Corks, involutions, and Heegaard Floer homology
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arXiv
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| Format: | Preprint |
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2020
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| author | Dai, Irving Hedden, Matthew Mallick, Abhishek |
| author_facet | Dai, Irving Hedden, Matthew Mallick, Abhishek |
| contents | Building on the algebraic framework developed by Hendricks, Manolescu, and Zemke, we introduce and study a set of Floer-theoretic invariants aimed at detecting corks. Our invariants obstruct the extension of a given involution over any homology ball, rather than a particular contractible manifold. Unlike previous approaches, we do not utilize any closed 4-manifold topology or contact topology. Instead, we adapt the formalism of local equivalence coming from involutive Heegaard Floer homology. As an application, we define a modification $Θ^τ_{\mathbb{Z}}$ of the homology cobordism group which takes into account an involution on each homology sphere, and prove that this admits a $\mathbb{Z}^\infty$-subgroup of strongly non-extendable corks. The group $Θ^τ_{\mathbb{Z}}$ can also be viewed as a refinement of the bordism group of diffeomorphisms. Using our invariants, we furthermore establish several new families of corks and prove that various known examples are strongly non-extendable. Our main computational tool is a monotonicity theorem which constrains the behavior of our invariants under equivariant negative-definite cobordisms, and an explicit method of constructing such cobordisms via equivariant surgery. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2002_02326 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Corks, involutions, and Heegaard Floer homology Dai, Irving Hedden, Matthew Mallick, Abhishek Geometric Topology 57M27, 57R58 Building on the algebraic framework developed by Hendricks, Manolescu, and Zemke, we introduce and study a set of Floer-theoretic invariants aimed at detecting corks. Our invariants obstruct the extension of a given involution over any homology ball, rather than a particular contractible manifold. Unlike previous approaches, we do not utilize any closed 4-manifold topology or contact topology. Instead, we adapt the formalism of local equivalence coming from involutive Heegaard Floer homology. As an application, we define a modification $Θ^τ_{\mathbb{Z}}$ of the homology cobordism group which takes into account an involution on each homology sphere, and prove that this admits a $\mathbb{Z}^\infty$-subgroup of strongly non-extendable corks. The group $Θ^τ_{\mathbb{Z}}$ can also be viewed as a refinement of the bordism group of diffeomorphisms. Using our invariants, we furthermore establish several new families of corks and prove that various known examples are strongly non-extendable. Our main computational tool is a monotonicity theorem which constrains the behavior of our invariants under equivariant negative-definite cobordisms, and an explicit method of constructing such cobordisms via equivariant surgery. |
| title | Corks, involutions, and Heegaard Floer homology |
| topic | Geometric Topology 57M27, 57R58 |
| url | https://arxiv.org/abs/2002.02326 |