Vector Field Index Euler Characteristic and Poincare Hopf

Let $M$ be a compact smooth $n$-manifold, possibly with boundary (Smooth manifolds and their smooth charts in the boundaryless case, Smooth charts, atlases, and structures with boundary in the boundary case). Its Euler characteristic is $\chi (M):={\sum }_{i=0}^n(-1{)}^i{\dim }_{\mathbb{Q}}{H}_i(M;\mathbb{Q})\in \mathbb{Z},$ the alternating sum of the rational Betti numbers of the singular homology of $M$ (The singular chain complex and singular homology, Finite-dimensional vector space, and its dimension ${\dim }_{F}V$; infinite-dimensional means having no finite basis, The rationals as equivalence classes of pairs of integers). The empty manifold has $\chi (\mathrm{\varnothing })=0$, the empty sum. The sum is finite because ${\dim }_{\mathbb{Q}}{H}_i(M;\mathbb{Q})<\mathrm{\infty }$ for every $i$ and $H_i(M;\mathbb{Q})=0$ for $i>n$; that finiteness is proved on this page in Finiteness and additivity of the Euler characteristic ↗, which is why it is recorded as the well-definedness pointer rather than assumed here. The definition itself uses no orientation and no choice of field or coefficient system.

When $M$ has a finite CW model, $\chi (M)$ agrees with its cell-count Euler characteristic (Euler characteristic of a finite CW complex): the finite rational cellular complex has one generator per cell, and alternating rank-nullity cancels boundary dimensions. Cellular homology (Cellular homology computes singular homology) therefore identifies that cell count with the rational Betti alternating sum. The integral Euler-Poincare formula (Euler–Poincare formula for finite CW complexes) gives the same count. The relative version is proved in Finiteness and additivity of the Euler characteristic ↗. The cited well-definedness proposition assumes the Axiom of Choice (The Axiom of Choice) to obtain an excellent Morse function. The formula for $\chi$ makes no selection of auxiliary data.

Assume the Axiom of Choice (The Axiom of Choice). Let $M$ be a compact smooth $n$-manifold, possibly with boundary.

(i) Each ${\dim }_{\mathbb{Q}}{H}_i(M;\mathbb{Q})$ is finite and $H_i(M;\mathbb{Q})=0$ for $i>n$, so $\chi (M)$ of Euler characteristic of a compact manifold is a well-defined integer.

(ii) If $A\subseteq M$ is a compact smooth submanifold, possibly with boundary, and $(M,A)$ is homotopy equivalent as a pair to a finite relative CW pair with $c_k$ relative $k$-cells, then $\chi (M)=\chi (A)+{\sum }_k(-1{)}^k{c}_k=\chi (A)+{\sum }_k(-1{)}^k{\dim }_{\mathbb{Q}}{H}_k(M,A;\mathbb{Q})\mathrm{.}$

(iii) If $M=M_1{\cup }_N{M}_2$ with $M_1,M_2$ compact smooth submanifolds, possibly with boundary, $N=M_1\cap M_2$ a common compact smooth submanifold, and the inclusions $N\hookrightarrow M_i$ cofibrations, then $\chi (M)=\chi (M_1)+\chi (M_2)-\chi (N)\mathrm{.}$

Write $S^0=\{-1,+1\}=\mathrm{\partial }[-1,1]$ for the $0$-sphere (Euclidean spheres and closed balls as subspaces of ${\mathbb{R}}^n$) and orient it by the boundary orientation of the interval $[-1,1]$ (Induced boundary orientation): the point $+1$ carries the sign $+1$ and the point $-1$ the sign $-1$, since an orientation of a $0$-manifold is a sign at each point (Oriented smooth manifolds and oriented charts). Let $(P,s)$ be a finite oriented $0$-manifold, that is, a finite set $P$ together with a sign $s(x)\in \{+1,-1\}$ for every $x\in P$ (Oriented smooth manifolds and oriented charts); it is balanced when ${\sum }_{x\in P}s(x)=0.$ For a balanced $(P,s)$ and a map $f:P\to S^0$ — every map is continuous because $P$ is discrete and $S^0$ is discrete — define the reduced degree $\mathrm{deg}(f):=\frac{1}{2}{\sum }_{x\in P}s(x)f(x)\in \mathbb{Z},$ the signed count of the values divided by two, where the values $f(x)\in \{-1,+1\}$ are read in $\mathbb{R}$. This is well defined: since $f$ takes only the two values $\pm 1$, balance gives ${\sum }_{f(x)=+1}s(x)=-{\sum }_{f(x)=-1}s(x)$, hence ${\sum }_{x}s(x)f(x)=2{\sum }_{f(x)=+1}s(x)$ is even, and replacing $s$ by $-s$ changes the sign of $\mathrm{deg}(f)$ exactly as changing an orientation changes the ordinary degree (Degree of a map between oriented closed manifolds).

