7Euler characteristic
III Algebraic Topology
7 Euler characteristic
There are many ways to define the Euler characteristic, and they are all equivalent.
So to define it, we pick a definition that makes it obvious it is a number.
Definition (Euler characteristic). Let X be a cell complex. We let
χ(X) =
X
n
(−1)
n
number of n-cells of X ∈ Z.
From this definition, it is not clear that this is a property of
X
itself, rather
than something about its cell decomposition.
We similarly define
χ
Z
(X) =
X
n
(−1)
n
rank H
n
(X; Z).
For any field F, we define
χ
F
(X) =
X
n
(−1)
n
dim
F
H
n
(X; F).
Theorem. We have
χ = χ
Z
= χ
F
.
Proof.
First note that the number of
n
cells of
X
is the rank of
C
cell
n
(
X
), which
we will just write as C
n
. Let
Z
n
= ker(d
n
: C
n
→ C
n−1
)
B
n
= im(d
n+1
: C
n+1
→ C
n
).
We are now going to write down two short exact sequences. By definition of
homology, we have
0 B
n
Z
n
H
n
(X; Z) 0 .
Also, the definition of Z
n
and B
n
give us
0 Z
n
C
n
B
n−1
0 .
We will now use the first isomorphism theorem to know that the rank of the
middle term is the sum of ranks of the outer terms. So we have
χ
Z
(X) =
X
(−1)
n
rank H
n
(X) =
X
(−1)
n
(rank Z
n
− rank B
n
).
We also have
rank B
n
= rank C
n+1
− rank Z
n+1
.
So we have
χ
Z
(X) =
X
n
(−1)
n
(rank Z
n
− rank C
n+1
+ rank Z
n+1
)
=
X
n
(−1)
n+1
rank C
n+1
= χ(X).
For χ
F
, we use the fact that
rank C
n
= dim
F
C
n
⊗ F.