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.