# nLab Poincaré–Hopf theorem

Contents

### Context

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

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

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 $v \in \Gamma(T X)$ with a finite set of isolated vanishing points $\{x_i\}$ on an orientable compact differential manifold $X$, the sum over the $x_i \in X$ of the degrees of the vector in the vicinity of these points, regarded as cohomotopy classes

$v/{\vert v\vert}_{\vert x_i} \;\colon\; \partial D_{x_i} \longrightarrow S(T_{x_i} X)$

and called the Poincaré–Hopf index of $f$ at $x_i$

$ind_{x_i}(v) \;\coloneqq\; deg\big( v/{\vert v\vert}_{\vert {x_i} } \big)$

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

$\chi(X) \coloneqq \chi[T X] \;=\; \underset{ {\text{isolated zero}} \atop { x_i } }{\sum} ind_{x_i}(v) \,.$ In particular, the existence of a nowhere vanishing vector field (for which the above sum is empty) implies that the Euler characteristic vanishes.

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)