The case $P=S^0$ with the standard orientation above is balanced, and then $\mathrm{deg}(f)=\frac{f(+1)-f(-1)}{2}\in \{-1,0,+1\}\mathrm{.}$ This is the induced map on the reduced group ${\tilde{H}}_0(S^0;\mathbb{Z})\cong \mathbb{Z}$ generated by the difference of the two point classes (Reduced homology theory and augmentation): the identity has reduced degree $+1$, the antipodal map $-1$, and the two constant maps $0$. It is the scalar replacement for the fundamental-class degree of Degree of a map between oriented closed manifolds, whose scalar definition is stated only for nonempty connected closed oriented manifolds and therefore does not apply to a source as disconnected as $S^0$.

The motivating balanced sources are boundaries: if $W$ is a compact oriented smooth $1$-manifold and $\mathrm{\partial }W$ carries the induced boundary orientation, then ${\sum }_{p\in \mathrm{\partial }W}\epsilon (p)=0$ (Oriented boundary counts of a compact oriented 1-manifold cancel), so $\mathrm{\partial }W$ is balanced and every map $\mathrm{\partial }W\to S^0$ has a reduced degree. No choice principle is used in the definition; the cited boundary-count lemma assumes ${\mathrm{A}\mathrm{C}}_{\omega }$.

Let $(P,s)$ be a balanced oriented finite $0$-manifold, so that every map $P\to S^0$ has a reduced degree, and let $f:P\to S^0$ be a map (The reduced degree of a map into the 0-sphere).

(i) If $F:P\times [0,1]\to S^0$ is continuous and $F_t(x):=F(x,t)$, then $F_t=F_0$ for every $t\in [0,1]$; in particular $\mathrm{deg}{F}_t=\mathrm{deg}{F}_0$.

(ii) If $g:S^0\to S^0$ is any map and $\mathrm{deg}g$ denotes its reduced degree as a self-map of $S^0$, then $\mathrm{deg}(g\circ f)=\mathrm{deg}(g)\mathrm{deg}(f)$.

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)) for the canonical smooth tangent-bundle structure.

Let $M$ be a smooth $n$-manifold without boundary, $n\ge 1$, and let $X$ be a smooth vector field on $M$ (A smooth vector field is a smooth section of the tangent bundle, Smooth manifolds and their smooth charts). A point $p\in M$ is an isolated zero of $X$ when $X(p)=0$ and some chart around $p$ contains no other zero of $X$; equivalently, the set of zeros of $X$ has $p$ as an isolated point in a chart around $p$ (Manifold charts, coordinate domains, and coordinate functions).

For an isolated zero choose a smooth chart $(\phi ,U)$ of the smooth structure of $M$ with $\phi (p)=0$ and write $X_{\phi }(u):=(d{\phi }_{{\phi }^{-1}(u)})X({\phi }^{-1}(u))\in {\mathbb{R}}^n$ for the chart representative of $X$ (The induced tangent bundle chart). Since $p$ is an isolated zero there is $\epsilon >0$ with $X_{\phi }\mathrm{\ne }0$ on ${\bar{B}}_{\epsilon }(0)\setminus \{0\}\subseteq \phi (U)$, so the normalized field $v\mapsto X_{\phi }(\epsilon v)\mathrm{/}\mathrm{\mid }{X}_{\phi }(\epsilon v)\mathrm{\mid }$ is a continuous map $S^{n-1}\to S^{n-1}$. The local index of $X$ at $p$ is ${\mathrm{ind}}_{p}X:=\mathrm{deg}(S^{n-1}\to S^{n-1},v\mapsto \frac{X_{\phi }(\epsilon v)}{\mathrm{\mid }{X}_{\phi }(\epsilon v)\mathrm{\mid }}),$ the degree of Degree of a map between oriented closed manifolds computed with the standard orientations of the two copies of $S^{n-1}$, for $n\ge 2$.

