arXiv subjects
Publications and source records attributed to Jonathan Clivio.
We construct a lift of the Goresky-Hingston coproduct on the based loop space. Using this lift, we produce formulas for the interaction of the coproduct with the $\pi_1$-action and with maps of manifolds. These results lets us compute the coproduct on the based loop space for manifolds of the form $S^n/G$.
We describe the Goresky-Hingston coproduct on the free loop space with real coefficients via the quasi-isomorphism $C_*(\Lambda M)\simeq C_*(M,C_*(\Omega M))$. This lets us describe the coproduct on the Leray-Serre spectral sequence as the diagonal on the manifold and the coproduct on the based loop space. With this description, we compute the coproduct with real coefficients for manifolds of the form $M=S^n/G$. The appendix contains results on inverting morphisms of $\mathcal{A}_\infty$-modules that are required for our computations.
We construct a PROP which encodes 2D-TQFTs with a grading. This defines a graded Frobenius algebra as algebras over this PROP. We also give a description of graded Frobenius algebras in terms of maps and relations. This structure naturally arises as the cohomology of manifolds, loop homology and Hochschild homology of Frobenius algebras. In addition, we give a comprehensive description of the signs that arise in suspending algebras over PROPs.