalized" because the fundamental group of $K_i$ may be non-trivial; "relative" because of the fixed subcomplex). The subcase $L = \phi$, i.e. the expectation that a simple-homotopy equivalence between finite $2$-dimensional complexes can always be replaced by a $3$-deformation, is called the generalized Andrews–Curtis conjecture, henceforth abbreviated AC'); see [a4].

Suppose $\mathcal{P} = \langle a _ { 1 } , \dots , a _ { g } | R _ { 1 } , \dots , R _ { n } \rangle$ and $\mathcal{Q} = \langle a _ { 1 } , \dots , a _ { g } | S _ { 1 } , \dots , S _ { n } \rangle$ are presentations of $\pi$ such that

D) each difference $R _ { i } S _ { i } ^ { - 1 }$ is a consequence of commutators $[ R _ { j } , R _ { k } ]$ ($1 \leq j , k \leq n$) of relators, then the corresponding $2$-dimensional complexes $K ^ { 2 }$ and $L^{2}$ are simple-homotopy equivalent. Furthermore, up to Andrews–Curtis moves the converse is true, see [a16].

Thus, in terms of presentations, AC') states that under the assumption D), $R_i$ can actually be made to coincide with $S _ { i }$ by Andrews–Curtis moves, for all $i$. Even though AC') is expected to be false, D) implies that the difference $R _ { i } S _ { i } ^ { - 1 }$ between the $i$th relators by Andrews–Curtis moves can be pushed to become a product of arbitrarily high commutators of relators, see [a17]. Furthermore, taking the one-point union not only with a finite number of $2$-spheres, but also with certain $2$-complexes of minimal Euler characteristic, eliminates any potential difference between simple-homotopy and $3$-deformations: A simple homotopy equivalence between finite connected $2$-complexes $K ^ { 2 }$, $L^{2}$ gives rise to a $3$-deformation between the one point union of $K ^ { 2 }$ (respectively, $L^{2}$) with a sufficiently large number of standard complexes of ${\bf Z} _ { 2 } \times {\bf Z} _ { 4 }$, see [a16]. For a detailed discussion on the status of the conjectures AC), AC') and relAC'), see [a4], Chap. XII.

There is a close relation between $2$-complexes and $3$-manifolds. (cf. Three-dimensional manifold): Every compact connected $3$-dimensional manifold with non-empty boundary collapses to a $2$-dimensional complex, called a spine (see [a4], Chap. I, §2.2), and thus determines a $3$-deformation class of $2$-complexes. A counterexample to AC) which is a $3$-manifold with spine $K ^ { 2 }$ would disprove the $3$-dimensional Poincaré conjecture (cf. Three-dimensional manifold)

Zeeman conjecture.

This prominent conjecture on $2$-complexes actually implies the $3$-dimensional Poincaré conjecture. The Zeeman conjecture states that (see [a23]):

Z) if $K ^ { 2 }$ is a compact contractible $2$-dimensional complex, then $K ^ { 2 } \times I \searrow \operatorname{pt}$, where $I$ is an interval. Note that Z) also implies AC), as $K ^ { 2 } \nearrow K ^ { 2 }\times I \searrow \operatorname {pt}$ would be a $3$-deformation. Examples which fulfil $K ^ { 2 } \times I \searrow \operatorname{pt}$ are the dunce hat, Bing's house and the house with one room, see [a4]. However, $K ^ { 2 } \times I \searrow \operatorname{pt}$ is not even established (as of 1999) for most of the standard $2$-complexes of presentations $\langle a , b | a ^ { p } b ^ { q } , a ^ { r } b ^ { s } \rangle$ where $p s - q r = \pm 1$, even though these are Andrews–Curtis equivalent to the empty presentation.

As for AC), there is a straightforward generalization to non-trivial groups; the generalized Zeeman conjecture:

Z') $K ^ { 2 } / \searrow L ^ { 2 }$ implies $K ^ { 2 } \times I \searrow L ^ { 2 }$ or $L ^ { 2 } \times I \searrow K ^ { 2 }$. Of course, Z') implies both Z) and AC'). It is open (as of 1999) whether AC') implies Z'), but given a $3$-deformation between finite $2$-complexes $K ^ { 2 }/ \stackrel { 3 } { \searrow } L ^ { 2 }$, then $K ^ { 2 }$ can be expanded by a sequence of $2$-expansions to a $2$-complex $K ^ { \prime 2 } \searrow K ^ { 2 }$ such that $K ^ { \prime 2 } \times I \searrow \operatorname{pt}$, see [a18].

