Goresky-Hingston product on cohomology of the relative free loop space on $S^1$

Question

I'm interested in the computations of the Goresky-Hingston product (defined https://arxiv.org/abs/0707.3486) on the cohomology of the relative free loop space on the circle (or better yet, their extension to absolute cohomology). They give a full computation for all $S^n, n\geq 3$, and give some description of all manifolds whose geodesics are closed. Yet somehow I cannot read off the computation for $S^1$ (as someone who is well outside this area). Any pointers would be appreciated!

Answer 1

Here is a sort of provocative, quick, partial answer. Take it as a pointer. Let $R$ be a commutative ring and consider $R[x]$ the polynomial ring on one variable $x$. For any $f \in R[x]$ we may consider the following discrete version of the derivative: $$D_q(f)= \frac{f(qx)-f(x)}{qx-x}$$ where $q$ here is thought of as a formal variable, so that $D_q(f)$ is an element in the polynomial ring $R[q,x]$ in two variables. In fact, we have an identification $R[q,x] \cong R[x] \otimes R[x]$ and under this identification $D_q$ gives rise to the coproduct $$\Delta \colon R[x] \to R[x] \otimes R[x]$$ determined by the formula $$\Delta(x^m)= \sum_{i=0}^{m-1} x^i \otimes x^{m-1-i}.$$ This is a coproduct of degree $-1$ sometimes called the "quantum derivative". It satisfies a Leibniz rule with respect to the multiplication of polynomials, namely, it defines an "infinitesimal bialgebra" structure on $R[x]$. I think after making certain identifications and extensions - you may want to add $x^{-1}$ as well - this construction should determine the Goresky-Hingston coproduct on $H_*(LS^1;R)$, the homology of the free loop space of $S^1$ and and consequently the product on $H^*(LS^1;R)$. Note there is no need to work modulo constant loops in this particular case.