What is the integral homology of the free commutative monoid on a space?

Question

The Dold-Thom construction on a space is its free commutative monoid. Its homotopy groups are isomorphic to the integral homology groups of the original space. What are the integral homology groups of the Dold-Thom construction?

Answer 1

The Dold-Thom construction of a connected CW-complex $X$ is a product of Eilenberg-Maclane spaces so its homology groups are those of a product of Eilenberg-Maclane space which are determined by a Kunneth formula.

One basic example to see is the Dold Thom construction of a circle which is homotopy equivalent to a circle and also the examples of spheres more generally.

Of course, the Dold-Thom construction which is also called the infinite symmetric product construction $SP^{\infty}$ is a homotopy functor, meaning that we can compute the integral homology of infinite symmetric products of "simpler" spaces if these spaces are homotopy equivalent.