Fiber products, Poincaré duality and $A_\infty$-ring spectra

Abstract:

For a Poincaré duality space $X^d$ and a map $X \to B$, consider the homotopy fiber product $X \times ^B X$. If $X$ is orientable with respect to a multiplicative cohomology theory $E$, then, after suitably regrading, it is shown that the $E$-homology of $X \times ^B X$ has the structure of a graded associative algebra. When $X \to B$ is the diagonal map of a manifold $X$, one recovers a result of Chas and Sullivan about the homology of the unbased loop space $LX$.