Editing Lefschetz fixed-point theorem (section)

==Lefschetz–Hopf theorem==
A stronger form of the theorem, also known as the '''Lefschetz–Hopf theorem''', states that, if <math>f</math> has only finitely many fixed points, then

:<math>\sum_{x \in \mathrm{Fix}(f)} \mathrm{ind}(f,x) = \Lambda_f,</math>

where <math>\mathrm{Fix}(f)</math> is the set of fixed points of <math>f</math>, and <math>\mathrm{ind}(f,x)</math> denotes the [[fixed-point index|index]] of the fixed point <math>x</math>.<ref>{{Cite book | last=Dold | first=Albrecht | authorlink=Albrecht Dold| title=Lectures on algebraic topology | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=2nd  | isbn=978-3-540-10369-1 |mr=606196 | year=1980 | volume=200 }}, Proposition VII.6.6.</ref> From this theorem one may deduce the [[Poincaré–Hopf theorem]] for vector fields as follows. Any [[vector field]] on a compact manifold induces a [[Flow (mathematics)|flow]] <math>\varphi(x,t)</math> in a natural way, and for every <math>t</math> the map <math>\varphi(x,t)</math> is homotopic to the identity (thus having the same Lefschetz number); moreover, for sufficiently small <math>t</math> the fixed points of the flow and the zeroes of the vector field have the same indices.