Lefschetz index for orientation reversing planar homeomorphisms (Q2782651)

\textit{M. Brown} [ibid. 108, No. 4, 1109-1114 (1990; Zbl 0686.58028)] proved that each integer occurs as the local fixed point index at the origin of an orientation preserving plane local homeomorphism. On the other hand, Brown [loc. cit.] stated without proof that in the orientation reversing case only \(-1\), 0, and 1 are possible. Drawing heavily on ideas of \textit{P. Le Calvez} and \textit{J. C. Yoccoz} [Ann. Math. (2) 146, No. 2, 241-293 (1997; Zbl 0895.58032)] the present author proves just this result. To be precise, he shows the following: Let \(V,W\) be two open connected neighbourhoods of \(0\) in \(\mathbb{R}^2\) and let \(h:V\to W\) be an orientation reversing homeomorphism which possesses \(0\) as an isolated fixed point. Then \(\text{ind}(h,0)\in\{-1,0,1\}\).