Index zero fixed points and 2-complexes with local separating points

Abstract

We study the situation of an isolated fixed point with local index zero at a local separating point in a $2$-complex. This fixed point sometimes can be removed and sometimes not, either locally or globally. Criteria are given for local removability in dimension at most two. Results are applied to finding fixed point minimal models on $S^2\vee S^2$ and $S^1 \vee S^2$. A non-existence result is given in the case of a wedge in which both factors are surfaces with non-positive Euler characteristic.