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Instanton homology and symplectic fixed points
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- By Stamatis Dostoglou, Mathematics Institute, University of Warwick, Coventry CV4 7AL, Great Britain, Dietmar Salamon, Mathematics Institute, University of Warwick, Coventry CV4 7AL, Great Britain
- Edited by Dietmar Salamon, University of Warwick
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- Book:
- Symplectic Geometry
- Published online:
- 16 October 2009
- Print publication:
- 03 February 1994, pp 57-94
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- Chapter
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Summary
Introduction
A gradient flow of a Morse function on a compact Riemannian manifold is said to be of Morse-Smale type if the stable and unstable manifolds of any two critical points intersect transversally. For such a Morse-Smale gradient flow there is a chain complex generated by the critical points and graded by the Morse index. The boundary operator has as its (x, y)-entry the number of gradient flow lines running from x to y counted with appropriate signs whenever the difference of the Morse indices is 1. The homology of this chain complex agrees with the homology of the underlying manifold M and this can be used to prove the Morse inequalities.
Around 1986 Floer generalized this idea and discovered a powerful new approach to infinite dimensional Morse theory now called Floer homology. He used this approach to prove the Arnold conjecture for monotone symplectic manifolds and discovered a new invariant for homology 3-spheres called instanton homology. This invariant can roughly be described as the homology of a chain complex generated by the irreducible representations of the fundamental group of the homology 3-sphere M in the Lie group SU(2). These representations can be thought of as flat connections on the principal bundle M × SU(2) and they appear as the critical points of the Chern-Simons functional on the infinite dimensional configuration space of connections on this bundle modulo gauge equivalence.
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