Abstract
Atiyah, Patodi and Singer [3] observed that the Fredholm index of the operator
with invertible limits \( {A^ \pm }\, = \,\mathop {\lim }\limits_{t \to \pm \infty } \,A\left( t \right) \) is given by the spectral flow of the self-adjoint operator family A(t) (the number of eigenvalues crossing 0 counted with signs). Such operators appear in infinite dimensional analogues of Morse theory as the linearisation of the gradient flow equation. The Fredholm index is the dimension of the space of gradient flow lines connecting two critical points. It can be thought of as the relative Morse index in cases where the absolute Morse index (the number of negative eigenvalues of the Hessian) is infinite.
This research has been partially supported by the SERC.
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References
M.F. Atiyah, New invariants of three and four dimensional manifolds, in: The Mathematical Heritage of Hermann Weyl, Proc. Sympos. Pure Math. 48, 1988, 295–299.
M.F. Atiyah and R. Bott, The Yang-Mills equations over Riemann surfaces, Phil Trans. R. Soc. Lond. A 308 (1982), 523–615.
M.F. Atiyah, V.K. Patodi, and I.M. Singer, Spectral asymmetry and Riemannian geometry III, Math. Proc. Camb. Phil. Soc. 79 (1976), 71–99.
S.E. Cappell, R. Lee, and E.Y. Miller, Self-adjoint elliptic operators and manifold decomposition I: General techniques and applications to Casson’s invariant, preprint, 1990.
C.C. Conley and E. Zehnder, Morse-type index theory for flows and periodic solutions of Hamiltonian equations, Commun. Pure Appl. Math. 37 (1984), 207–253.
G. D. Daskalopoulos and K.K. Uhlenbeck, An Application of Transversality to the Topology of the Moduli Space of Stable Bundles, preprint,1990.
S. Donaldson, M. Furuta, and D. Kotschick, Floer Homology Groups in Yang-Mills Theory, in preparation.
S. Dostoglou and D.A. Salamon, Instanton Homology and Symplectic Fixed Points, in: Symplectic Geometry, D. Salamon (ed.), LMS Lecture Note Series 192 (1993), Cambridge University Press, pp. 57–93.
S. Dostoglou and D.A. Salamon, Self-Dual instantons and holomorphic curves, Annals of Mathematics, to appear.
A. Floer, An instanton invariant for 3-manifolds, Commun. Math. Phys. 118 (1988), 215–240.
A. Floer, Symplectic fixed points and holomorphic spheres, Commun. Math. Phys. 120 (1989), 575–611.
A. Floer, Instanton Homology and Dehn Surgery, preprint, 1991.
A. Floer and H. Hofer, Coherent orientations for periodic orbit problems in symplectic geometry, Mathematische Zeitschrift, to appear.
M. Gromov, Pseudoholomorphic curves in symplectic manifolds, Invent. Math. 82 (1985), 307–347.
H. Hofer and D.A. Salamon, Floer Homology and Novikov Rings, to appear in: Gauge Theory, Symplectic Geometry and Topology; Essays in Memory of Andreas Floer, H. Hofer, C. Taubes, and E. Zehnder (eds.)
J. Jones, J. Rawnsley, and D. Salamon, Instanton Homology, preprint 1991.
T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, 1976.
D. McDuff, Elliptic methods in symplectic geometry. Bull. AMS 23 (1990), 311–358.
P.E. Newstead, Topological properties of some spaces of stable bundles, Topology 6 (1967), 241–262.
J.W. Robbin and D. Salamon, The Maslov index for paths. Topology 32 (1993), 827–844.
J.W. Robbin and D. Salamon, The spectral flow and the Maslov index, Bull. LMS, to appear.
D. Salamon, Morse theory, the Conley index and Floer homology, Bull LMS 22 (1990). 113–140.
D. Salamon and E. Zehnder. Morse theory for periodic orbits of Hamiltonian systems and the Maslov index, Comm. Pure Appl. Math. 45 (1992), 1303–1360.
C.H. Taubes, Casson’s invariant and Gauge theory. J. Diff. Geom. 31 (1990), 547–599.
C. Viterbo, Intersections de sous-variétés Lagrangiennes. fonctionelles d’action et indice des systèmes Hamiltoniens, Bull Soc. Math. France 115 (1987), 361–390.
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© 1994 Birkhäuser Verlag
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Dostoglou, S., Salamon, D.A. (1994). Cauchy-Riemann Operators, Self—Duality, and the Spectral Flow. In: Joseph, A., Mignot, F., Murat, F., Prum, B., Rentschler, R. (eds) First European Congress of Mathematics . Progress in Mathematics, vol 3. Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-9110-3_17
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