Pin(2)‐monopole Floer homology, higher compositions and connected sums

Abstract

We study the behavior of Pin(2)-monopole Floer homology under connected sums. After constructing a (partially defined) A∞-module structure on the Pin(2)-monopole Floer chain complex of a three-manifold (in the spirit of Baldwin and Bloom's monopole category), we identify up to quasi-isomorphism the Floer chain complex of a connected sum with a version of the A ∞-tensor product of the modules of the summands. There is a naturally associated spectral sequence converging to the Floer groups of the connected sum whose E 2 page is the Tor of the Floer groups of the summands. We discuss in detail a simple example, and use this computation to show that the Pin(2)-monopole Floer homology of S 3 has non-trivial Massey products.

Connected Heegaard Floer homology of sums of Seifert fibrations

“…Further variants and applications of Manolescu's construction have been studied extensively by several authors; see e.g. [10], [11], [12], [18], [19], [1].…”

Connected sums and involutive knot Floer homology

“…The modules from are expected to correspond to the ‐equivariant theory for the subgroup . Lin proves a connected sum formula for ‐equivariant monopole Floer homology in terms of an tensor product . Our connected sum formula for involutive link Floer homology is different, since the involutive complex is determined entirely by the chain homotopy type of the map .…”

“…Here θ w i and ξ w i , respectively, denote the higher and lower gr w -graded intersection points of α i ∩ β i . Using the grading change formulas from equations (19), (20) and (21), one sees that both F S 1 ×D 3 ,M1,t and F D 2 ×S 2 ,M2,t0 induce gr w and gr z -grading changes of 0. For example, to compute the gr w -grading changes of the two maps, one computes that χ(S 1,w ) = 0 and χ(S 1 × D 3 ) = 0, while χ(S 2,w ) = 1 and χ(D 2 × S 2 ) = 2, and that the other terms in the grading change formula vanish.…”

“…The Alexander grading change of F W,F ,s is equal to zero, by the Alexander grading change formula from equation (19). The map F W,F ,s on HFL ∞ thus becomes simply the regular cobordism map F ∞ W,s in each Alexander grading.…”

PIN(2)-monopole Floer homology and the Rokhlin invariant

“…We claim that in this case we have Indeed, our main theorem implies that, up to grading shift, where is the unique torsion spin structure. The latter can be computed for example using the connected sum spectral sequence (see [Lin17]). Indeed, we know that and, as this is a free module over , the invariant of the connected sum is simply the tensor product over of the invariants (as the spectral sequence collapses at the page).…”