Non-Formality of $S^2$ via the free loop space
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866911882127867904 |
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| author | McGowan, Ryan Naef, Florian O'Callaghan, Brian |
| author_facet | McGowan, Ryan Naef, Florian O'Callaghan, Brian |
| contents | We show that the $E_1$-equivalence $C^\bullet(S^2) \simeq H^\bullet(S^2)$ does not intertwine the inclusion of constant loops into the free loop space $S^2 \to LS^2$. That is, the isomorphism $HH_\bullet(H^\bullet(S^2)) \cong H^\bullet(LS^2)$ does not preserve the obvious maps to $H^\bullet(S^2)$ that exist on both sides. We give an explicit computation of the defect in terms of the $E_\infty$-structure on $C^\bullet(S^2)$. Finally, we relate our calculation to recent work of Poirier-Tradler on the string topology of $S^2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_12047 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Non-Formality of $S^2$ via the free loop space McGowan, Ryan Naef, Florian O'Callaghan, Brian Algebraic Topology We show that the $E_1$-equivalence $C^\bullet(S^2) \simeq H^\bullet(S^2)$ does not intertwine the inclusion of constant loops into the free loop space $S^2 \to LS^2$. That is, the isomorphism $HH_\bullet(H^\bullet(S^2)) \cong H^\bullet(LS^2)$ does not preserve the obvious maps to $H^\bullet(S^2)$ that exist on both sides. We give an explicit computation of the defect in terms of the $E_\infty$-structure on $C^\bullet(S^2)$. Finally, we relate our calculation to recent work of Poirier-Tradler on the string topology of $S^2$. |
| title | Non-Formality of $S^2$ via the free loop space |
| topic | Algebraic Topology |
| url | https://arxiv.org/abs/2405.12047 |