In the special case of expansion of a single $3$-ball, followed by a $3$-collapse, $K ^ { 2 } \nearrow K ^ { 2 } \cup _ { B ^ { 2 } } B ^ { 3 } \searrow L ^ { 2 }$, it is true that $K ^ { 2 } \times I \searrow L ^ { 2 }$, see [a19], [a20], [a21]. This can be viewed as a first step in proving Z') modulo AC'), as every $3$-deformation between finite $2$-complexes can be replaced by one where each $3$-ball is transient, i.e. is collapsed (in general from a different free face) immediately after its expansion, see [a22]. For $L ^ { 2 } = \operatorname {pt}$, this method is called collapsing by adding a cell and works for all above-mentioned examples for $K ^ { 2 } \times I \searrow \operatorname{pt}$.

A second general method for collapsing $K ^ { 2 } \times I$ was proposed by A. Zimmermann (see [a24]) and is called prismatic collapsing. At first one gets rid of the $3$-dimensional part of $K ^ { 2 } \times I$ as follows: For each $2$-cell $C ^ { 2 }$ of $K ^ { 2 }$ one collapses $C ^ { 2 } \times I$ to the union of $\partial C ^ { 2 } \times I$ and a $2$-cell $C ^ { * } \subset C ^ { 2 } \times I$ such that the direct product projection maps $\operatorname { lnt } C ^ { * }$ onto $\operatorname { lnt } C ^ { 2 }$ homeomorphically. Then one looks for a collapse of the resulting $2$-complex.

One may say that prismatic collapsing is a very rough method, but exactly this roughness allows one to give an algebraic criterion for the prismatic collapsibility of $K ^ { 2 } \times I$: Attaching mappings for $2$-cells of $K ^ { 2 }$ have to determine a basis-up-to-conjugation in the free fundamental group of the $1$-dimensional skeleton (see [a7]) of $K ^ { 2 }$.

Z) becomes true if one admits multiplication of $K ^ { 2 }$ by the $n$-fold product of $I$: For each contractible $K ^ { 2 }$ there exists an integer $n$ such that $K ^ { 2 } \times I ^ { n } \searrow \operatorname{pt}$, see [a19], [a20]. In fact, $n = 6$ suffices for all $K ^ { 2 }$, see [a25]. It is surprising that there is such a large gap between the presently (1999) known ($n = 6$) and Zeeman's conjectured ($n = 1$) values of $n$.

On the other hand, a generalization of Z) to higher-dimensional complexes is false, since for any $n > 2$ there exists a contractible complex $K ^ { n }$ of dimension $n$ such that $K ^ { n } \times 1$ is not collapsible, see [a26]. The proof of non-collapsibility is based on a very specific (one may say "bad" ) local structure of $K ^ { n }$. So, the idea to investigate Z) for $2$-dimensional polyhedra with a "nice" local structure (such polyhedra are called special) seems to be very promising.

In fact, if $K ^ { 2 }$ is a special spine of a homotopy $3$-ball $M ^ { 3 }$, then $K ^ { 2 } \times I$ collapses onto a homeomorphic copy of $M ^ { 3 }$, see [a27]. It follows that Z) is true for all special spines of a genuine $3$-ball and that for special spines of $3$-manifolds, Z) is equivalent to the $3$-dimensional Poincaré conjecture. Surprisingly, for special polyhedra that cannot be embedded in a $3$-manifold, Z) turns out to be equivalent to AC) (see [a28]), so that for special polyhedra, Z) is equivalent to the union of AC) and the $3$-dimensional Poincaré conjecture.

$\operatorname{Wh} ^ { * }$-question.

Another situation where dimension $2$ presents a severe difficulty in passing from chain complexes to geometry concerns the Whitehead group and the Whitehead torsion of a pair $( K , L )$, where $L$ is a strong deformation retraction of $K$ (cf. Whitehead group, Whitehead torsion). All elements of $\operatorname{Wh} ( \pi )$ can be realized by $\operatorname { dim } K = 3$. Let $\operatorname{Wh} ^ { * } ( \pi ) \subseteq \operatorname{Wh} ( \pi )$ be the set of those torsion values that can be realized by a $2$-dimensional extension, i.e. $\operatorname { dim } ( K - L ) \leq 2$. The $\operatorname{Wh} ^ { * }$-question is whether $\operatorname{Wh} ^ { * } ( \pi ) \neq \{ 0 \}$ can happen; see [a4]. If so, another related question is whether $\operatorname{ Wh} ^ { * } ( \pi )$ is a subgroup.

A famous result of O.S. Rothaus is that there exist examples $\tau \in \operatorname{Wh} ( \pi )$ for dihedral groups $\pi$ with $\tau \notin \operatorname{Wh} ^ { * } ( \pi )$; see [a29]. This result was the basis for work by M.M. Cohen [a26] on the generalization of Z) to higher dimensions.

Whitehead's asphericity question.

