Open-Closed Hochschild Homology and the Relative Disk Mapping Space
It is known that a model for the differential graded algebra (dga) of differential forms on the free loop space $LN$ of a simply connected smooth manifold $N$ is given by the Hochschild chain complex of the dga $Ω(N)$ of differential forms on $N$, as shown by K.-T. Chen via his theory of iterated integrals. We develop a relative version of Chen's model. Given a smooth map $f\colon N\to M$ between smooth manifolds, we consider the ``relative disk mapping space'' consisting of pairs $(Φ,γ)$ of maps $Φ\colon \mathbb D\to M$ and $γ\colon S^1\to N$ such that $Φ|_{\partial\mathbb D}=f\circγ$. We construct iterated integral models for this mapping space through an open-closed homotopy algebra (OCHA) naturally associated to $f$ and the theory of open-closed Hochschild homology, which may be of independent interest. Our main theorem states that the resulting map is a quasi-isomorphism when $M$ is contractible or 2-connected with the rational homotopy type of an odd sphere, and $N$ is simply connected. This result generalizes Chen's classical theorem for free loop spaces and, in the above special cases, extends the theorem of Getzler-Jones for double loop spaces.