Obstructions to homotopy invariance of loop coproduct via parametrised fixed-point theory
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866918280038449152 |
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| author | Kenigsberg, Lea Porcelli, Noah |
| author_facet | Kenigsberg, Lea Porcelli, Noah |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_13662 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Obstructions to homotopy invariance of loop coproduct via parametrised fixed-point theory Kenigsberg, Lea Porcelli, Noah Algebraic Topology Geometric Topology K-Theory and Homology Symplectic Geometry 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. |
| title | Obstructions to homotopy invariance of loop coproduct via parametrised fixed-point theory |
| topic | Algebraic Topology Geometric Topology K-Theory and Homology Symplectic Geometry |
| url | https://arxiv.org/abs/2407.13662 |