Models for the Cohomology of Certain Polyhedral Products
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866912198378389504 |
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| author | Bendersky, Martin Grbić, Jelena |
| author_facet | Bendersky, Martin Grbić, Jelena |
| contents | For a commutative ring $\mathbf k$ with unit, we describe and study various differential graded $\mathbf k$-modules and $ \mathbf k$-algebras which are models for the cohomology of polyhedral products $(\underline{CX},\underline X)^K$. Along the way, we prove that the integral cohomology $H^*((D^1, S^0)^K; \mathbb Z)$ of the real moment-angle complex is a Tor module, the one that does not come from a geometric setting. We also reveal that the apriori different cup product structures in $H^*((D^1, S^0)^K;\mathbb Z)$ and in $H^*((D^n, S^{n-1})^K; \mathbb Z)$ for $n\geq 2$ have the same origin.
As an application, this work sets the stage for studying the based loop space of $(\underline{CX}, \underline X)^K$ in terms of the bar construction applied to the differential graded $\mathbb Z$-algebras $B(\mathcal C^*(\underline X; \mathbb Z), K) $ quasi-isomorphic to the singular cochain algebra $\mathcal C^*((\underline{CX},\underline X)^K;\mathbb Z)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_12433 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Models for the Cohomology of Certain Polyhedral Products Bendersky, Martin Grbić, Jelena Algebraic Topology For a commutative ring $\mathbf k$ with unit, we describe and study various differential graded $\mathbf k$-modules and $ \mathbf k$-algebras which are models for the cohomology of polyhedral products $(\underline{CX},\underline X)^K$. Along the way, we prove that the integral cohomology $H^*((D^1, S^0)^K; \mathbb Z)$ of the real moment-angle complex is a Tor module, the one that does not come from a geometric setting. We also reveal that the apriori different cup product structures in $H^*((D^1, S^0)^K;\mathbb Z)$ and in $H^*((D^n, S^{n-1})^K; \mathbb Z)$ for $n\geq 2$ have the same origin. As an application, this work sets the stage for studying the based loop space of $(\underline{CX}, \underline X)^K$ in terms of the bar construction applied to the differential graded $\mathbb Z$-algebras $B(\mathcal C^*(\underline X; \mathbb Z), K) $ quasi-isomorphic to the singular cochain algebra $\mathcal C^*((\underline{CX},\underline X)^K;\mathbb Z)$. |
| title | Models for the Cohomology of Certain Polyhedral Products |
| topic | Algebraic Topology |
| url | https://arxiv.org/abs/2206.12433 |