Topology of manifolds

The branch of the theory of manifolds (cf. Manifold) concerned with the study of relations between different types of manifolds.

The most important types of finite-dimensional manifolds and relations between them are illustrated in (1).

$$ \tag{1 } Here $ \mathop{\rm Diff} $ is the category of differentiable (smooth) manifolds; $ \mathop{\rm PL} $ is the category of piecewise-linear (combinatorial) manifolds; $ \mathop{\rm TRI} $ is the category of topological manifolds that are polyhedra; $ \mathop{\rm Handle} $ is the category of topological manifolds admitting a topological decomposition into handles; $ \mathop{\rm Lip} $ is the category of Lipschitz manifolds (with Lipschitz transition mappings between local charts); $ \mathop{\rm TOP} $ is the category of topological manifolds (Hausdorff and with a countable base); $ \mathop{\rm H} $ is the category of polyhedral homology manifolds without boundary (polyhedra, the boundary of the star of each vertex of which has the homology of the sphere of corresponding dimension); $ \mathop{\rm H} ( \mathop{\rm ANR} ) $ is the category of generalized manifolds (finite-dimensional absolute neighbourhood retracts $ X $ that are homology manifolds without boundary, i.e. with the property that for any point $ x \in X $ the group $ H ^ {*} ( X, X \setminus x; \mathbf Z ) $ is isomorphic to the group $ H ^ {*} ( \mathbf R ^ {n} , \mathbf R ^ {n} \setminus 0; \mathbf Z ) $); $ \mathop{\rm P} ( \mathop{\rm ANR} ) $ is the category of Poincaré spaces (finite-dimensional absolute neighbourhood retracts $ X $ for which there exists a number $ n $ and an element $ \mu \in H _ {n} ( X) $ such that $ H _ {r} ( X, \mathbf Z ) = 0 $ when $ r \geq n + 1 $, and the mapping $ \mu \cap : H ^ {r} ( X) \rightarrow H _ {n - r } ( X) $ is an isomorphism for all $ r $); and $ \mathop{\rm P} $ is the category of Poincaré polyhedra (the subcategory of the preceding category consisting of polyhedra). The arrows of (1), apart from the 3 lower ones and the arrows $ \mathop{\rm H} \rightarrow \mathop{\rm TOP} \rightarrow \mathop{\rm P} $, denote functors with the structure of forgetting functors. The arrow $ \mathop{\rm Diff} \rightarrow \mathop{\rm PL} $ expresses Whitehead's theorem on the triangulability of smooth manifolds. In dimensions $ < 8 $ this arrow is reversible (an arbitrary $ \mathop{\rm PL} $- manifold is smoothable) but in dimensions $ \geq 8 $ there are non-smoothable $ \mathop{\rm PL} $- manifolds and even $ \mathop{\rm PL} $- manifolds that are homotopy inequivalent to any smooth manifold. The imbedding $ \mathop{\rm PL} \subset \mathop{\rm TRI} $ is also irreversible in the same strong sense (there exist polyhedral manifolds of dimension $ \geq 5 $ that are homotopy inequivalent to any $ \mathop{\rm PL} $- manifold). Here already for the sphere $ S ^ {n} $, $ n \geq 5 $, there exist triangulations in which it is not a $ \mathop{\rm PL} $- manifold. The arrow $ \mathop{\rm PL} \rightarrow \mathop{\rm Handle} $ expresses the fact that every $ \mathop{\rm PL} $- manifold has a handle decomposition. The arrow $ \mathop{\rm PL} \rightarrow \mathop{\rm Lip} $ expresses the theorem on the existence of a Lipschitz structure on an arbitrary $ \mathop{\rm PL} $- manifold. The arrow $ \mathop{\rm Handle} \rightarrow \mathop{\rm TOP} $ is reversible if $ n \neq 4 $ and irreversible if $ n = 4 $( an arbitrary topological manifold of dimension $ n \neq 4 $ admits a handle decomposition and there exist four-dimensional topological manifolds for which this is not true). Similarly, if $ n \neq 4 $ the arrow $ \mathop{\rm Lip} \rightarrow \mathop{\rm TOP} $ is reversible (and moreover in a unique way). The question on the reversibility of the arrow $ \mathop{\rm TRI} \rightarrow \mathop{\rm TOP} $ gives the classical unsolved problem on the triangulability of arbitrary topological manifolds. The arrow $ \mathop{\rm H} \rightarrow \mathop{\rm P} $ is irreversible in the strong sense (there exist Poincaré polyhedra that are homotopy inequivalent to any homology manifold). The arrow $ \mathop{\rm H} \rightarrow \mathop{\rm TOP} $ expresses a theorem on the homotopy equivalence of an arbitrary homology manifold of dimension $ n \geq 5 $ to a topological manifold. The arrow $ \mathop{\rm TOP} \rightarrow \mathop{\rm P} $ expresses the theorem on the homotopy equivalence of an arbitrary topological manifold to a polyhedron. The imbedding $ \mathop{\rm TOP} \subset \mathop{\rm H} ( \mathop{\rm ANR} ) $ expresses that an arbitrary topological manifold is an $ \mathop{\rm ANR} $. The similar question for the arrows $ \mathop{\rm Diff} \rightarrow \mathop{\rm PL} \rightarrow \mathop{\rm TOP} \rightarrow \mathop{\rm P} $ has been solved using the theory of stable bundles (respectively, vector, piecewise-linear, topological, and spherical bundles), i.e. by examining the homotopy classes of mappings of a manifold $ X $ into the corresponding classifying spaces BO, BPL, BTOP, BG. There exist canonical composition mappings $$ \tag{2 }

