Let
be a Morse–Bott function on a finite-dimensional closed smooth manifold
.
Choosing an appropriate Riemannian metric on
and Morse-Smale
functions
on the
critical submanifolds
,
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
that scales the
Morse-Smale functions
,
can be used to define an explicit perturbation of the Morse-Bott function
to a Morse-Smale
function
[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
is
the same as the Morse chain complex defined using cascades for any
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
is isomorphic to the
singular homology
.
