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Pin(2)-equivariant Seiberg–Witten Floer homology of Seifert fibrations

Part of: Low-dimensional topology

Published online by Cambridge University Press:  09 December 2019

Matthew Stoffregen
Matthew Stoffregen*
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA02142, USA email mstoff@mit.edu
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Abstract

We compute the $\text{Pin}(2)$-equivariant Seiberg–Witten Floer homology of Seifert rational homology three-spheres in terms of their Heegaard Floer homology. As a result of this computation, we prove Manolescu’s conjecture that $\unicode[STIX]{x1D6FD}=-\bar{\unicode[STIX]{x1D707}}$ for Seifert integral homology three-spheres. We show that the Manolescu invariants $\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD},$ and $\unicode[STIX]{x1D6FE}$ give new obstructions to homology cobordisms between Seifert fiber spaces, and that many Seifert homology spheres $\unicode[STIX]{x1D6F4}(a_{1},\ldots ,a_{n})$ are not homology cobordant to any $-\unicode[STIX]{x1D6F4}(b_{1},\ldots ,b_{n})$. We then use the same invariants to give an example of an integral homology sphere not homology cobordant to any Seifert fiber space. We also show that the $\text{Pin}(2)$-equivariant Seiberg–Witten Floer spectrum provides homology cobordism obstructions distinct from $\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD},$ and $\unicode[STIX]{x1D6FE}$. In particular, we identify an $\mathbb{F}[U]$-module called connected Seiberg–Witten Floer homology, whose isomorphism class is a homology cobordism invariant.

Keywords

monopole Floer homologyhomology cobordismSeifert spaces

MSC classification

Primary: 57M27: Invariants of knots and 3-manifolds
Type
Research Article
Information
Compositio Mathematica , Volume 156 , Issue 2 , February 2020 , pp. 199 - 250
Copyright
© The Author 2019

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Footnotes

MS was supported by NSF Grant DMS-1702532.

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