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weak homotopy equivalences

Proceedings of the Edinburgh Mathematical Society (1)

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Spaces which Invert Weak Homotopy Equivalences
Part of
Jonathan Ariel Barmak
Journal:

Proceedings of the Edinburgh Mathematical Society / Volume 62 / Issue 2 / May 2019

Published online by Cambridge University Press:

03 December 2018, pp. 553-558
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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.