Obstructions to homotopy invariance of loop coproduct via parametrised fixed-point theory

Abstract

Given $f: M \to N$ a homotopy equivalence of compact manifolds with boundary, we use a construction of Geoghegan and Nicas to define its Reidemeister trace $[T] \in π_1^{st}(\mathcal{L} N, N)$. We realize the Goresky-Hingston coproduct as a map of spectra, and show that the failure of $f$ to entwine the spectral coproducts can be characterized by Chas-Sullivan multiplication with $[T]$. In particular, when $f$ is a simple homotopy equivalence, the spectral coproducts of $M$ and $N$ agree.

Figures (6)

• Figure 1: Heuristic picture of the coproduct in the case $*=1$, $n=2$: left shows a $1$-parameter family of loops, right shows the output of the coproduct, a $0$-parameter family of pairs of loops.
• Figure 2: Coproduct in the closed case: the figure on the left shows a triple $(v,\gamma, t)$ in the domain of the coproduct. The figure on the right shows the output: the first component is the sum of the two vectors indicated, scaled appropriate by a factor of ${\sqrt L}/\varepsilon$. The incidence condition holds because $\gamma(t)$ lies in the ball $B_\varepsilon(v)$.
• Figure 3: Some choices in the definition of the coproduct: $e(M)$, ${\color{purple}e(M^{ext})}$, ${\color{blue}U}$, ${\color{olive}\tilde{U}}$ and ${\color{red}V}$ are shown.
• Figure 4: Coproduct: the figure on the left shows a triple $(v,\gamma, t)$ in the domain of the coproduct. The figure on the right shows the output.
• Figure 5: Collars.

Theorems & Definitions (183)

• Remark 1.1
• Theorem 1: Theorem \ref{['thm: cop Xi']}
• Theorem 2: Theorem \ref{['thm: T Xi']}
• Theorem 3: Theorem \ref{['thm: stability']} and Proposition \ref{['prop: thom cop']}
• Corollary 1.2
• Remark 1.3
• Theorem 4
• Corollary 1.4
• Remark 1.5
• Remark 1.6