Topology in Molecular Biology

9 Euler, Dehn–Sommerville Characteristics, and Their Applications 175

vector at a pointx∈Mcan be written as ann-tuple (ξαj). The twon-tuples
corresponding to different coordinate systems are connected by the formula:

ξαj=

∑n

k=1

(

∂fαj

∂xkβ

)

ξβk.

The set of all tangent vectors toMat a pointxforms ann-dim linear space
Tx=TxM, called thetangent spacetoMatx. A smooth mapF:M→N
induces a linear map
F∗:Tx→TF(x)

at anyx∈M.
Denote byTMthetangent bundleofM. It consists of pairs (x, η), where
x∈Mandηis a tangent vector toMatx. A (tangent)vector fieldonM
is a smooth mapξ:M→TMsuch thatξ(x)∈TxM. It follows thatξis an
embedding; we denote its image byM(ξ). We identify the manifoldMwith
its imageM(0)⊂TMunder zero vector field. Note that dimTM=2nif
dimM=n.
A vector fieldξonMis said to be ofgeneral positionifM(ξ)andM=
M(0) intersect transversally, that is, in a finite number of points. These points
are calledsingular. All singular pointsxjof a general positioned vector field
on an oriented manifoldMarenon-degeneratein the sense that

(

det

∂ξk

∂xj

)

xj

=0.

Theindexof a singular pointxjis

(

det∂ξ

k

∂xj

)

xj

.

Theorem 2 (Hopf–Poincar ́e).The Euler characteristic of a closed oriented
manifoldMnequals the sum of indices of singular points of any general posi-
tioned vector fieldξ. In particular, the latter sum does not depend on a vector
field.

An important case of vector fields arises from a smooth functionfonM
with only non-degenerate critical pointsxj. (A pointx∈Mis calledcritical
if (df)x= 0; a critical point isnon-degenerateif det(d^2 f)x= 0.) Denote by
i(x) the number of negative squares in the canonical representation of the
quadratic form (d^2 f)x. Then the sum

∑m

j=1

(−1)i(xj),

over the critical pointsx 1 ,...,xmoffequalsχ(M); in particular, it is inde-
pendent off.