Topology in Molecular Biology

172 V.M. Buchstaber

Fig. 9.8.Mμ^2 =1:realprojectiveplaneRP^2

Fig. 9.9.Mμ^2 =2: Klein bottleK^2

M1 # M2

Fig. 9.10.Connected sum

9.6 Minimal and Neighbourly Triangulations

A triangulationT∗of a manifoldMnis calledminimaliff 0 (T∗)f 0 (T)for
any triangulationT ofMn, wheref 0 denotes the number of vertices.
Now assume thatM^2 is a 2-dim triangulated manifold withmvertices.
Letχ=χ(M^2 ) be its Euler characteristic. Then using the Dehn–Sommerville
equation we obtain

f(M^2 )=

(

m,3(m−χ),2(m−χ)

)

.

For any simplicial complexKonmvertices there exists a simplicial inclu-
sionK↪→σm−^1 (see Example 2). Therefore,