\mathop{\rm BO}  \rightarrow   \mathop{\rm BPL}  \rightarrow   \mathop{\rm BTOP}  \rightarrow   \mathop{\rm BG} ,

$$ of which the homotopy fibres and the homotopy fibres of their compositions are denoted, respectively, by the symbols $$

\mathop{\rm PL} /  \mathop{\rm O} ,  \mathop{\rm TOP} /  \mathop{\rm O} ,  \mathop{\rm G} /
\mathop{\rm O} ,  \mathop{\rm TOP} /  \mathop{\rm PL} ,\ 
\mathop{\rm G} /  \mathop{\rm PL} ,  \mathop{\rm G} /  \mathop{\rm TOP} .

$$ For every manifold $ X $ from a category $ \mathop{\rm Diff} $, $ \mathop{\rm PL} $, $ \mathop{\rm TOP} $, $ \mathop{\rm P} $ there exists a normal stable bundle, i.e. a canonical mapping $ \tau _ {X} $ from $ X $ into the corresponding classifying space. In the transition from a "narrow" category of manifolds to a "wider" one, for example, from smooth to piecewise-linear, the mapping $ \tau _ {X} $ is composed with the corresponding mappings (2). Hence, for example, for a PL-manifold $ X $ there exists a smooth manifold PL-homeomorphic to it ( $ X $ is said to be smoothable) only if the lifting problem (3), the homotopy obstruction to the solution of which lies in the groups $ H ^ {i + 1 } ( X, \pi _ {i} ( \mathop{\rm PL} / \mathop{\rm O} )) $, is solvable: $$ \tag{3 }

Here it turns out that the solvability of (3) is not only necessary but also sufficient for the smoothability of a PL-manifold $ X $( and all non-equivalent smoothings are in bijective correspondence with the set $ [ X, \mathop{\rm PL} / \mathop{\rm O} ] $ of homotopy classes of mappings $ X \rightarrow \mathop{\rm PL} / \mathop{\rm O} $).

By replacing $ \mathop{\rm PL} / \mathop{\rm O} $ by $ \mathop{\rm TOP} / \mathop{\rm O} $, the same holds for the smoothability of topological manifolds $ X $ of dimension $ \geq 5 $, and also (by replacing $ \mathop{\rm PL} / \mathop{\rm O} $ by $ \mathop{\rm TOP} / \mathop{\rm O} $) for their $ \mathop{\rm PL} $- triangulations. The homotopy group $ \Gamma _ {k} = \pi _ {k} ( \mathop{\rm PL} / \mathop{\rm O} ) $ is isomorphic to the group of classes of oriented diffeomorphic smooth manifolds obtained by glueing the boundaries of two $ k $- dimensional spheres. This group is finite for all $ k $( and is even trivial for $ k \leq 6 $). Therefore, the number of non-equivalent smoothings of an arbitrary PL-manifold $ X $ is finite and is bounded above by the number

$$ \mathop{\rm ord} \sum _ { k } H ^ {k} ( X, \pi _ {k} ( \mathop{\rm PL} / \mathop{\rm O} )). $$

The homotopy group $ \pi _ {k} ( \mathop{\rm TOP} / \mathop{\rm PL} ) $ vanishes, with one exception: $ \pi _ {3} ( \mathop{\rm TOP} / \mathop{\rm PL} ) = \mathbf Z /2 $. Thus, the existence of a $ \mathop{\rm PL} $- triangulation of a topological manifold $ X $ of dimension $ \geq 5 $ is determined by the vanishing of a certain cohomology class $ \Delta ( X) \in H ^ {4} ( X, \mathbf Z /2) $, while the set of non-equivalent $ \mathop{\rm PL} $- triangulations of $ X $ is in bijective correspondence with the group $ H ^ {3} ( X, \mathbf Z /2) $.

The group $ \pi _ {k} ( \mathop{\rm TOP} / \mathop{\rm O} ) $ coincides with the group $ \Gamma _ {k} $ if $ k \neq 3 $ and differs from $ \Gamma _ {k} $ for $ k = 3 $ by the group $ \mathbf Z /2 $. The number of non-equivalent smoothings of a topological manifold $ X $ of dimension $ \geq 5 $ is finite and is bounded above by the number $ \mathop{\rm ord} \sum _ {k} H ^ {k} ( X, \pi _ {k} ( \mathop{\rm TOP} / \mathop{\rm O} )) $.

