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A remark on \(\operatorname{Pin}(2)\)-equivariant Floer homology. (English) Zbl 1394.57031
Gauge theoretic techniques have given many numerical homology cobordism invariants. In this paper, Stoffregen studies six Frøyshov-type invariants, \(\alpha\), \(\beta\), \(\gamma\), \(\underline{\delta}\), \(\overline{\delta}\) and \(\delta\), defined using variations of Manolescu’s \(\text{Pin}(2)\)-equivariant Seiberg-Witten Floer homology [C. Manolescu, J. Am. Math. Soc. 29, No. 1, 147–176 (2016; Zbl 1343.57015)]. The invariants \(\alpha\), \(\beta\) and \(\gamma\) are correction terms defined from the \(\text{Pin}(2)\)-equivariant version, and \(\delta\) is defined from the \(S^1\)-equivariant version. The invariants \(\underline{\delta}\) and \(\overline{\delta}\) are defined using the \(\mathbb{Z}_4\)-equivariant version, and are expected to coincide up to a factor of 2 with Manolescu and Hendricks’ involutive correction terms \(\underline{d}\) and \(\overline{d}\) from Heegaard Floer homology [K. Hendricks and C. Manolescu, Duke Math. J. 166, No. 7, 1211–1299 (2017; Zbl 1383.57036)].
In this article, Stoffregen shows that \(\underline{\delta}\), \(\overline{\delta}\) and \(\delta\) cannot in general be recovered from the \(\text{Pin}(2)\)-equivariant version of Seiberg-Witten Floer homology, though \(\delta\) can be recovered from the \(\mathbb{Z}/4\)-equivariant version. Topologically, \(\text{Pin}(2)\) is a disjoint union of two copies of \(S^1\), and there are many other subgroups of \(\text{Pin}(2)\) other than \(\text{Pin}(2)\), \(S^1\) and \(\mathbb{Z}_4\). One of Stoffregen’s main theorems is that if \(H\subset \text{Pin}(2)\) is a Lie subgroup, then the Frøyshov-type invariant defined using \(H\)-equivariant Seiberg-Witten Floer homology must be one of \(\alpha,\) \(\beta,\) \(\gamma,\) \(\underline{\delta},\) \(\overline{\delta},\) or \(\delta\). As an example application of Stoffregen’s techniques, he shows that if \(m\) is even and \(Q_{4m}\) denotes the generalized quaternion group generated by \(j\) and \(e^{\pi i/m}\), then the \(\text{Pin}(2)\)-equivariant version of Seiberg-Witten Floer homology can be recovered from the \(Q_{4m}\)-equivariant version.

57R58 Floer homology
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
55N91 Equivariant homology and cohomology in algebraic topology
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