Small-time flow fixed point indices and vector field zero indices

The expansion. Work in a smooth chart $(\phi ,U)$ at $p$ with $\phi (p)=0$, write $\tilde{X}$ for the chart representative of $X$ and ${\widehat{f}}_t$ for the chart representative of a family as in (i) or of the flow. By [F1], ${\widehat{f}}_t(u)=u+t\tilde{X}(u)+t^{2}r(t,u)$ and ${\widehat{f}}_{-t}(u)=u-t\tilde{X}(u)+t^2{r}_{-}(t,u)$ with $r,r_{-}$ smooth near $(0,0)$; for the flow, ${\widehat{\phi }}_t(0)=0$ for all $t$, so the expansion gives $t^{2}r(t,0)=0$ and hence $r(t,0)=0$ near $t=0$. Choose $\epsilon >0$ with $\tilde{X}\mathrm{\ne }0$ on $0<\mathrm{\mid }u\mathrm{\mid }\le \epsilon$; then $c:={\min }_{\mathrm{\mid }u\mathrm{\mid }=\epsilon }\mathrm{\mid }\tilde{X}(u)\mathrm{\mid }>0$.

The index identities for a tangent family. Let the family of (i) satisfy its isolation hypothesis on a neighbourhood containing the closed ball $\mathrm{\mid }u\mathrm{\mid }\le \epsilon$. For small $t>0$ the normalized displacement $v\mapsto (u-{\widehat{f}}_t(u))(\epsilon v)\mathrm{/}\mathrm{\mid }(u-{\widehat{f}}_t(u))(\epsilon v)\mathrm{\mid }$ is defined, and by step 1.1 it equals $(-\tilde{X}(\epsilon v)-tr(t,\epsilon v))\mathrm{/}\mathrm{\mid }-\tilde{X}(\epsilon v)-tr(t,\epsilon v)\mathrm{\mid }$. Since $\mathrm{\mid }\tilde{X}\mathrm{\mid }\ge c$ on the sphere and $r$ is bounded there, for $\mathrm{\mid }t\mathrm{\mid }$ small every vector $-\tilde{X}(\epsilon v)-str(t,\epsilon v)$, $s\in [0,1]$, has norm at least $c\mathrm{/}2>0$; hence the straight-line homotopy in $s$ is one of nowhere-zero maps of $S^{n-1}$, and [F2] gives ${\mathrm{ind}}_p(f_t)=\mathrm{deg}(v\mapsto -\tilde{X}(\epsilon v)\mathrm{/}\mathrm{\mid }\tilde{X}(\epsilon v)\mathrm{\mid })=(-1{)}^n{\mathrm{ind}}_{p}X$ by [F3]. The same computation with ${\widehat{f}}_{-t}$ gives ${\mathrm{ind}}_p(f_{-t})=\mathrm{deg}(\tilde{X}\mathrm{/}\mathrm{\mid }\tilde{X}\mathrm{\mid })={\mathrm{ind}}_{p}X$, again by [F2] and [F3].

The flow at a nondegenerate zero. The flow is tangent to $X$ at time zero and fixes $p$, so it satisfies all hypotheses of (i) except possibly the isolation one. Suppose $p$ is nondegenerate, so that $D{\tilde{X}}_0$ is invertible, and consider $G(t,u):=(t,\tilde{X}(u)+tr(t,u))$ near $(0,0)$; its differential at $(0,0)$ is block triangular with diagonal blocks $1$ and $D{\tilde{X}}_0$, hence invertible. By [F4] $G$ is a local diffeomorphism at $(0,0)$, so there are $\alpha ,\beta >0$ such that for $\mathrm{\mid }t\mathrm{\mid }<\alpha$ every solution of $\tilde{X}(u)+tr(t,u)=0$ with $\mathrm{\mid }u\mathrm{\mid }<\beta$ is unique; the fixed points of ${\phi }_t$ in the chart are exactly these solutions, and $u=0$ is one of them because $r(t,0)=0$ by step 1.1. Therefore for every $0<\mathrm{\mid }t\mathrm{\mid }<\alpha$ the point $p$ is the only fixed point of ${\phi }_t$ in the ball $\mathrm{\mid }u\mathrm{\mid }<\beta$, the isolation hypothesis of (i) holds, and step 2.1 applied to ${\phi }_t$ gives ${\mathrm{ind}}_p({\phi }_t)=(-1{)}^n{\mathrm{ind}}_{p}X$ and ${\mathrm{ind}}_p({\phi }_{-t})={\mathrm{ind}}_{p}X$. No orientation of $M$ is used, and no further choice is used after the vector-field index suppliers.