Fixed points of local actions of nilpotent Lie groups on surfaces

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$\unicode[STIX]{x2202}M$. 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$\mathsf{Fix}(G)\cap K\neq \varnothing$.