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Spaces which Invert Weak Homotopy Equivalences

Part of: Homotopy groups Homotopy theory Peculiar spaces

Published online by Cambridge University Press:  03 December 2018

Jonathan Ariel Barmak
Jonathan Ariel Barmak*
Affiliation:
Departamento de Matemática, Universidad de Buenos Aires, Facultad de Ciencias Exactas y Naturales, Buenos Aires, Argentina (jbarmak@dm.uba.ar) CONICET-Universidad de Buenos Aires, Instituto de Investigaciones Matemáticas Luis A. Santaló (IMAS), Buenos Aires, Argentina
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Abstract

It is well known that if X is a CW-complex, then for every weak homotopy equivalence f : A → B, the map f* : [X, A] → [X, B] induced in homotopy classes is a bijection. In fact, up to homotopy equivalence, only CW-complexes have that property. Now, for which spaces X is f* : [B, X] → [A, X] a bijection for every weak equivalence f? This question was considered by J. Strom and T. Goodwillie. In this note we prove that a non-empty space inverts weak equivalences if and only if it is contractible.

Keywords

weak homotopy equivalenceshomotopy typesnon-Hausdorff spaces

MSC classification

Primary: 55Q05: Homotopy groups, general; sets of homotopy classes
Secondary: 55P15: Classification of homotopy type 54G05: Extremally disconnected spaces, $F$-spaces, etc. 54G10: $P$-spaces
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 62 , Issue 2 , May 2019 , pp. 553 - 558
Copyright
Copyright © Edinburgh Mathematical Society 2018 

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References

1.Goodwillie, T., Strickland, N. and Strom, J., Spaces that invert weak homotopy equivalences, available at https://mathoverflow.net/q/47042.Google Scholar
2.Hatcher, A., Algebraic topology (Cambridge University Press, 2002).Google Scholar