Floer homology groups for homology three-spheres

Abstract

Recently Floer [Fl] has defined eight new homology groups for oriented 3-dimensional homology spheres. The definition of these groups makes heavy use of the theory of gauge fields on threeand four-manifolds, as it has been developed during the last decade by Uhlenbeck, Taubes, Donaldson, and others. The purpose of this note is to give an introductory, rather detailed description of these interesting groups. We also discuss some further structure on these groups, the computations of Floer homology for the Brieskorn homology 3-spheres by Fintushel and Stern, and the relation with Donaldson polynomials for 4-manifolds. Here and there some further conjectures have been inserted. Apart from a few remarks in 94, nothing in these notes is due to the author, and the credit for this work should go to Floer, the designers of gauge theory, mentioned above, and Fintushel and Stern. Very interestingly, Witten [W2] has recently given a quantum field theory approach to Floer homology and Donaldson polynomials; this we shall not review here. Also we shall not enter into a discussion on how to relate Floer homology to two dimensional gauge theory through a Heegaard splitting of the homology sphere. This last point, as well as much other beautiful scenery around Floer’s theory, can be found in Atiyah [At].