Andrea Bianchi: String topology and graph cobordisms

Abstract.

Let \(M\) be an oriented, closed manifold; the collection of homology groups \(H_*(\operatorname{map}(X, M))\), for varying topological space \(X\), carries additional operations; these come from functoriality of \(\operatorname{map}(-, M)\), and from Poincare' duality of \(M\). The most famous such operation is perhaps the Chas-Sullivan string product on \(H_*(\operatorname{map}(S^1, M))\). I will report on my current attempt to define quite general operations \(H_*(\operatorname{map}(X, M)) \to H_*(\operatorname{map}(Y, M))\) using the homology of a suitable moduli space of graph cobordisms between \(X\) and \(Y\).