Instantons and rational homology spheres
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arXiv
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| Format: | Preprint |
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2022
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| author | Daemi, Aliakbar Eismeier, Mike Miller |
| author_facet | Daemi, Aliakbar Eismeier, Mike Miller |
| contents | In previous work, the second author defined 'equivariant instanton homology groups' $I^\bullet(Y,π;R)$ for a rational homology 3-sphere $Y$, a set of auxiliary data $π$, and a PID $R$. These objects are modules over the cohomology ring $H^{-*}(BSO_3;R)$.
We prove that the equivariant instanton homology groups $I^\bullet(Y;R)$ are independent of the auxiliary data $π$, and thus define topological invariants of rational homology spheres. Further, we prove that these invariants are functorial under cobordisms of 3-manifolds with a path between the boundary components.
For any rational homology sphere $Y$, we may also define an analogue of Floer's irreducible instanton homology group of integer homology spheres $I_*(Y, π; R)$ which now depends on the auxiliary data $π$, unlike the equivariant instanton homology groups. However, our methods allow us to prove a precise "wall-crossing formula'' for $I_*(Y, π; R)$ as the auxiliary data $π$ moves between adjacent chambers. We use this to define an instanton invariant $λ_I(Y) \in \Bbb Q$ of rational homology spheres, conjecturally equal to the Casson-Walker invariant.
Our approach to invariance uses a novel technique known as a suspended flow category. Given an obstructed cobordism $W: Y \to Y'$, which supports reducible instantons which can neither be cut out transversely nor be removed by a small change to the perturbation, we remove and replace a neighborhood of obstructed solutions in the moduli space of instantons. The resulting moduli spaces have a new type of boundary component, so do not define a chain map between the instanton chain complexes of $Y$ and $Y'$. However, it does define a chain map between the instanton chain complex of $Y$ and a sort of suspension of the instanton chain complex of $Y'$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_14071 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Instantons and rational homology spheres Daemi, Aliakbar Eismeier, Mike Miller Geometric Topology 57M27, 57R58 In previous work, the second author defined 'equivariant instanton homology groups' $I^\bullet(Y,π;R)$ for a rational homology 3-sphere $Y$, a set of auxiliary data $π$, and a PID $R$. These objects are modules over the cohomology ring $H^{-*}(BSO_3;R)$. We prove that the equivariant instanton homology groups $I^\bullet(Y;R)$ are independent of the auxiliary data $π$, and thus define topological invariants of rational homology spheres. Further, we prove that these invariants are functorial under cobordisms of 3-manifolds with a path between the boundary components. For any rational homology sphere $Y$, we may also define an analogue of Floer's irreducible instanton homology group of integer homology spheres $I_*(Y, π; R)$ which now depends on the auxiliary data $π$, unlike the equivariant instanton homology groups. However, our methods allow us to prove a precise "wall-crossing formula'' for $I_*(Y, π; R)$ as the auxiliary data $π$ moves between adjacent chambers. We use this to define an instanton invariant $λ_I(Y) \in \Bbb Q$ of rational homology spheres, conjecturally equal to the Casson-Walker invariant. Our approach to invariance uses a novel technique known as a suspended flow category. Given an obstructed cobordism $W: Y \to Y'$, which supports reducible instantons which can neither be cut out transversely nor be removed by a small change to the perturbation, we remove and replace a neighborhood of obstructed solutions in the moduli space of instantons. The resulting moduli spaces have a new type of boundary component, so do not define a chain map between the instanton chain complexes of $Y$ and $Y'$. However, it does define a chain map between the instanton chain complex of $Y$ and a sort of suspension of the instanton chain complex of $Y'$. |
| title | Instantons and rational homology spheres |
| topic | Geometric Topology 57M27, 57R58 |
| url | https://arxiv.org/abs/2210.14071 |