Topology in Molecular Biology

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.