8 2. ALGEBRAIC DISCRETE MORSE THEORY
For an acyclic matching M on the graph G(C•) = (V, E) we introduce the
following notation, which is an adaptation of the notation introduced in [17] to our
situation.
(1) We call a vertex c V critical with respect to M if c does not lie in any
edge e M; we write
Xi
M
:= {c Xi | c critical }
for the set of all critical vertices of homological degree i.
(2) We write c c if c Xi, c Xi−1, and [c : c ] = 0.
(3) Path(c, c ) is the set of paths from c to c in the graph GM(C•).
(4) The weight w(p) of a path p = c1 ··· cr Path(c1, cr) is given by
w(c1 . . . cr) :=
r−1
i=1
w(ci ci+1),
w(c c ) :=






1
[c : c ]
, c c ,
[c : c ] , c c.
(5) We write Γ(c, c ) =
p∈Path(c,c )
w(p) for the sum of weights of all paths
from c to c .
Now we are in position to define a new complex C•
M,
which we call the Morse
complex of C• with respect to M. The complex C•
M
= (Ci
M,
∂i
M)i≥0
is defined by
Ci
M
:=
c∈XiM
R c,
∂i
M
:



Ci
M

Ci−1M
c
c ∈Xi−1M
Γ(c, c )c , .
Theorem 2.2. C•
M
is a complex of free R-modules and is homotopy equivalent
to the complex C•; in particular, for all i 0
Hi(C•)

= Hi(C•
M).
The maps defined below are chain homotopies between C• and C•
M:
f :



C•
C•M
c Xi f(c) :=
c ∈XiM
Γ(c, c )c ,
g :



C• M C•
c Xi M gi(c) :=
c ∈Xi
Γ(c, c )c .
The proof of Theorem 2.2 is given in Appendix B. Note that if C• is the
cellular chain complex of a regular CW-complex and X is the set of cells of the
regular CW-complex, then algebraic discrete Morse theory is the part of Forman’s
[17] discrete Morse theory which describes the impact of a discrete Morse matching
on the cellular chain complex of the CW-complex.
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