Floer homology and splittings of manifolds (Q1185447)

Floer homology \(I_ *(M)\) for an oriented integral homology 3-sphere is a \(\mathbb{Z}/8\)-graded abelian group coming from Morse theory of the Chern- Simons functional on the space of irreducible \(SU(2)\)-connections modulo gauge transformations. The Floer chain complex is generated by (the gauge equivalence classes) of smooth irreducible flat connections \(A\) such that the associated Fredholm operator \(D_ A\) has trivial kernel. Given to such connections \(A_ 0\) and \(A_ 1\) and a smooth path \(\{A_ t\}\) of connections from \(A_ 0\) to \(A_ 1\), the difference of the degrees \(d(A_ 1)-d(A_ 0)\) is the spectral flow of the family of operators \(\{D_{A_ t}\}\) modulo 8. The paper gives a practical method of calculation of the spectral flow above if the oriented closed 3-manifold \(M\) splits as \(M_ 1\cup M_ 2\) where \(\partial M_ 1=\partial M_ 2=M_ 1\cap M_ 2=\Sigma\) is a surface of genus \(g\geq 2\). In the situation above the main result of the article says: Theorem: There is Riemannian metric on \(M\) and a smooth generic path \(\{A_ t\}\) from \(A_ 0\) to \(A_ 1\) such that \(A_ t\) restricts to a product \(B_ 1\times 1\) in a neighborhood of \(\Sigma\) for an irreducible flat connection \(B_ t\) on \(\Sigma\) and the trivial connection 1 in the normal direction of \(\Sigma\) and there is an invariant \(\gamma(A_ t)\) depending only on the homotopy type of the path derived from \(\{A_ t\}\) in the space of all Lagrangian pairs in a \(6g-6\)-dimensional symplectic vector space such that \(SF(M,\{A_ t\})=\gamma(\{A_ t\})\) holds for the spectral flow \(SF(M,\{A_ t\})\) of \(\{D_{A_ t}\}\). As an application the Floer homology of the homology 3-sphere obtained by \((1/k)\)-Dehn surgery on the figure eight knot is computed.