Symplectic Instanton Homology: twisting, connected sums, and Dehn surgery

Abstract

We define a twisted version of Manolescu and Woodward's Symplectic Instanton homology, prove that this invariant fits into the framework of Wehrheim and Woodward's Floer Field theory, and describe its behaviour for connected sum and Dehn surgery.

Bordered theory for pillowcase homology

“…For a thorough introduction to the above Floer-theoretic invariants, connections between them, and their applications see [28], and references there. See also more recent papers [6], [14], [15] on symplectic instanton Floer homology, and [7] on Atiyah-Floer conjecture. Now we turn our attention to knot invariants.…”

Symplectic instanton homology: naturality, and maps from cobordisms

“…These satisfy the assumptions of definition 2.2, and the generalized Lagrangian correspondence L(W, P ), as a morphism in Symp, is independent from the decompositions of W , see [Caz16,Prop. 3.18, Th.…”

“…To keep notations short we will simply write (W, P ) from (Σ 0 , P 0 ) to (Σ 1 , P 1 ), or just W from Σ 0 to Σ 1 when P = W × SO(3). Notice that in [Caz16] we used a slightly different category (using cohomology classes instead of bundles).…”

“…This is not an elementary cobordism anymore, but a compression body. What matters is that after applying the FFT functor, the Lagrangian correspondences still compose in an embedded + way (see [Caz16,Th. 3.22]).…”

“…First, to provide a geometric interpretation of the maps appearing in the surgery exact sequence of [Caz16]: Theorem 1.3. (see Theorem 3.7 and Corollary 3.8 for precise statements) With suitable choices of principal bundles, the maps from cobordisms associated to 2-handle attachments provide a long exact sequence for a Dehn surgery triad.…”