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# Non-contractible space with trivial homotopy groups

What is an example of a non-contractible space $X$ with $\pi_n(X) = 0$ for all $n\geq 0$ (note in particular $X$ is path connected)?

Motivation: Whitehead's theorem implies that no such CW complex $X$ exists. I'd like to know a counterexample to the "general Whitehead theorem".

You can find many examples among finite topological spaces. For information on these, you can browse Jonathan Barmak's LNM book in the subject.

Quasi-circle, which is defined in exercise 7 of page 79 in Hatcher’s book Algebraic topology

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# Non-contractible space with trivial homotopy groups

What is an example of a non-contractible space $X$ with $\pi_n(X) = 0$ for all $n\geq 0$ (note in particular $X$ is path connected)?

Motivation: Whitehead's theorem implies that no such CW complex $X$ exists. I'd like to know a counterexample to the "general Whitehead theorem".

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You can find many examples among finite topological spaces. For information on these, you can browse Jonathan Barmak's LNM book in the subject.

I would add for those unfamiliar with finite topological spaces that every finite space is locally contractible. - Eric Wofsey Feb 18, 2018 at 2:23

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