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Lefschetz index for orientation reversing planar homeomorphisms. (English) Zbl 0986.55003
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 P. Le Calvez and 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 $$\text{ind}(h,0)\in\{-1,0,1\}$$.

MSC:
 55M20 Fixed points and coincidences in algebraic topology 54H25 Fixed-point and coincidence theorems (topological aspects)
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References:
 [1] Morton Brown, On the fixed point index of iterates of planar homeomorphisms, Proc. Amer. Math. Soc. 108 (1990), no. 4, 1109 – 1114. · Zbl 0686.58028 [2] Patrice Le Calvez and Jean-Christophe Yoccoz, Un théorème d’indice pour les homéomorphismes du plan au voisinage d’un point fixe, Ann. of Math. (2) 146 (1997), no. 2, 241 – 293 (French, with English summary). · Zbl 0895.58032 · doi:10.2307/2952463 · doi.org
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