We develop a machinery of Chen iterated integrals for higher Hochschild complexes. These are complexes whose differentials are modeled on an arbitrary simplicial set much in the same way the ordinary Hochschild differential is modeled on the circle. We use these to give algebraic models for general mapping spaces and define and study the surface product operation on the homology of mapping spaces of surfaces of all genera into a manifold. This is an analogue of the loop product in string topology....

In this work we compute the Chen–Ruan cohomology of the moduli spaces of smooth and stable $n$-pointed curves of genus $1$. In the first part of the paper we study and describe stack theoretically the twisted sectors of ${ℳ}_{1,n}$ and ${\overline{ℳ}}_{1,n}$. In the second part, we study the orbifold intersection theory of ${\overline{ℳ}}_{1,n}$. We suggest a definition for an orbifold tautological ring in genus $1$, which is a subring of both the Chen–Ruan cohomology and of the stringy Chow ring.

The homotopy fiber of the inclusion from the long embedding space to the long immersion space is known to be an iterated based loop space (if the codimension is greater than two). In this paper we deloop the homotopy fiber to obtain the topological Stiefel manifold, combining results of Lashof and of Lees. We also give a delooping of the long embedding space, which can be regarded as a version of Morlet-Burghelea-Lashof's delooping of the diffeomorphism group of the disk relative to the boundary....

In this short note we compute the Chas-Sullivan BV-algebra structure on the singular homology of the free loop space of complex projective spaces. We compare this result with computations in Hochschild cohomology.

We use the computational power of rational homotopy theory to provide an explicit cochain model for the loop product and the string bracket of a simply connected closed manifold $M$. We prove that the loop homology of $M$ is isomorphic to the Hochschild cohomology of the cochain algebra $C^{*}\left(M\right)$ with coefficients in $C^{*}\left(M\right)$. Some explicit computations of the loop product and the string bracket are given.