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Algebraic Topology
· Problem 2.2.41
Homology — Allen Hatcher
Mathematics — Topology · euler-characteristic
Problem
For
X
X
X
a finite CW complex and
F
F
F
a field, show that the Euler characteristic
χ
(
X
)
\chi(X)
χ
(
X
)
can also be computed by the formula
χ
(
X
)
=
∑
n
(
−
1
)
n
dim
H
n
(
X
;
F
)
\chi(X) = \sum_n (-1)^n \dim H_n(X; F)
χ
(
X
)
=
∑
n
(
−
1
)
n
dim
H
n
(
X
;
F
)
, the alternating sum of the dimensions of the vector spaces
H
n
(
X
;
F
)
H_n(X; F)
H
n
(
X
;
F
)
.
Related theory
Euler S Totient Function And Euler S Theorem
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