For $n=1$ the same formula is read in dimension zero: the sphere $S^0=\{\pm 1\}$ parametrizes $\mathrm{\partial }[-\epsilon ,\epsilon ]$ by $v\mapsto \epsilon v$, and the displayed map is a map $S^0\to S^0$ whose reduced degree in the sense of The reduced degree of a map into the 0-sphere is declared to be the index, ${\mathrm{ind}}_{p}X=\frac{f(+1)-f(-1)}{2}\in \{-1,0,+1\},f(v)=\frac{X_{\phi }(\epsilon v)}{\mathrm{\mid }{X}_{\phi }(\epsilon v)\mathrm{\mid }}\mathrm{.}$ This case is well posed for every $0<{\epsilon }^{\mathrm{\prime }}<\epsilon$: the two signs of $X_{\phi }$ on $(0,\epsilon )$ and on $(-\epsilon ,0)$ are each constant, because $X_{\phi }$ is continuous and nowhere zero on either half-interval, so the two values $f(\pm 1)$ do not depend on ${\epsilon }^{\mathrm{\prime }}$; for $n\ge 2$ the value is independent of $\epsilon$ by the homotopy $v\mapsto X_{\phi }(((1-t)\epsilon +t{\epsilon }^{\mathrm{\prime }})v)\mathrm{/}\mathrm{\mid }\cdot \mathrm{\mid }$ on the zero-free annulus (Degree is invariant under proper smooth homotopy). The value is also independent of the smooth chart and admissible radius, and of trivializations whose fibre orientation matches the chosen base orientation (The local index is independent of chart, ball and trivialization ↗); in particular no orientation of $M$ is required, because a chart change multiplies both the source and the target orientation by the same sign, and the $n=1$ case is the same statement read on the $0$-sphere. Every statement on this page assumes $n\ge 1$; the index is a signed integer, ${\mathrm{ind}}_{p}X\in \mathbb{Z}$.

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)) for the canonical smooth tangent-bundle structure.

Let $M$ be a smooth $n$-manifold without boundary, $n\ge 1$, let $X$ have an isolated zero $p$, and let $\phi ,\psi$ be smooth charts of the given smooth structure centered at $p$, with admissible radii $\epsilon ,\delta$ as in Isolated zero and local index of a vector field. Then the maps $f_{\phi }(v)=\frac{X_{\phi }(\epsilon v)}{\mathrm{\mid }{X}_{\phi }(\epsilon v)\mathrm{\mid }},f_{\psi }(v)=\frac{X_{\psi }(\delta v)}{\mathrm{\mid }{X}_{\psi }(\delta v)\mathrm{\mid }}$ have the same degree, using reduced degree when $n=1$. Thus the local index is independent of chart and radius. On a coordinate ball it is also unchanged by a smooth fibre trivialization preserving the coordinate orientation. Equivalently, base and fibre orientations must be chosen consistently; reversing only the fibre orientation reverses the degree. No orientation of $M$ is required.

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)) for the canonical smooth tangent-bundle structure.

Let $M$ be a smooth $n$-manifold, $X$ a smooth vector field on $M$ and $p\in M$ a zero of $X$ (A smooth vector field is a smooth section of the tangent bundle). View $X$ as a smooth section $X:M\to TM$ (Smoothness of a section is equivalent to smooth local components). The zero section $0_M$ is a smooth embedding (The zero section is a smooth embedding), so its differential $d(0_M{)}_p:T_{p}M\to T_{(p,0)}TM$ is injective (The differential of a smooth map), and the vertical quotient at $p$ is $N_p:=T_{(p,0)}TM/d(0_M{)}_p(T_{p}M)\mathrm{.}$ Since $X$ is a section, both $d{X}_p$ and $d(0_M{)}_p$ are right inverses of the projection $d{\pi }_{(p,0)}$; hence the class of $d{X}_p$ in $\mathrm{Hom}(T_{p}M,N_p)$ is the vertical derivative $D{X}_p\in \mathrm{End}(T_{p}M),$ read through the canonical identification $N_p\cong T_{p}M$: explicitly, the difference $d{X}_p-d(0_M{)}_p$ takes values in the vertical space $\ker d{\pi }_{(p,0)}$, and $D{X}_p$ is that difference followed by the canonical isomorphism from the vertical space $V_{(p,0)}=\ker d{\pi }_{(p,0)}$ to $T_{p}M$. In an induced tangent-bundle chart $\tilde{\phi }$ over a chart $\phi$ with $\phi (p)=0$ the section reads $u\mapsto (u,X_{\phi }(u))$ and $D{X}_p$ corresponds to the ordinary derivative $D{X}_{\phi }(0)$ of the chart representative $X_{\phi }$ (The induced tangent bundle chart, Local and global frames of a vector bundle); a change of chart conjugates $D{X}_p$ by an isomorphism, so invertibility and the determinant $\det D{X}_p$ are independent of the chart (The tangent bundle as a disjoint union).

