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Poincaré–Hopf theorem

Contents

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

graded differential cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Contents

Idea

The Poincaré–Hopf theorem says that for any vector field vΓ(TX)v \in \Gamma(T X) with a finite set of isolated vanishing points {x i}\{x_i\} on an orientable compact differential manifold XX, the sum over the x iXx_i \in X of the degrees of the vector in the vicinity of these points, regarded as cohomotopy classes

v/|v| |x i:D x iS(T x iX) v/{\vert v\vert}_{\vert x_i} \;\colon\; \partial D_{x_i} \longrightarrow S(T_{x_i} X)

and called the Poincaré–Hopf index of ff at x ix_i

ind x i(v)deg(v/|v| |x i) ind_{x_i}(v) \;\coloneqq\; deg\big( v/{\vert v\vert}_{\vert {x_i} } \big)

is given by the Euler characteristic, hence by the value of the Euler class on the tangent bundle:

χ(X)χ[TX]=isolated zerox iind x i(v). \chi(X) \coloneqq \chi[T X] \;=\; \underset{ {\text{isolated zero}} \atop { x_i } }{\sum} ind_{x_i}(v) \,.

In particular, the existence of a nowhere vanishing vector field (for which the above sum is empty) implies that the Euler characteristic vanishes.

References

Named after Henri Poincaré and Heinz Hopf.

Textbook accounts:

Review includes

  • Alex Wright, Kael Dixon, The Poincaré–Hopf theorem (pdf)

  • Ariel Hafftka, Differential topology and the Poincaré–Hopf theorem (pdf)

Discussion in a broader perspective of K-theory and index theorems:

  • Omar Mohsen, Poincaré–Hopf theorem and groupoids (pdf, pdf)

A comment on the version for complex vector fields is in

  • Howard Jacobowitz, Non-vanishing complex vector fields and the Euler characteristic (arXiv:0901.0893)

Generalization to orbifolds:

  • Christopher Seaton, Two Gauss–Bonnet and Poincaré–Hopf theorems for orbifolds with boundary, Differential Geometry and its Applications Volume 26, Issue 1, February 2008, Pages 42-51 (doi:10.1016/j.difgeo.2007.11.002)

See also

Last revised on May 30, 2019 at 17:24:56. See the history of this page for a list of all contributions to it.