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If $X$ is a pointed space and $A$ is an abelian group, then we can form the space $A[X]$ whose points are finite formal sums $\sum a_i x_i$ with $a_i \in A, x_i \in X$ subject to some natural relations involving $0,+,$ and the basepoint of $X$. We topologize this space as a quotient of $\bigsqcup A^n \times X^n$ where $A$ has discrete topology. Evidently this is true: $$\pi_n(A[X],0) \cong \tilde{H}_n(X;A)$$ if $X$ has the homotopy type of a CW-complex. Can anyone point me to a proof of this fact?

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up vote 6 down vote accepted

This result appears in M. C. McCord, Classifying Spaces and Infinite Symmetric Products published in Transactions of the American Mathematical Society, Vol. 146, (Dec., 1969), pp. 273-298.

Also, this can be generalized to other generalized homology theories: given any spectrum $E$ one can build out of it a partial monoid, $A$, for which a variant of the construction you describe produces, given a space $X$, a new space whose homotopy groups are the reduced $E$-homology of $X$. This is proved in Jacob Mostovoy's paper Partial monoids and Dold-Thom functors.

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