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Seifert-fibred homology 3-spheres, V-surfaces and the Floer index

Published online by Cambridge University Press:  24 October 2008

Emile Ben Nasatyr
Emile Ben Nasatyr
Affiliation:
Mathematical Institute, 24–29 St Giles', Oxford OX1 3LB
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Extract

Let Σ be a Seifert-fibred homology 3-sphere. We are interested in the chain complex for the Floer homology of Σ. This is generated by the critical points of the ChernSimons functional acting on the moduli space of irreducible SU(2)-connections modulo gauge-equivalence, i.e. the equivalence classes of flat connections: see [6]. Specifically, we ask the question: given the holonomy ρ of a flat connection Cρ, what is the index of Cρ in the chain complex? By definition this Floer index is given by the spectral flow of a family of twisted signature operators with coefficients in the adjoint bundle (with fibre su(2)) corresponding to any path of connections joining the trivial connection to Cρ. We will calculate this spectral flow by an almost entirely direct method, obtaining a formula in terms of the dimensions of spaces of π1(Σ)- automorphic functions. These dimensions will be evaluated to give the numerical result previously obtained using a different method by Fintushel and Stern: see [5].

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 111 , Issue 3 , May 1992 , pp. 461 - 485
Copyright
Copyright © Cambridge Philosophical Society 1992

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