String topology and graph cobordisms

Abstract

We introduce a symmetric monoidal $\infty$-category $\mathrm{GrCob}$ of graph cobordisms between spaces, and use the homology of its morphism spaces to define string operations. Precisely, for an $E_\infty$-ring spectrum $R$ and an oriented $d$-dimensional $R$-Poincar\'e duality space $M$, we construct a "graph field theory" $\mathrm{GFT}_M$, i.e. a symmetric monoidal functor from a suitable $R$-linearisation of $\mathrm{GrCob}^\mathrm{op}$ to the category $\mathrm{Mod}_R$ of $R$-modules in spectra; the graph field theory takes an object $X\in\mathrm{GrCob}^\mathrm{op}$, i.e. a space, to the $R$-module $\Sigma_+^\infty\mathrm{map}(X,M)\otimes R$ of $R$-chains on the mapping space from $X$ to $M$; by selecting suitable graph cobordisms we recover the basic string operations given by restriction, cross product with the fundamental class, and the Chas-Sullivan operations. The construction is natural with respect to oriented homotopy equivalences of $R$-Poincar\'e duality spaces; in particular, restricting to the endomorphisms of $\emptyset\in\mathrm{GrCob}^\mathrm{op}$, we obtain characteristic classes of $R$-oriented $M$-fibrations parametrised by the suitably twisted homology of $\mathbf{B}\mathrm{Out}(F_n)$, recovering results of Berglund and Barkan-Steinebrunner. Finally, we describe explicitly the morphism spaces in $\mathrm{GrCob}$, answering along the way a question by Hatcher. This allows us to construct a symmetric monoidal functor from the open-closed cobordism $\infty$-category $\mathcal{OC}$ to $\mathrm{GrCob}$. Composing with $\mathrm{GFT}_M$, we obtain an open-closed field theory with values in $\mathrm{Mod}_R$, attaining values $\Sigma^\infty_+LM\otimes R$ and $\Sigma^\infty_+M\otimes R$ at the circle and at the interval, respectively. We expect this to recover and extend constructions of Cohen, Godin and others.