A $2$-complex $K$ is called aspherical if its second homotopy group $\pi_2 ( K )$ is trivial (or equivalently, if all $\pi _ { n } ( K )$ for $n \geq 2$ are trivial). J.H.C. Whitehead asked, (see [a30]), whether subcomplexes of aspherical $2$-complexes are themselves aspherical. An affirmative answer to this question is called the Whitehead conjecture:

WH) A subcomplex $K$ of an aspherical $2$-complex $L$ is aspherical.

A lot of work has already been done in trying to solve this conjecture and there are about six false results in the literature which would imply WH).

WH) is known to be true if $K$ has at most one $2$-cell and also in the case where $\pi_1 ( L )$ is either finite, Abelian or free, see [a31]. If $K$ is a subcomplex of an aspherical $2$-complex, then one can show that the second homology of the covering $\overline { K } \rightarrow K$ corresponding to the commutator subgroup is trivial. In fact, J.F. Adams has shown [a32] that $K$ has an acyclic regular covering $K ^ { * } \rightarrow \overline { K } \rightarrow K$ (i.e. $H _ { 2 } ( K ^ { * } ) = H _ { 1 } ( K ^ { * } ) = 0$). A counterexample to WH) can thus be covered by an acyclic complex, but not by a contractible one.

In any counterexample $K \subset L$ to WH), the kernel of the inclusion induced mapping $\pi _ { 1 } ( K ) \rightarrow \pi _ { 1 } ( L )$ has a non-trivial, finitely generated, perfect subgroup, [a33].

J. Howie has shown [a34] that if WH) is false, then there exists a counterexample $K \subset L$ satisfying either

a) $L$ is finite and contractible, and $K = L - e$ for some $2$-cell $e$ of $L$; or

b) $L$ is the union of an infinite ascending chain of finite non-aspherical subcomplexes $K = K _ { 0 } \subset K _ { 1 } \subset \ldots$ such that each inclusion mapping is nullhomotopic.

This result has been sharpened by E. Luft, who showed that if WH) is false, then there must even exist an infinite counterexample of type b).

Let $\mathcal{P} = \langle x _ { 1 } , \dots , x _ { g } | R _ { 1 } , \dots , R _ { n } \rangle$ be a finite presentation where each relator is of the form $x _ { i } = x _ { j } x _ { k } x _ { j } ^ { - 1 }$. Such a presentation may be represented by a graph $T _ { \mathcal{P} }$ in the following way: For each generator $x_{i}$ of $\mathcal{P}$, define a vertex labelled $i$ and for each relator $x _ { i } = x _ { j } x _ { k } x _ { j } ^ { - 1 }$ define an edge oriented from the vertex $i$ to the vertex $k$ labelled by $j$. If $T _ { \mathcal{P} }$ is a tree, then $\mathcal{P}$ or $T _ { \mathcal{P} }$ or the standard-$2$-complex $K _ { \mathcal{P} }$ modelled on $\mathcal{P}$ is called a labelled oriented tree.

Now Howie showed [a34] that if the Andrews–Curtis conjecture is true and all labelled oriented trees are aspherical, then there are no counterexamples of type a) to WH). Conversely, if there are no counterexamples of type a) to WH), then all labelled oriented trees are aspherical, which is easy to see since adding an extra relator $x _ { 1 } = 1$ to a labelled oriented tree yields a balanced presentation of the trivial group and hence a contractible complex.

So the finite case of WH) can be reduced to the study of the asphericity of labelled oriented trees. Every knot group has a labelled oriented tree presentation (the Wirtinger presentation, see, e.g., [a6]) and by a theorem of C.D. Papakyriakopoulos, [a36], it is known that these labelled oriented trees are aspherical. Every labelled oriented tree satisfying the small cancellation conditions $C ( 4 )$, $T ( 4 )$ or a more refined curvature condition such as the weight or cycle test, [a37], is aspherical. Apart from that, there are not many classes of aspherical labelled oriented trees known: Howie, [a35], shows the asphericity of labelled oriented trees of diameter at most $3$ and G. Huck and S. Rosebrock have two other classes of aspherical labelled oriented trees satisfying certain conditions on the relators.

An overview on WH), where further aspects of this conjecture are treated, can be found in [a4], Chap. X.

Wall's domination problem.

