S : ∞-Grpd₀ ⭢ ∞-Grpd₀ such that Homology of X is the homotopy of SX?

Question

Let $X$ be a based CW-complex, and write ∞-Grpd₋₁ for the category of based CW-complexes.

I am wondering if there is some operation S or functor on spaces, remaining in the homotopy category, which from X produces a space SX such that πₙSX is Hₙ(X,ℤ), or even Hₙ(X,A).

Recall the notion of an H-space, which produces an internal abelian group in the homotopy category of based connected CW-complexes.

Here are some structures in higher algebra that I might like to consider:

• Recall the notion of an H-space, which I here take to be an internal abelian group in the derived category.
• A${}^{∞}$ and E${}^{∞}$ spaces are also of potential interest here.

Overall I wanted some kind of free construction, maybe even something like this: form a simplicial set S = Sₙ out of the CW complex X, then take the free abelian group on Sₙ, making Tₙ := ℤSₙ. Then view T as a simplicial set again, but now also as an H-space.

Question:

Is it then true that πₙ T is HₙS?

Answer 1

Sounds like you're looking for the Dold-Thom theorem. For a space $X$, define the $n$-th symmetric product $\operatorname{SP}^n(X) = X^n / \Sigma_n$, and let the infinite symmetric product be $\operatorname{SP}(X) = \varinjlim \operatorname{SP}^n(X)$. Then,

Dold-Thom theorem. If $X$ is a connected CW complex, then $\pi_n \operatorname{SP}(X) \cong \tilde{H}_n(X; \mathbb{Z})$.

The symmetric product construction is essentially what you've described: it's a kind of free commutative monoid and linearization. There is also a version of this construction adding labels in an abelian group $A$ that gives homology with coefficients in $A$.