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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.

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