Topology in Molecular Biology

168 V.M. Buchstaber

Theboundaryof an arbitraryk-chainck=

∑

igiσiis then given by

∂ck:=

∑

i

gi∂σi.

Again, we have∂∂ck=0.
Thek-cyclesofKare thosek-chainscksatisfying∂ck= 0. They form a
subgroup denotedZk.
Theboundaryk-cycles are those that are “homologous to zero”, i.e. are of
the form∂ck+1for some (k+ 1)-chainck+1. The subgroup they comprise is
denoted byBk.
We say that two chainsc′kandc′′karehomologousif

c′k=c′′k+∂ck+1

for some (k+ 1)-chainck+1ofK.Thek-dimhomologygroupHk(K;G)isthe
quotient ofZk/Bk.
The fundamental group is not commutative in general, and its abelianisa-
tion is the first homology group:

H 1 (K;Z)∼=π 1 (K)

/[

π 1 (K),π 1 (K)

]

,

where [π 1 (K),π 1 (K)] is thecommutator subgroupof the fundamental group.
WhenG=Ris the group of real numbers, the groupHk(K;G)isareal
vector space. By definition, thekthBetti numberbkofMis dimHk(K;R).
A theorem of Poincar ́e states that

χ(K)=

∑

i≥ 0

(−1)ifi=

∑

i≥ 0

(−1)ibi.

IfMnis a closed and connected manifold admitting a finite triangulation,
then
Hn(Mn;Z 2 )=Z 2 ,

whereZ 2 is the 2-element group of residues modulo 2. The generator of
Hn(Mn;Z 2 ) is given by the homology class of the chain

∑

iσ

n
i, where the sum
is taken over all simplices ofK. This generator is called theZ 2 -fundamental
classofMn.
A closed connected triangulated manifoldMnis calledorientableif there
is a choice of signsεi=±1 such that then-chain

∑

iεiσiis a cycle. This
choice of signs is called anorientationofMn, and the homology class of the
cycle

∑

iεiσiinHn(M

n;Z) is called thefundamental classof the oriented

manifoldMn. It generates the groupHn(Mn;Z). The property of orientability
does not depend on a choice of triangulation, and there are two different
orientations in total. The corresponding fundamental classes differ by a sign.
For every closed connected orientable triangulated manifoldMnwe have
Hn(Mn;G)=Gfor allG.IfMnis non-orientable then