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A Morse complex for infinite dimensional manifolds Alberto Abbondandolo

Summary: A Morse complex for infinite dimensional manifolds
Part I
Alberto Abbondandolo
and Pietro Majer
October 23, 2004
In this paper and in the forthcoming Part II we introduce a Morse complex for a class of
functions f defined on an infinite dimensional Hilbert manifold M, possibly having critical
points of infinite Morse index and co-index. The idea is to consider an infinite dimensional
subbundle - or more generally an essential subbundle - of the tangent bundle of M, suitably
related with the gradient flow of f. This Part I deals with the following questions about
the intersection W of the unstable manifold of a critical point x and the stable manifold of
another critical point y: finite dimensionality of W, possibility that different components of
W have different dimension, orientability of W and coherence in the choice of an orientation,
compactness of the closure of W, classification, up to topological conjugacy, of the gradient
flow on the closure of W, in the case dim W = 2.
Morse theory [Mor25] relates the topology of a compact differentiable manifold M to the combi-
natorics of the critical points of a smooth Morse function f : M R: if q(M) = rank Hq(M)
denotes the q-th Betti number of M, and cq(f) is the number of critical points x of f with Morse


Source: Abbondandolo, Alberto - Scuola Normale Superiore of Pisa


Collections: Mathematics