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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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2 Answers
You can find many examples among finite topological spaces. For information on these, you can browse Jonathan Barmak's LNM book in the subject.
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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".
5,99323 silver badges52 bronze badges
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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.
137k11 gold badges247 silver badges381 bronze badges
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