Function spaces and shape theories

Jerzy Dydak¹ ; Sławomir Nowak²

¹ Department of Mathematics University of Tennessee Knoxville, TN 37996, U.S.A.
² Institute of Mathematics University of Warsaw Banacha 2 02-097 Warszawa, Poland and Institute of Mathematics Polish Academy of Sciences Sniadeckich 8 00-950 Warszawa, Poland

Fundamenta Mathematicae, Tome 171 (2002) no. 2, pp. 117-154

Résumé

The purpose of this paper is to provide a geometric explanation of strong shape theory and to give a fairly simple way of introducing the strong shape category formally. Generally speaking, it is useful to introduce a shape theory as a localization at some class of “equivalences". We follow this principle and we extend the standard shape category Sh(HoTop) to Sh(pro-HoTop) by localizing pro-HoTop at shape equivalences. Similarly, we extend the strong shape category of Edwards–Hastings to sSh(pro-Top) by localizing pro-Top at strong shape equivalences. A map $f:X\to Y$ is a shape equivalence if and only if the induced function $f^*:[Y,P]\to [X,P]$ is a bijection for all $P\in {\rm ANR}$. A map $f:X\to Y$ of $k$-spaces is a strong shape equivalence if and only if the induced map $f^*:{\rm Map}(Y,P)\to {\rm Map}(X,P)$ is a weak homotopy equivalence for all $P\in {\rm ANR}$. One generalizes the concept of being a shape equivalence to morphisms of pro-HoTop without any problem and the only difficulty is to show that a localization of pro-HoTop at shape equivalences is a category (which amounts to showing that the morphisms form a set). Due to peculiarities of function spaces, extending the concept of strong shape equivalence to morphisms of pro-Top is more involved. However, it can be done and we show that the corresponding localization exists. One can introduce the concept of a super shape equivalence $f:X\to Y$ of topological spaces as a map such that the induced map $f^*:{\rm Map}(Y,P)\to {\rm Map}(X,P)$ is a homotopy equivalence for all $P\in {\rm ANR}$, and one can extend it to morphisms of pro-Top. However, the authors do not know if the corresponding localization exists. Here are applications of our methods:Theorem. A map $f:X\to Y$ of $k$-spaces is a strong shape equivalence if and only if $f\times \mathop{\rm id}_Q:X\times_k Q\to Y\times_k Q$ is a shape equivalence for each CW complex $Q$. Theorem. Suppose $f:X\to Y$ is a map of topological spaces. (a) $f$ is a shape equivalence if and only if the induced function $f^\ast:[Y,M]\to [X,M]$ is a bijection for all $M={\rm Map}(Q,P)$, where $P\in {\rm ANR}$ and $Q$ is a finite CW complex. (b) If $f$ is a strong shape equivalence, then the induced function $f^\ast:[Y,M]\to [X,M]$ is a bijection for all $M={\rm Map}(Q,P)$, where $P\in {\rm ANR}$ and $Q$ is an arbitrary CW complex.(c) If $X$, $Y$ are $k$-spaces and the induced function $f^\ast:[Y,M]\to [X,M]$ is a bijection for all $M={\rm Map}(Q,P)$, where $P\in {\rm ANR}$ and $Q$ is an arbitrary CW complex, then $f$ is a strong shape equivalence.

Détail

DOI : 10.4064/fm171-2-2

Keywords: purpose paper provide geometric explanation strong shape theory fairly simple introducing strong shape category formally generally speaking useful introduce shape theory localization class equivalences follow principle extend standard shape category hotop pro hotop localizing pro hotop shape equivalences similarly extend strong shape category edwards hastings ssh pro top localizing pro top strong shape equivalences map shape equivalence only induced function * bijection anr map k spaces strong shape equivalence only induced map * map map weak homotopy equivalence anr generalizes concept being shape equivalence morphisms pro hotop without problem only difficulty localization pro hotop shape equivalences category which amounts showing morphisms form set due peculiarities function spaces extending concept strong shape equivalence morphisms pro top involved however done corresponding localization exists introduce concept super shape equivalence topological spaces map induced map * map map homotopy equivalence anr extend morphisms pro top however authors know corresponding localization exists here applications methods theorem map k spaces strong shape equivalence only times mathop times times shape equivalence each complex theorem suppose map topological spaces shape equivalence only induced function ast bijection map where anr finite complex strong shape equivalence induced function ast bijection map where anr arbitrary complex k spaces induced function ast bijection map where anr arbitrary complex strong shape equivalence