Question

For a topological group G, we have a Pontryagin product in homology by multiplying representative cycles. This gives the homology the structure of an associative graded algebra. Am I correct in thinking we can prove this by seeing a topological group as an algebra over the Ass-operad and applying homology everywhere?

Answer 1

You can do that. If an operad O acts on a space X, then the structure maps O(n) x Xⁿ -> X induce homology operations H_*O(n) ⊗ H_*(X)^⊗n -> H_*(X). In particular, any path component in O(2) produces a multiplication on H_*X, if it's in the same path component as its own image under the symmetric group action it's commutative, if the two composites of it are in the same path component of O(3) it's associative, et cetera.

In particular if O is the associative operad (so O(n) are discrete) then this structure reduces to the Pontrjagin ring structure.