An $A_\infty$-version of the Eilenberg-Moore theorem
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866912314727333888 |
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| author | Franz, Matthias |
| author_facet | Franz, Matthias |
| contents | We construct an $A_\infty$-structure on the two-sided bar construction involving homotopy Gerstenhaber algebras (hgas). It extends the non-associative product defined by Carlson and the author and generalizes the dga structure on the one-sided bar construction due to Kadeishvili-Saneblidze. As a consequence, the multiplicative cohomology isomorphism from the Eilenberg-Moore theorem is promoted to a quasi-isomorphism of $A_\infty$-algebras.
We also show that the resulting product on the differential torsion product involving cochain algebras agrees with the one defined by Eilenberg-Moore and Smith, for all triples of spaces. This is a consequence of the following result, which is of independent interest: The strongly homotopy commutative (shc) structure on cochains inductively constructed by Gugenheim-Munkholm agrees with the one previously defined by the author for all hgas. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_16947 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | An $A_\infty$-version of the Eilenberg-Moore theorem Franz, Matthias Algebraic Topology 55R20 (Primary), 16E45, 55T20 (Secondary) We construct an $A_\infty$-structure on the two-sided bar construction involving homotopy Gerstenhaber algebras (hgas). It extends the non-associative product defined by Carlson and the author and generalizes the dga structure on the one-sided bar construction due to Kadeishvili-Saneblidze. As a consequence, the multiplicative cohomology isomorphism from the Eilenberg-Moore theorem is promoted to a quasi-isomorphism of $A_\infty$-algebras. We also show that the resulting product on the differential torsion product involving cochain algebras agrees with the one defined by Eilenberg-Moore and Smith, for all triples of spaces. This is a consequence of the following result, which is of independent interest: The strongly homotopy commutative (shc) structure on cochains inductively constructed by Gugenheim-Munkholm agrees with the one previously defined by the author for all hgas. |
| title | An $A_\infty$-version of the Eilenberg-Moore theorem |
| topic | Algebraic Topology 55R20 (Primary), 16E45, 55T20 (Secondary) |
| url | https://arxiv.org/abs/2311.16947 |