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Given an involution on a rational homology 3-sphere Y with quotient the 3-sphere, we prove a formula for the Lefschetz num- ber of the map induced by this involution in the reduced mono- pole Floer homology. This formula is motivated by a variant of Witten’s conjecture relating the Donaldson and Seiberg–Witten invariants of 4-manifolds. A key ingredient is a skein-theoretic ar- gument, making use of an exact triangle in monopole Floer homol- ogy, that computes the Lefschetz number in terms of the Murasugi signature of the branch set and the sum of Frøyshov invariants as- sociated to spin structures on Y . We discuss various applications of our formula in gauge theory, knot theory, contact geometry, and 4-dimensional topology.more » « less
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In studying the “11/8-Conjecture” on the Geography Problem in 4-dimensional topology, Furuta proposed a question on the existence of Pin ( 2 ) \operatorname {Pin}(2) -equivariant stable maps between certain representation spheres. A precise answer of Furuta’s problem was later conjectured by Jones. In this paper, we completely resolve Jones conjecture by analyzing the Pin ( 2 ) \operatorname {Pin}(2) -equivariant Mahowald invariants. As a geometric application of our result, we prove a “10/8+4”-Theorem. We prove our theorem by analyzing maps between certain finite spectra arising from B Pin ( 2 ) B\operatorname {Pin}(2) and various Thom spectra associated with it. To analyze these maps, we use the technique of cell diagrams, known results on the stable homotopy groups of spheres, and the j j -based Atiyah–Hirzebruch spectral sequence.more » « less
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