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Abstract
Let
f
:
M
→
ℝ
be a Morse–Bott function on a finite-dimensional closed smooth manifold
M .
Choosing an appropriate Riemannian metric on
M and Morse-Smale
functions
f j : C j
→
ℝ on the
critical submanifolds
C j ,
one can construct a Morse chain complex whose boundary operator is defined by counting
cascades [Int. Math. Res. Not. 42 (2004) 2179–2269]. Similar data, which also includes a parameter
ε
> 0 that scales the
Morse-Smale functions
f j ,
can be used to define an explicit perturbation of the Morse-Bott function
f to a Morse-Smale
function
h ε :
M
→
ℝ
[Progr. Math. 133 (1995) 123–183; Ergodic Theory Dynam. Systems 29 (2009)
1693–1703]. In this paper we show that the Morse–Smale–Witten chain complex of
h ε is
the same as the Morse chain complex defined using cascades for any
ε
> 0
sufficiently small. That is, the two chain complexes have the same generators, and
their boundary operators are the same (up to a choice of sign). Thus, the Morse
Homology Theorem implies that the homology of the cascade chain complex of
f
:
M
→
ℝ is isomorphic to the
singular homology
H ∗ ( M ;
ℤ ) .
Keywords
Morse homology, Morse–Bott, critical submanifold, cascade,
exchange lemma
Mathematical Subject Classification 2010
Primary: 57R70
Secondary: 37D05, 37D15, 58E05
Publication
Received: 22 March 2012
Accepted: 30 August 2012
Published: 16 February 2013