The zero $p$ is nondegenerate when $D{X}_p$ is invertible. A nondegenerate zero is isolated: in a chart, $X_{\phi }(0)=0$ and $D{X}_{\phi }(0)$ is invertible, so $X_{\phi }$ is a diffeomorphism near $0$ by the inverse function theorem (The smooth inverse function theorem on manifolds) and its only zero near $0$ is $0$ itself. In particular a nondegenerate zero is an isolated zero and $X$ has no other zero in some neighbourhood of $p$. The vertical derivative itself requires no further choice once the smooth tangent-bundle structure is supplied.

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)) for the canonical smooth tangent-bundle structure.

Let $M$ be a smooth $n$-manifold, $n\ge 1$, let $X$ be a smooth vector field and let $p$ be a nondegenerate zero of $X$ (Nondegenerate zero of a vector field). Then ${\mathrm{ind}}_{p}X=\mathrm{sign}\det (D{X}_p:T_{p}M\to T_{p}M)\in \{+1,-1\}\mathrm{.}$ In particular every nondegenerate zero has index $+1$ or $-1$.

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Let $M$ be a smooth $n$-manifold, $n\ge 1$, let $X$ be a smooth vector field with an isolated zero at $p$, and let $B\subseteq M$ be an embedded closed ball with $p\in \mathrm{int}B$, $X\mathrm{\ne }0$ on $\mathrm{\partial }B$, and no zero of $X$ in $\mathrm{int}B$ other than $p$ (such a ball exists in a chart around $p$).

(i) Choose an orientation of $B$ and a smooth trivialization of $TM{\mathrm{\mid }}_B$ preserving that orientation, and orient $\mathrm{\partial }B$ as the boundary of $B$; then ${\mathrm{ind}}_{p}X$ equals the degree of $x\mapsto X(x)\mathrm{/}\mathrm{\mid }X(x)\mathrm{\mid }$ from $\mathrm{\partial }B$ to $S^{n-1}$ (the reduced degree for $n=1$).

(ii) If $X^{\mathrm{\prime }}$ is a smooth vector field on $M$ with $X^{\mathrm{\prime }}=X$ outside $\mathrm{int}B$ and only nondegenerate zeros $q_1,\dots ,q_k$ in $\mathrm{int}B$, then ${\sum }_{i=1}^k{\mathrm{ind}}_{q_i}{X}^{\mathrm{\prime }}={\mathrm{ind}}_{p}X$.

(iii) Consequently every isolated zero can be perturbed, supported in an arbitrarily small ball around it, to finitely many nondegenerate zeros with the same index sum.

Assume countable choice ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Let $M$ be a closed oriented smooth $n$-manifold, $n\ge 1$, and give $TM$ the orientation which along the zero section is the direct sum of the horizontal (tangent) orientation of the base and the vertical (fibre) orientation, first factor first (Product orientations, The tangent bundle as a disjoint union), orient both submanifolds by their projections to $M$, and let $X_0\subset TM$ be the zero section (The zero section is a smooth embedding) and let ${\mathrm{\Gamma }}_X$ be the graph of a smooth vector field $X$ transverse to $X_0$, so that the zeros of $X$ are nondegenerate. Then, with the oriented intersection number of The oriented intersection number, $I(X_0,{\mathrm{\Gamma }}_X)={\sum }_{p:X(p)=0}{\mathrm{ind}}_{p}X,$ and at each zero the local oriented intersection sign of The local oriented intersection sign satisfies ${\epsilon }_{(X_0,{\mathrm{\Gamma }}_X)}(p)={\mathrm{ind}}_{p}X$.