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

Abstract

We compute the $\mathrm{Pin}(2)$-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 $\beta =-\bar{\mu }$ for Seifert integral homology three-spheres. We show that the Manolescu invariants $\alpha ,\beta ,$ and $\gamma$ give new obstructions to homology cobordisms between Seifert fiber spaces, and that many Seifert homology spheres $\Sigma (a_1,...,a_n)$ are not homology cobordant to any $-\Sigma (b_1,...,b_n)$. 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 $\mathrm{Pin}(2)$-equivariant Seiberg-Witten Floer spectrum provides homology cobordism obstructions distinct from $\alpha ,\beta ,$ and $\gamma$. In particular, we identify an $\mathbb{F}[U]$-module called connected Seiberg-Witten Floer homology, whose isomorphism class is a homology cobordism invariant.

49 pages, 11 figures. Corrected an error in Lemma 3.8, slightly changing the statement of Theorem 1.1

References in corpus (1)

• The surgery exact triangle in Pin(2)-monopole Floer homology

Cited by in corpus (8)

• Connected sums and involutive knot Floer homology
• A splitting theorem for the Seiberg-Witten invariant of a homology $S^1\times S^3$
• An infinite-rank summand of the homology cobordism group
• A survey of the homology cobordism group
• Lectures on the triangulation conjecture
• Irreducible 3-manifolds that cannot be obtained by 0-surgery on a knot
• Instantons, special cycles, and knot concordance
• Involutions, knots, and Floer K-theory