Given a CW-complex, it is natural to ask whether it can be replaced by a simpler one having the same homotopy type. Questions of this kind were first considered by J.H.C. Whitehead, who posed in particular the question: When is a CW-complex homotopy equivalent to a finite dimensional one? In [a38], C.T.C. Wall answered this by giving an algebraic characterization of finiteness. He also showed that a finite complex $X$ dominated by a finite $n$-complex $Y$ has the homotopy type of a finite $\operatorname{max}( 3 , n )$-complex if and only if a certain algebraic obstruction vanishes. ($X$ is dominated by $Y$ if the "homotopy of X survives passing through Y" , i.e. if there are mappings $f : X \rightarrow Y$, $g : Y \rightarrow X$ such that the composition $X \stackrel { f } { \rightarrow } Y \stackrel { g } { \rightarrow } X$ is homotopic to the identity). Whether "max3,n" can simply be replaced by "n" is still (1999) unanswered, due to difficulties when attempting to geometrically realize an algebraic $2$-complex.

In order to explain this in more detail, assume $B$ is a chain complex of free ${\bf Z} G$-modules,

\begin{equation*} B _ { 2 } \stackrel { d } { \rightarrow } B _ { 1 } \stackrel { d _ { 1 } } { \rightarrow } B _ { 0 } \rightarrow 0, \end{equation*}

where $B_0$ is freely generated by a single element $e_0$, $B _ { 1 }$ by $\{ e _ { 1 } ^ { i } \}$, $B _ { 2 }$ by $\{ e _ { 2 } ^ { j } \}$, $d _ { 1 } ( e _ { 1 } ^ { i } ) = g _ { i } e _ { 0 } - e _ { 0 }$ for some group element $g_i$, and $H _ { 1 } ( B ) = 0$, $H _ { 0 } ( B ) = \mathbf{Z}$. Wall asked if $B$ is necessarily the cellular chain complex of the universal covering $\widetilde { K }$ of a $2$-complex $K$ with fundamental group $G$. An affirmative answer would resolve the difficulties in dimension two mentioned above.

This topological set-up can also be rephrased in terms of combinatorial group theory. Let $F$ be the free group generated by $\{ x _ { i } \}$ and let $N$ be the kernel of the homomorphism from $F$ to $G$, sending $x_{i}$ to $g_i$. The image of the second boundary mapping $d _ { 2 }$ can be shown to be isomorphic to the relation ${\bf Z} G$-module $N / [ N , N ]$. Wall's question of geometric realizability now translates to asking whether the relation module generators $d _ { 2 } ( e _ { 2 } ^ { j } )$ lift to give a set of normal generators for $N$. This was answered negatively by M. Dunwoody (see [a39]).

Relation gap question.

M. Dyer showed that a more serious failure of this lifting problem, the relation gap question, would actually show that there does exist a finite $3$-complex dominated by a finite $2$-complex, with vanishing obstruction, that is not homotopically equivalent to a finite $2$-complex. Here, a finite presentation $F / N$ of a group $G$ is said to have a relation gap if no normal generating set of $N$ gives a minimal generating set for the relation module $N / [ N , N ]$. There have been many attempts to construct a relation gap in finitely presented groups (see [a4], p. 50). The existence of an infinite relation gap for a certain finitely-generated infinitely-related group was established in the influential paper of M. Bestvina and N. Brady [a1].

Eilenberg–Ganea conjecture.

Another problem revolving around geometric realizability, connected to the relation gap problem and the Whitehead conjecture, is the Eilenberg–Ganea conjecture. A group $G$ is of cohomological dimension $n$ if there exists a projective resolution of length $n$

\begin{equation*} 0 \rightarrow P _ { n } \rightarrow \ldots \rightarrow P _ { 0 } \rightarrow \mathbf{Z} \rightarrow 0 \end{equation*}

but no shorter one (see [a2] for a good reference on these matters). It was shown by S. Eilenberg, T. Ganea and J. Stallings ([a40], [a41]) that a group of cohomological dimension $n \neq 2$ admits an $n$-dimensional $K ( G , 1 )$ complex $K$. In particular, there is a geometric resolution of length $n$ arising as the augmented cellular chain complex of the universal covering of $K$.

The Eilenberg–Ganea conjecture states that this is true in dimension $2$ as well. This conjecture is widely believed to be wrong; promising potential counterexamples have been exhibited by Bestvina and also by Bestvina and Brady [a1]. If the group in question does not have a relation gap, then J.A. Hillman showed that a weaker version of the conjecture is true, see [a3]. In particular, if the group $G$ does not have a relation gap and acts freely and co-compactly on an acyclic $2$-complex, then it also admits a co-compact free action on a contractible $2$-complex.

A perhaps unsuspected connection between the Eilenberg–Ganea and the Whitehead conjecture was found by Bestvina and Brady in [a1]: at least one of the conjectures must be wrong!

References

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How to Cite This Entry:
Low-dimensional topology, problems in. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Low-dimensional_topology,_problems_in&oldid=18980
This article was adapted from an original article by Jens HarlanderCynthia Hog-AngeloniWolfgang MetzlerStephan Rosebrock (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article