Fixed points of local actions of nilpotent Lie groups on surfaces. (28th January 2016) - HIRSCH, MORRIS W.
Abstract : Let $G$ be a connected nilpotent Lie group with a continuous local action on a real surface $M$, which might be non-compact or have non-empty boundary $\unicode[STIX]x2202M$ . The action need not be smooth. Let $\unicode[STIX]{x1D711}$ be the local flow on $M$ induced by the action of some one-parameter subgroup. Assume $K$ is a compact set of fixed points of $\unicode[STIX]{x1D711}$ and $U$ is a neighborhood of $K$ containing no other fixed points. Theorem. If the Dold fixed-point index of $\unicode[STIX]{x1D711}_{t}|U$ is non-zero for sufficiently small $t>0$, then $\mathsf{Fix}(G)\cap K≠ \varnothing$ .