Does Singular complex determine a topological space

Question

This question comes from thinking about the singular geometric realization adjunction $Sing \dashv |\cdot|$. I suspect that this adjunction is not monadic, so using the Monadicity theorem I tried to cook up two topological spaces with isomorphic singular complexes, but are not homeomorphic. However, I could not think of such a space! I feel like I am missing something. The closest I got was a torus and a Klein bottle have the same chain complex groups but different maps. So my question is, is it true that two spaces have isomorphic chain complexes iff they are homeomorphic?

Answer 1

The singular simplicial set of a space $X$ knows only about paths in $X$, which in general are not enough to determine the topology of $X$. For instance, let $X=\mathbb{Q}$ with the usual topology and let $Y=\mathbb{Q}$ with the discrete topology. Then the identity map $Y\to X$ induces an isomorphism of singular sets $Sing(Y)\to Sing(X)$ (and hence also an isomorphism of singular chain complexes), since every map from a simplex to either space is constant. But $Y$ and $X$ are not homeomorphic.