A combinatorial approach to Kontsevich's Swiss cheese conjecture
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866908730382090240 |
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| author | De Leger, Florian |
| author_facet | De Leger, Florian |
| contents | From a coloured operad $\mathcal{P}$ and a $\mathcal{P}$-algebra $A$, we construct a new operad $\mathrm{SC}(\mathcal{P})$ and a Hochschild object $\mathrm{Hoch}(A)$ together with an $\mathrm{SC}(\mathcal{P})$-action on the pair $(\mathrm{Hoch}(A),A)$. We prove that if $\mathcal{P}$ is the little $n$-disks operad, then $\mathrm{SC}(\mathcal{P})$ is equivalent to the Swiss cheese operad $\mathcal{SC}_n$. This gives us a weak version of Kontsevich's Swiss cheese conjecture (without the universal property). We apply our result to prove the existence of an $E_{n+1}$-action on the Hochschild-Pirashvili cochain of order $n$ of a commutative algebra. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_20167 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A combinatorial approach to Kontsevich's Swiss cheese conjecture De Leger, Florian Algebraic Topology From a coloured operad $\mathcal{P}$ and a $\mathcal{P}$-algebra $A$, we construct a new operad $\mathrm{SC}(\mathcal{P})$ and a Hochschild object $\mathrm{Hoch}(A)$ together with an $\mathrm{SC}(\mathcal{P})$-action on the pair $(\mathrm{Hoch}(A),A)$. We prove that if $\mathcal{P}$ is the little $n$-disks operad, then $\mathrm{SC}(\mathcal{P})$ is equivalent to the Swiss cheese operad $\mathcal{SC}_n$. This gives us a weak version of Kontsevich's Swiss cheese conjecture (without the universal property). We apply our result to prove the existence of an $E_{n+1}$-action on the Hochschild-Pirashvili cochain of order $n$ of a commutative algebra. |
| title | A combinatorial approach to Kontsevich's Swiss cheese conjecture |
| topic | Algebraic Topology |
| url | https://arxiv.org/abs/2512.20167 |