Recall that a chain complex $(C_*,d)$ of abelian groups is contractible if it is homotopic to the zero map. Or equivalently: there exists a degree 1 map $F: C_* \to C_*$ such that $\operatorname{Id}= dF+ Fd$.

Question: does there exist a topological space $X$ which is not contractible (in the sense of topology), but whose complex of singular chain $C_*(X)$ is contractible?

More generally, one can ask whether the functor $X \mapsto C_*(X)$ from the category of topological spaces to the homotopy category of chain complexes of abelian groups forgets any information. I assume the answer is "yes", but I can't seem to come up with a counterexample.

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    $\begingroup$ I've decided to interpret the question as asking whether the reduced singular complex is trivial, or equivalently if the canonical map $C_*(X)→\mathbb{Z}$ is a homotopy equivalence, since that's the property that's analogue to the contractibility of $X$ (i.e. $X\to \ast$ is a homotopy equivalence). $\endgroup$ Apr 5, 2020 at 8:20

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These are known as acyclic spaces (note that since $\tilde C_*(X)$ is a bounded below complex of projectives, it being contractible is equivalent to its homology being trivial).

There's an extensive literature about them, starting with this Emanuel Farjoun's paper.

In general, yes $C_*(X)$ does forget some information about $X$: first of all it factors through the stable homotopy type $\Sigma^\infty_+X$ of $X$, and it forgets further information from there (the slogan is that it remembers only the "$\mathbb{Z}$-linear" information contained in $\Sigma^\infty_+X$). For example it is unable to distinguish between $\mathbb{CP}^2$ and $S^2\vee S^4$ (since the attaching map $\eta:S^3\to S^2$ is sent to a map homotopic to 0 by $C_*(-)$).

You can get much closer to reconstructing the full homotopy type of $X$, by remembering the coalgebra structure on $C_*(X)$ induced by the diagonal, although in general that is still not enough.

  • $\begingroup$ For more information relating to the last paragraph, one can look up Mandell's theorem that gives some conditions for $C^*(X)$ to lose no information $\endgroup$ Apr 5, 2020 at 8:27
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    $\begingroup$ @MaximeRamzi I think this paper by Allen Yuan is the state-of-the art on the subject (reconstructing unstable data from stable data plus algebra structures), although of course there's a vast literature on this too :). $\endgroup$ Apr 5, 2020 at 9:47

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