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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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References

REFERENCES

[1]Atiyah, M. F. and Bott, R.. A Lefschetz fixed-point formula for elliptic complexes, II: Applications. Ann. of Math. 88 (1968), 451–491.CrossRefGoogle Scholar
[2]Atiyah, M. F. and Bott, R.. The Yang-Mills equations over Riemann surfaces. Philos. Trans. Roy. Soc. London Ser. A308 (1982), 523–615.Google Scholar
[3]Atiyah, M. F., Patodi, V. K. and Singer, I. M.. Spectral asymmetry and Riemannian geometry: III. Math. Proc. Cambridge. Philos. Soc. 79 (1976), 71–99.Google Scholar
[4]Braam, P. J.. Floer homology groups for homology three spheres. University of Utrecht mathematics preprint 484 (1987).Google Scholar
[5]Fintushel, R. and Stern, R. J.. Instanton homology of Seifert fibred homology three-spheres. Proc. London Math. Soc. (3) 61 (1990), 109–137.Google Scholar
[6]Floer, A.. An instanton-invariant for 3-manifolds. Comm. Math. Phys. 118 (1988), 215–240.Google Scholar
[7]Fox, R. H.. On Fenchel's conjecture about F-groups. Math. Tiddskrift B (1952), 61–65.Google Scholar
[8]Furuta, M. and Steer, B.. Seifert fibred homology 3-spheres and the Yang-Mills equations on Riemann surfaces with marked points. Adv. in Math., in the press.Google Scholar
[9]Kawasaki, T.. The Riemann–Roch theorem for complex V-manifolds. Osaka J. Math. 16 (1979), 151–159.Google Scholar
[10]Milnor, J.. On the 3-dimensional Brieskorn manifolds M(p, q, r). In Knots, Groups and 3-Manifolds – Papers Dedicated to the Memory of R.H. Fox. Ann. of Math. Stud, no. 84, ed. Neuwirth, L. P. (Princeton University Press, 1975), pp. 175–225.Google Scholar
[11]Nasatyr, K. B.. On Floer homology and the Brieskorn homology 3-spheres. Dissertation, Oxford University (1988).Google Scholar
[12]Orlik, P.. Seifert Manifolds. Lecture Notes in Math. vol. 291 (Springer-Verlag, 1972).Google Scholar
[13]Parthasarathy, R.. Dirac operators and the discrete series. Ann. of Math. 96 (1972), 1–30.Google Scholar
[14]Pukánszky, L.. The Plancherel formula for the universal covering group of Sl 2 ℝ. Math. Ann. 156 (1964), 96–143.Google Scholar
[15]Raymond, F. and Vasquez, A. T.. 3-Manifolds whose universal coverings are Lie groups. Topology Appl. 12 (1981), 161–179.Google Scholar
[16]Sally, P. J. Jr. Analytic Continuation of the Irreducible Unitary Representations of the universal covering group of Sl2ℝ It Mem. Amer. Math. Soc. no. 69 (American Mathematical Society, 1967).Google Scholar
[17]Schmid, W.. Representations of semi-simple Lie groups. In Representation Theory of Lie Groups, London. Math. Soc. Lecture Note Ser. no. 34, ed. Atiyah, M. F. (Cambridge, 1979), pp. 185–235.Google Scholar
[18]Seade, J. and Steer, B.. A note on the eta function for quotients of PSL 2(ℝ) by co-compact Fuchsian groups. Topology 26 (1987), 79–91.Google Scholar