If $X$ is a manifold and any continuous mapping $f:Y\to X$ is null-homotopic for any compact space $Y$, then is $X$ necessarily contractible?

Asked Apr 16 '23 at 22:29

Active Apr 16 '23 at 22:29

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From Whitehead's theorem, we can know the statement is correct when $X$ is a CW complex. For general manifold $X$, it seems not clear whether there is a CW structure on $X$. Also, I don't know whether allowing $Y$ to be any compact space, rather than just $S^n$, will help.

asked Apr 16 '23 at 22:29

qdmj

4

Each manifold is homotopy-equivalent to a CW complex. See here. – Moishe Kohan Apr 16 '23 at 22:34

If $X$ is a manifold and any continuous mapping $f:Y\to X$ is null-homotopic for any compact space $Y$, then is $X$ necessarily contractible?

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