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Floer Homology Groups in Yang-Mills Theory (Cambridge Tracts in Mathematics): S. K. Donaldson, M. Furuta, D. Kotschick

From the introduction:

In 1985 Andreas Floer discovered new topological invariants of certain 3-manifolds, the ‘Floer homology groups’. This book originated from a series of seminars on this subject held in Oxford in 1988, the manuscript for the book beingwritten sporadically over the intervening12 years. The original plan of the project has been modified over time, but the basic aims have remained largely the same: these are, first, to givea thorough exposition of Floer’s original work, and, second, to develop some further aspects of the theory which have not appeared in detail in the literature before. The author can only apologise for the long delay in completingthis project.

Floer’s original motivation for introducing his groups - beyond the intrinsic interest and beauty of the construction - seems to have been largely as a source of new invariants in 3-manifold theory, refining the Casson invariant which had been discovered shortly before. It was soon realised however that Floer’s conception fitted in perfectly with the ‘instanton invariants’ of 4-dimensional manifolds, which date from much the same period. Roughly speaking, the Floer groups are the data required to extend this theory from closed 4-manifolds to man- ifolds with boundary. From another point of view the Floer groups appear, formally, as the homology groups in the ‘middle dimension’ of an infinite-dimensional space (the space of connections modulo equiva- lence) associated to a 3-manifold. This picture is obtained by carrying certain aspects of the Morse theory description of the homology of a finite-dimensional manifold over to infinite dimensions. All of this is closely related to ideas from quantum field theory - indeed, one of Floer’s startingpoints was the renowned paper of Witten, [49], which inter alia forged a link between quantum mechanics and Morse theory - and the connection with mathematical physics permeates the whole subject.

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