How is a graph not like a manifold?
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866912886078570496 |
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| author | Ayzenberg, Anton Masuda, Mikiya Solomadin, Grigory |
| author_facet | Ayzenberg, Anton Masuda, Mikiya Solomadin, Grigory |
| contents | For an equivariantly formal action of a compact torus $T$ on a smooth manifold $X$ with isolated fixed points we investigate the global homological properties of the graded poset $S(X)$ of face submanifolds. We prove that the condition of $j$-independency of tangent weights at each fixed point implies $(j+1)$-acyclicity of the skeleta $S(X)_r$ for $r>j+1$. This result provides a necessary topological condition for a GKM graph to be a GKM graph of some GKM manifold. We use particular acyclicity arguments to describe the equivariant cohomology algebra of an equivariantly formal manifold of dimension $2n$ with an $(n-1)$-independent action of $(n-1)$-dimensional torus, under certain colorability assumptions on its GKM graph. This description relates the equivariant cohomology algebra to the face algebra of a simplicial poset. Such observation underlines certain similarity between actions of complexity one and torus manifolds. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2203_10641 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | How is a graph not like a manifold? Ayzenberg, Anton Masuda, Mikiya Solomadin, Grigory Algebraic Topology K-Theory and Homology 57S12, 55N91, 13F55, 06A06 (Primary) 55P91, 55U10, 55T25, 57R91, 13H10, 55R20 (Secondary) For an equivariantly formal action of a compact torus $T$ on a smooth manifold $X$ with isolated fixed points we investigate the global homological properties of the graded poset $S(X)$ of face submanifolds. We prove that the condition of $j$-independency of tangent weights at each fixed point implies $(j+1)$-acyclicity of the skeleta $S(X)_r$ for $r>j+1$. This result provides a necessary topological condition for a GKM graph to be a GKM graph of some GKM manifold. We use particular acyclicity arguments to describe the equivariant cohomology algebra of an equivariantly formal manifold of dimension $2n$ with an $(n-1)$-independent action of $(n-1)$-dimensional torus, under certain colorability assumptions on its GKM graph. This description relates the equivariant cohomology algebra to the face algebra of a simplicial poset. Such observation underlines certain similarity between actions of complexity one and torus manifolds. |
| title | How is a graph not like a manifold? |
| topic | Algebraic Topology K-Theory and Homology 57S12, 55N91, 13F55, 06A06 (Primary) 55P91, 55U10, 55T25, 57R91, 13H10, 55R20 (Secondary) |
| url | https://arxiv.org/abs/2203.10641 |