FANR space

These spaces were introduced by K. Borsuk [a1] as a shape-theoretic analogue of ANR spaces (cf. Retract of a topological space). FANR is an abbreviation of fundamental absolute neighbourhood retract, where "fundamental" refers to the particular technique used by Borsuk in his construction of the shape category $\operatorname{Sh}$ (cf. Shape theory). A metric compactum (cf. Metric space; Compact space) $X$ is an FANR space provided that for every metric compactum $Y$ containing $X$ there exist a neighbourhood $U$ of $X$ in $Y$ and a shape retraction $R : U \rightarrow X$, i.e., a shape morphism such that $R S [ i ] = id_X$. Here $i : X \rightarrow U$ denotes the inclusion mapping and $S [ i ]$ is the induced shape morphism. Clearly, every compact ANR is a FANR. For $U = Y$ one obtains FAR spaces (fundamental absolute retracts).

If $X$ is shape dominated by $X ^ { \prime }$, i.e., if there exist shape morphisms $F : X \rightarrow X ^ { \prime }$ and $G : X ^ { \prime } \rightarrow X$ such that $G F = \operatorname {id}_X$, and $X ^ { \prime }$ is an FANR space, then so is $X$. Consequently, FANR spaces coincide with metric compacta which are shape dominated by compact ANR spaces, or equivalently, by compact polyhedra. FANR spaces are characterized by a form of movability, called strong movability [a2]. In particular, a FANR is a movable space.

In various constructions and theorems, FANR spaces must be pointed. E.g., if the intersection of two pointed FANR spaces is a pointed FANR, then their union is also a pointed FANR [a3]. Connected pointed FANR spaces coincide with stable continua, i.e., have the shape of an ANR (equivalently, of a polyhedron) [a5]. A FANR $X$ has the shape of a compact ANR (equivalently, of a compact polyhedron) if and only if its Wall obstruction $\sigma( X ) = 0$. This obstruction is an element of the reduced projective class group $\widetilde{ K } ^ { 0 } ( \check{\pi} _ { 1 } ( X , * ) )$ of the first shape group $\check{\pi} _ { 1 } ( X , * )$. There exist FANR spaces for which $\sigma( X ) \neq 0$ [a4]. A pointed metric continuum of finite shape dimension is a pointed FANR if and only if its homotopy pro-groups are stable, i.e. are isomorphic to groups.

All FANR spaces are pointed FANR spaces [a6]. The crucial step in the proof of this important theorem is the following homotopy-theoretic result: On a finite-dimensional polyhedron $X$ every homotopy idempotent $f : X \rightarrow X$ splits, i.e., $f ^ { 2 } \simeq f$ implies the existence of a space $Y$ and of mappings $u : Y \rightarrow X$, $v : X \rightarrow Y$, such that $v u \simeq 1_{Y}$, $u v \simeq f$.

References

[a1]	K. Borsuk, "Fundamental retracts and extensions of fundamental sequences" Fund. Math. , 64 (1969) pp. 55–85
[a2]	K. Borsuk, "A note on the theory of shape of compacta" Fund. Math. , 67 (1970) pp. 265–278
[a3]	J. Dydak, S. Nowak, S. Strok, "On the union of two FANR-sets" Bull. Acad. Polon. Sci. Ser. Sci. Math. Astr. Phys. , 24 (1976) pp. 485–489
[a4]	D.A. Edwards, R. Geoghegan, "Shapes of complexes, ends of manifolds, homotopy limits and the Wall obstruction" Ann. Math. , 101 (1975) pp. 521–535 (Correction: 104 (1976), 389)
[a5]	D.A. Edwards, R. Geoghegan, "Stability theorems in shape and pro-homotopy" Trans. Amer. Math. Soc. , 222 (1976) pp. 389–403
[a6]	H.M. Hastings, A. Heller, "Homotopy idempotents on finite-dimensional complexes split" Proc. Amer. Math. Soc. , 85 (1982) pp. 619–622