9 Euler, Dehn–Sommerville Characteristics, and Their Applications 171
g=1
g=0
g=2
g=1
a a
a a
a a
a
a
b
b
b
b
b
b
b
b
c
c
c c
1
1
1
1
2
2
2
2
N
N
2
2
g,1
g,2
Fig. 9.6.Ng,^21 andNg,^22
g=1
g=2
g=0
Fig. 9.7.3-Dim body
The first homology of 2-dim surfaces is given by
H 1 (Mg^2 ;Z)=Z︸⊕···⊕︷︷ Z︸
2 g
;
H 1 (Mμ^2 ;Z)=Z︸⊕···⊕︷︷ Z︸
μ− 1
⊕Z 2.
It follows that
χ(Mg^2 )=2− 2 g, χ(Mμ^2 )=2−μ.
Any closed non-orientable surfaceMμ^2 can be obtained from the orientable
surfaceMg^2 withg=μ−1 by taking the orbit space of a certain involution.