String topology and graph cobordisms

Short introductory lectures.

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