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Homotopy Non-Invariance of the String Cobracket
Abstract
We prove that the integral string cobracket is not invariant under homotopy equivalences. Using Naef's computation arXiv:2106.11307 of the Goresky--Hingston coproduct and the Leray--Serre spectral sequence for the Borel construction, we compute the degree-two string cobracket for the three-dimensional lens spaces L(9;1) and L(9;4). The rank profiles of the cobracket on their nontrivial free-loop components differ, although the two lens spaces are homotopy equivalent. We also discuss the defect of naturality in relation to the Dennis trace of Whitehead torsion.
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BibTeX
@misc{sumoto2026homotopy,
title = {{Homotopy Non-Invariance of the String Cobracket}},
author = {Isana Sumoto},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2605.10273},
eprint = {2605.10273},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.10273},
}
RIS
TY - UNPB
TI - Homotopy Non-Invariance of the String Cobracket
AU - Isana Sumoto
PY - 2026
DA - 2026/10/10
PB - arXiv
DO - 10.48550/arxiv.2605.10273
UR - https://arxiv.org/abs/2605.10273
AB - We prove that the integral string cobracket is not invariant under homotopy equivalences. Using Naef's computation arXiv:2106.11307 of the Goresky--Hingston coproduct and the Leray--Serre spectral sequence for the Borel construction, we compute the degree-two string cobracket for the three-dimensional lens spaces L(9;1) and L(9;4). The rank profiles of the cobracket on their nontrivial free-loop components differ, although the two lens spaces are homotopy equivalent. We also discuss the defect of naturality in relation to the Dennis trace of Whitehead torsion.
ER -