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 (cf. Shape theory). A metric compactum (cf. Metric space; Compact space) is an FANR space provided that for every metric compactum containing there exist a neighbourhood of in and a shape retraction , i.e., a shape morphism such that . Here denotes the inclusion mapping and is the induced shape morphism. Clearly, every compact ANR is a FANR. For one obtains FAR spaces (fundamental absolute retracts).

If is shape dominated by , i.e., if there exist shape morphisms and such that , and is an FANR space, then so is . 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 has the shape of a compact ANR (equivalently, of a compact polyhedron) if and only if its Wall obstruction . This obstruction is an element of the reduced projective class group of the first shape group . There exist FANR spaces for which [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 every homotopy idempotent splits, i.e., implies the existence of a space and of mappings , , such that , .

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