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I realise this is a very vague question! I've heard people say that A∞ algebras are the right homotopy-theoretic generalization of usual associative algebras, because you can deform them. What ...

Let $X$ be a topological space, and $Homeo(X)$ the group of self-homeomorphisms of $X$. (1) What is the exact meaning of: $H^*(X)$ is a an $A_\infty$-module over $Homeo(X)$? (2) Does $H_*(X)$ also ...