Similar results are not valid for Poincaré polyhedra.

$$ \tag{4 }

Of course, the existence of a lifting, for example, in (4) is a necessary condition for the existence of a PL-manifold homotopy equivalent to the Poincaré polyhedron $ X $, but, generally speaking, it ensures (for $ n \geq 5 $) only the existence of a PL-manifold $ M $ and a mapping $ f: M \rightarrow X $ of degree 1 such that $ \tau _ {M} = f \circ \tau _ {x} ^ \prime $. The transformation of this manifold into a manifold that is homotopy equivalent to $ X $ requires the technique of surgery (reconstruction), initially developed by S.P. Novikov for the case when $ X $ is a simply-connected smooth manifold of dimension $ \geq 5 $. For simply-connected $ X $ this technique has been generalized to the case under consideration. Thus, for a simply-connected Poincaré polyhedron $ X $ a PL-manifold of dimension $ \geq 5 $ homotopy equivalent to it exists if and only a lifting (4) exists. The problem of the existence of topological or smooth manifolds that are homotopy equivalent to an (even simply-connected) Poincaré polyhedron is still more complicated.

References

[1]	S.P. Novikov, "On manifolds with free abelian group and their application" Izv. Akad. Nauk SSSR Ser. Mat. , 30 (1966) pp. 207–246 (In Russian)
[2]	J. Madsen, R.J. Milgram, "The classifying spaces for surgery and cobordism of manifolds" , Princeton Univ. Press (1979)
[3]	F. Latour, "Double suspension d'une sphere d'homologie [d'après R. Edwards]" , Sem. Bourbaki Exp. 515 , Lect. notes in math. , 710 , Springer (1979) pp. 169–186
[4]	M.H. Freedman, "The topology of four-dimensional manifolds" J. Differential Geom. , 17 (1982) pp. 357–453
[5]	F. Quinn, "Ends of maps III. Dimensions 4 and 5" J. Differential Geom. , 17 (1982) pp. 503–521
[6]	R. Mandelbaum, "Four-dimensional topology: an introduction" Bull. Amer. Math. Soc. , 2 (1980) pp. 1–159
[7]	R. Lashof, "The immersion approach to triangulation and smoothing" A. Liulevicius (ed.) , Algebraic Topology (Madison, 1970) , Proc. Symp. Pure Math. , 22 , Amer. Math. Soc. (1971) pp. 131–164
[8]	R.D. Edwards, "Approximating certain cell-like maps by homeomorphisms" Notices Amer. Math. Soc. , 24 : 7 (1977) pp. A649
[9]	F. Quinn, "The topological characterization of manifolds" Abstracts Amer. Math. Soc. , 1 : 7 (1980) pp. 613–614
[10]	J.W. Cannon, "The recognition problem: what is a topological manifold" Bull. Amer. Math. Soc. , 84 : 5 (1978) pp. 832–866
[11]	M. Spivak, "Spaces satisfying Poincaré duality" Topology , 6 (1967) pp. 77–101
[12]	N.H. Kuiper, "A short history of triangulation and related matters" P.C. Baayen (ed.) D. van Dulst (ed.) J. Oosterhoff (ed.) , Bicentennial Congress Wisk. Genootschap (Amsterdam 1978) , Math. Centre Tracts , 100 , CWI (1979) pp. 61–79

Comments

It was found recently [a1] that the behaviour of smooth manifolds of dimension $ 4 $ is radically different from those in dimensions $ \geq 5 $. Among very numerous recent results one has:

i) There is a countably infinite family of smooth, compact, simply-connected $ 4 $- manifolds, all mutually homeomorphic but with distinct smooth structure.

ii) There is an uncountable family of smooth $ 4 $- manifolds, each homeomorphic to $ \mathbf R ^ {4} $ but with mutually distinct smooth structure.

iii) There are simply-connected smooth $ 4 $- manifolds which are $ h $- cobordant (cf. $ h $- cobordism) but not diffeomorphic.

For the lifting problem (3) see [a2]–[a3].

For the Kirby–Siebenmann theorem, the arrow $ \mathop{\rm TOP} \rightarrow \mathop{\rm P} $, see also [a4].

References

[a1]	S.M. Donaldson, "The geometry of 4-manifolds" A.M. Gleason (ed.) , Proc. Internat. Congress Mathematicians (Berkeley, 1986) , Amer. Math. Soc. (1987) pp. 43–54
[a2]	M.W. Hirsch, B. Mazur, "Smoothings of piecewise-linear manifolds" , Princeton Univ. Press (1974)
[a3]	R. Lashof, M. Rothenberg, "Microbundles and smoothing" Topology , 3 (1965) pp. 357–388
[a4]	R.C. Kirby, L.C. Siebenmann, "Foundational essays on topological manifolds, smoothings, and triangulations" , Princeton Univ. Press (1977)