The operations on an H-space $X$ equip its homology with the structure of ring. At least for ordinary homology this is known as the Pontrjagin ring $H_*(X)$ of $X$.
The homological version of the group completion theorem relates the Pontrjagin ring of a topological monoid $A$ to that of its group completion $\Omega B A$.
Lev Pontrjagin, Homologies in compact Lie groups, Rec. Math. [Mat. Sbornik] N.S., 1939 Volume 6(48), Number 3, Pages 389–422 (mathnet:5835)
William Browder, Homology Rings of Groups, American Journal of Mathematics
Vol. 90, No. 1 (Jan., 1968) (jstor:2373440)
See also
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