Topology in Molecular Biology

170 V.M. Buchstaber

g=1 g=2

g=3

Fig. 9.4.Sphere withghandles

g=1

a a

b

b

g=2

a a

a a

b

b

b

b

1

1

1

1

2

2

2

2

Fig. 9.5.Mg^2 :4gedges

An orientable closed surfaceMg^2 (a sphere withghandles) can be smoothly
embedded inR^3 as the boundary of a 3-dim body.
A model of the 3-body with boundaryMg^2 can be obtained by taking the
small closed smooth neighbourhood of the wedge ofgcircles inR^3.
Any non-orientable closed surface can be obtained as follows. Take a sphere
S^2 , removeμdisjoint open disksD^2 , and identify the diametrically opposite
points on the boundary of each hole. This is equivalent to filling all theμ
holes by M ̈obius bands (crosscaps). Denote the resulting surfaces byMμ^2 ,
μ=1, 2 ,....
The diffeomorphism classes of connected closed manifoldsM^2 form acom-
mutative semigroupwith respect to the connected sum operation #. This
semigroup has two generatorsa(the torusT^2 )andb(the projective plane
RP^2 ) with a single defining relation

a#b=b#b#b.

Every connected closed smooth manifoldM^2 admits a finite triangulation,
i.e. can be subdivided by means of smooth curves into finitely many smooth
triangles in such a way that any two triangles either do not intersect, have a
single common vertex (0-face), or share a single common edge (1-dim face).