Topology in Molecular Biology

9 Euler, Dehn–Sommerville Characteristics, and Their Applications 169

Hn(Mn;Z)=0,Hn(Mn;Z 2 )=Z 2.

A simplicial mapf:K 1 →K 2 induces group homomorphisms

f∗:Hk(K 1 ;G)→Hk(K 2 ;G),k=0, 1 ,....

In particular, for a simplicial mapf:M 1 n→M 2 nbetween two oriented man-
ifolds we havef∗[M 1 n]=m[M 2 n], wheremis called thedegreeoff(the same
is true for non-orientable manifolds if we work withZ 2 -homology).
Consider the barycentric subdivisionM′of a manifoldM=Mnand define

wk=

∑

i

ωki,

where the sum is taken over allk-dim simplicesωki ofM′. It is remarkable
that for anykthe chainwkis a cycle with coefficientsZ 2. The homology class
[wk] is a homotopy invariant ofMn, called itskthhomology Stiefel–Whitney
class. In particular, [wn]=[Mn]andw 0 =χ(Mn)mod2.

9.5 Classification of 2-Manifolds

Any closed orientable surface is homeomorphic to a sphere withghandles.
The integergis called thegenus. A model of a closed orientable surface of
genusgis given by a non-singularhyperelliptic curve:

V={(x, y)∈C^2 :y^2 =F(x)},

where
F(x)=4x^2 g+1+λ 2 gx^2 g+···+λ 1 x+λ 0

is a polynomial with all distinct roots.
A closed orientable surfaceMg^2 of genusgcan be obtained by a suitable
identification of edges in a 4g-gon.
The fundamental groupπ 1 (Mg^2 )has2ggeneratorsa 1 ,...,ag,b 1 ,...,bg
with a single defining relation

a 1 b 1 a 1 −^1 b− 11 ···agbga−g^1 b−g^1 =1,

coming from the identification of edges.
A closed non-orientable surface can be obtained by a suitable identification
of edges in a 4g-or(4g+ 2)-gon. Therefore, there are two families of non-
orientable closed surfaces,Ng,^21 andNg,^22.
The corresponding relations in the fundamental group are:

Ng,^21 :

(∏g− 1

i=1aibia

− 1

i b

− 1

i

)

agbga−g^1 bg=1,

Ng,^22 :

(∏g

i=1aibia

− 1

i b

− 1

i

)

c^2 =1.