Floer homology for 2-torsion instanton invariants

Instanton Floer homology for lens spaces

“…Note that i ρσ ≡ i στ mod 2 since we assumed i ρτ ≡ 0 mod 2. This lemma can be proved in the same way as [19,Lemma 3.14]. We give outline of the proof.…”

“…As in the previous case, to extend this construction to lens spaces Y , we must change the definition of the boundary using twisted Dirac operators over Y × R. Furthermore, as we will see later, we need to look out the contribution of the trivial connection differently from [12], [4] and the previous subsection. The discussion which involves the trivial flat connection is similar to that in [19].…”

“…To define the matrix element, we use the determinant line bundle of twisted ∂-operators over γ × R. Take a spin structure of γ which represent the trivial spin bordism class. ( See [19,Remark 2.3] for the reason why we choose the spin structure. ) The spin structure induces a spin structure of γ × R (i.e.…”

“…The variant Ψ u 1 X is a function on a subspace A ′ (X) of A(X) with values in Z 2 . Variants of instanton Floer homology are defined for oriented homology 3-spheres in [13] and [19], and there is a similar gluing formula for Ψ u 1 X . A natural problem is how to define instanton Floer homology for more general 3-manifolds Y , which enable us to construct gluing formulas for more general decompositions of X.…”

“…The calculation of Floer homologies requires counting the number of instantons over L(p, 2) × R. This was done in [2], [15] and [14]. The non-vanishing of Ψ u 1 X was proved in [19].…”