Involutions, links, and Floer cohomologies (Q6564520)

The authors define invariants for closed spin\(^c\) three-manifolds \((Y,\mathfrak{s})\) with \(b_1=0\) equipped with a real structure, i.e. an involution \(\iota\) of \(Y\) (assumed to have non-empty fixed locus) lifting to an isomorphism \(\iota^*\mathfrak{s}\cong \bar{\mathfrak{s}}\) where \(\bar{\mathfrak{s}}\) denotes the conjugate spin\(^c\) structure. Under these circumstances, there is an involution induced on the space of Seiberg-Witten configurations, and the invariant is constructed by analyzing its fixed locus. \N\NThe framework of the paper involves Manolescu's Floer homotopy type for three-manifolds with \(b_1=0\) [\textit{C. Manolescu}, Geom. Topol. 7, 889--932 (2003; Zbl 1127.57303)], and the construction can be thought of as a homotopy refinement of Jiakai Li's recent real monopole Floer homology groups (which are defined with no \(b_1\) assumption, [\textit{J. Li}, ``Monopole Floer Homology and Real Structures'', Preprint, \url{arXiv:2211.10768}]). Let us remark that these constructions (which are conjectured to be equivalent, in the same spirit as [\textit{T. Lidman} and \textit{C. Manolescu}, The equivalence of two Seiberg-Witten Floer homologies. Paris: Société Mathématique de France (SMF) (2018; Zbl 1415.57002)]) lead to \(\mathbb{Z}/2\)-equivariant homology groups, where \(\mathbb{Z}/2\) corresponds to the constant gauge transformations \(\pm1\). \N\NThe authors provide several applications of these invariants to questions of current interest in topology such as link concordance (via branched double covers), non-orientable genus in the four-ball and Nielsen realization in four-dimensions.