The exactness theorem for Floer homology
The author illustrates some aspects of the work of A. Floer on the surgery formula for instanton Floer homology, known as the exact triangle. A detailed overview of Floer's work can be found in [\textit{P. J. Braam} and \textit{S. K. Donaldson}, Floer's work on instanton homology, knots and surgery, in `The Floer Memorial Volume', Progr. Math. 133 (1995)]. In this paper the author concentrates on the part concerning the proof of the exactness of the sequence determined by the surgery cobordisms. A suitable perturbation of the Chern-Simons functional on a homology sphere \(Y\) provides a decomposition of the moduli space of irreducible critical points (perturbed flat connections) on \(Y\) as \(R_Y = R_{Y'} \cup R_{Y''}\), where \(Y'\) is the homology sphere obtained by 1-surgery on a knot \(K\) in \(Y\) and \(Y''\) is the 3-manifold, with the homology of \(S^1\times S^2\), obtained by 0-surgery on \(K\). This induces exact sequences of abelian groups relating the Floer complexes of \(Y\), \(Y'\), and \(Y''\). Using the surgery cobordisms, it is possible to define chain homomorphisms between the same Floer complexes. Considering the natural filtration of the Floer complexes, it is possible to relate these chain homomorphisms to the exact sequence of groups induced by the identification of the flat connections. The difference between the maps induced by the cobordism and the exact sequence of groups is given by maps that are strictly monotonic with respect to the filtration. This is the essential ingredient in the proof of the exactness of the cobordism sequence. The author then proves that the connecting homomorphism in the exact triangle is also a map induced geometrically by counting solutions on a cobordism: in fact, the homomorphism is induced by the three-dimensional moduli space of solutions of the anti-self-dual equation on the composite cobordism, connecting \(Y''\) to \(Y'\). The composite cobordism can be written as a connected sum \(W_3 \# CP^2\), where \(W_3\) is also a cobordism between \(Y''\) and \(Y'\). The three-dimensional moduli space on the composite cobordism corresponds to the zero-dimensional moduli space on \(W_3\), hence the connecting homomorphism can be thought of as counting solutions of the anti-self-dual equation on \(W_3\).
scientific article; zbMATH DE number 1195594
scientific article