The Lefschetz index formula recovers Poincare-Hopf

The derivation. Assume AC (The Axiom of Choice). Let $M$ be a closed smooth $n$-manifold, $n\ge 1$, and let $X$ be a smooth vector field with isolated zeros (Isolated zero and local index of a vector field). Embed $M$ as a closed smooth submanifold of a Euclidean space (The weak Whitney proper embedding theorem), let $N$ be an open tubular neighbourhood with its normal-fibre retraction $r:N\to M$ (The Euclidean tubular neighbourhood theorem, A closed Euclidean submanifold has a smooth neighborhood retraction), and for small $t>0$ define $f_t(x):=r(x-tX(x))\mathrm{.}$ Compactness gives a uniform tube margin around $M$ (take a finite cover by balls whose doubled balls lie in $N$), while the Euclidean norm of $X$ is bounded by A continuous image of a compact space is compact; a continuous real-valued map on a nonempty compact space attains a maximum and a minimum; and a continuous bijection from a compact space to a Hausdorff space is a homeomorphism, clause 2. Hence $f_t$ and the same formula for every $s\in [0,t]$ are defined for a common small $t>0$. This is Guillemin and Pollack's normal-projection approximation to the flow of $-X$, and it has the following properties.

• The fixed points of $f_t$ are exactly the zeros of $X$. If $r(x-tX(x))=x$, put $z:=x-tX(x)$; then $z-x=-tX(x)$ is perpendicular to $T_{x}M$ because $r$ is the normal-fibre projection, while $X(x)\in T_{x}M$, so $tX(x)=0$ and $X(x)=0$; conversely $X(x)=0$ gives $f_t(x)=r(x)=x$. Hence the fixed points of $f_t$ are the isolated zeros of $X$, for every sufficiently small $t>0$ for which the family is defined.

• $f_t$ is homotopic to the identity. The formula $(s,x)\mapsto r(x-sX(x))$, $s\in [0,t]$, is a homotopy from $f_0={\mathrm{i}\mathrm{d}}_M$ to $f_t$, so $L(f_t)=L({\mathrm{i}\mathrm{d}}_M)=\chi (M)$ by The Lefschetz number is a homotopy invariant and The Lefschetz number of the identity is the Euler characteristic.

• The index identification. Since $r$ restricts to the identity on $M$ with identity differential along $T_{x}M$, the family satisfies $f_t(x)=x-tX(x)+O(t^2)$: it is tangent to $-X$ at time zero, and its fixed points are isolated. The tangent-family part of Small-time flow fixed point indices and vector field zero indices, applied to the field $-X$, gives ${\mathrm{ind}}_p(f_t)=(-1{)}^n{\mathrm{ind}}_p(-X)$, and the negation law ${\mathrm{ind}}_p(-X)=(-1{)}^n{\mathrm{ind}}_{p}X$ (Negation scales the local index by $(-1{)}^n$) leaves ${\mathrm{ind}}_p(f_t)={\mathrm{ind}}_{p}X$ at every zero $p$ of $X$, for all sufficiently small $t>0$.

The Lefschetz–Hopf index formula Lefschetz-Hopf index formula applies to the smooth map $f_t$, whose fixed points are exactly the isolated zeros of $X$, and combines the three properties into ${\sum }_{p:X(p)=0}{\mathrm{ind}}_{p}X={\sum }_{p\in \mathrm{Fix}(f_t)}{\mathrm{ind}}_p(f_t)=L(f_t)=\chi (M),$ which is precisely the Poincaré–Hopf theorem Poincare-Hopf for closed manifolds — recovered here as a corollary of the Lefschetz–Hopf index formula. The normal-projection family gives the fixed-set description directly, including at degenerate zeros; it does not require a periodic-orbit analysis of the actual flow. Compactness makes the isolated zero set finite (it is closed and discrete), by A closed discrete subset of a compact space is finite: locally, continuity makes the nonzero locus open. Thus the finitely many local small-time bounds have a common positive bound. For odd $n$ the conclusion is also consistent with the vanishing of $\chi$ recorded in Closed odd-dimensional manifolds have zero Euler characteristic.

What is used. The argument uses a proper embedding of $M$, a tubular neighbourhood with its normal-fibre retraction, the tangent-family index computation, the Lefschetz–Hopf index formula and the homotopy invariance of $L$; no countability or orientation hypothesis on $M$ is added beyond the ones already carried by those suppliers.