Let $X$ be a path-connected topological space such that its reduced cohomology vanishes, i.e. $\tilde{H}^i(X)=0$. Does it follow that $X$ is contractible (i.e homotopy equivalent to a point)?
(This question was motivated by this question of mine.)
2026-04-07 06:30:35.1775543435
Cohomology determining homotopy type
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The answer is no. For simply connected spaces, examples that fail would not be CW complexes (e.g. the Warsaw Circle). Within CW complexes, there are examples $X$ with $\pi_1(X)$ non-trivial (so not contractible) with reduced integral (co)homology vanishing. See the link in the comment above to avoid duplicative answers.