Topology in Molecular Biology

9 Euler, Dehn–Sommerville Characteristics, and Their Applications 173

f 1 (K)≤

(

m

2

)

.

For 2-dim triangulations we obtain:

6(m−χ)≤m(m−1).

This leads to the following lower bounds for the number of verticesf 0 =min
triangulations of particular 2-dim manifolds, e.g.

2-dim torusT^2 :χ(T^2 )=0andm≥7;

Projective planeRP^2 :χ(RP^2 )=1andm≥ 6.

A triangulationT of a manifoldM^2 is calledneighbourlyif its 1-skeleton
is a complete graph (i.e. any two vertices are joined by an edge).

Theorem 1.LetTbe a minimal triangulation of a closed manifoldM^2 .Then
T is neighbourly if and only if

6(m−χ)=m(m−1),

wherem=f 0 (M^2 )andχ=χ(M^2 ). In this case:

(a) IfM^2 is orientable of genusg, then

g∈{(3q−1)(4q−1),(3q+1)(4q+1),q(12q−1),q(12q+1),q=0, 1 ,...}.

(b) IfM^2 is non-orientable withμcrosscaps, then

μ∈{(2q−1)(3q−2),(2q−1)(3q−1),q(6q−1),q(6q+1),q=0, 1 ,...}.

Example 6.Forq= 0 the possible values ofgare 1,0, and the possible values
ofμare 2, 1 ,0. Forq=1wegetg∈{ 6 , 20 , 11 , 13 }andμ∈{ 1 , 2 , 5 , 7 }.

Minimal neighbourly triangulations exist for the sphereS^2 (g=0,m= 4),
torusT^2 (g=1,m= 7), and real projective planeRP^2 (μ=1,m= 6).
However for most values ofχthere is no minimal neighbourly triangulation
(see the previous theorem). For example, minimal triangulations of orientable
surfaces of genus 2 to 5 are not neighbourly, and minimal triangulations of
non-orientable surfaces withμcrosscaps are not neighbourly forμ=3,4.

9.7 Smooth Manifolds...........................................

Asmoothn-manifoldMnis covered by open subsetsUα:

Mn=∪αUα.