I am trying to understand what the singular chain complex $C(X)$ of a topological space $X$ remembers about $X$. For example, suppose that $C(X)= C(Y)$. Does this imply that $X,Y$ are homotopy equivalence? Are there counterexamples?

I ask because I know that a related construction to the chain complex does remember the homotopy type of $X$. We form the simplicial set of singular chains and its geometric realization will be homotopy equivalent to $X$. From this simplicial set we can form the singular chain complex of $X$ (with differential equal to the alternating sum of face maps) but it seems to be a strictly weaker notion (and I am trying to understand the relationship between the two).

Is the main difference between the simplicial set and the singular chain complex that the former remembers all the face maps (in giving order) but the latter forgets the order of the face maps? I can try to construct the following chain complex that does remember all the faces: $$ \oplus^2_{i=1} C_0(X) \leftarrow \oplus^3_{i=1} C_1(X) \leftarrow \oplus^4_{i=1} C_2(X) \leftarrow \oplus^5_{i=1} C_3(X) \leftarrow \oplus^6_{i=1} C_4(X) \leftarrow \cdots $$ For example, each component $\oplus^4_{i=1} C_2(X) \leftarrow C_3(X)$ of the map $\oplus^4_{i=1} C_2(X) \leftarrow \oplus^5_{i=1} C_3(X)$ is given by taking a 3-chain to its four 2-chain faces. I think this is still a chain complex. Does this new chain complex remember the homotopy type of $X$?

In general, I am wondering if there are any algebraic objects (chain complexes, dga) that trivially remember the homotopy type of a space. I know that all simplicial spaces are the classifying spaces of certain categories so this is an example.

  • $\begingroup$ Are you familiar with whitehead theorem ? en.wikipedia.org/wiki/Whitehead_theorem, $\endgroup$ – jeanmfischer Jun 29 '17 at 17:02
  • $\begingroup$ I don't really see the relevance of Whitehead here. Essentially all the chain complex remembers is the homology of $X$, and the critical difference with a simplicial set is the fact that chain compresses are made from abelian groups. $\endgroup$ – Kevin Carlson Jun 29 '17 at 20:50
  • $\begingroup$ The simplicial set that remembers the homotopy type is the singular simplicial complex, not the simplicial complex whose terms are the singular chains. $\endgroup$ – Hurkyl Jun 30 '17 at 21:44
  • $\begingroup$ Once you move into the land of abelian groups, you've lost the game; chain complexes are then the best you can do. E.g. in regard to simplicial abelian groups, the Dold-Kan correspondence proves something much stronger than that: the category of simplicial abelian groups is equivalent to the category of chain complexes of abelian groups. $\endgroup$ – Hurkyl Jun 30 '17 at 21:46

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