Fixed points for orientation preserving homeomorphisms of the plane which interchange two points. (English) Zbl 0728.55001

Brown, Morton

Pac. J. Math. 143, No. 1, 37-41 (1990).

In this note the author proves: “Let h be an orientation preserving homeomorphism of the plane which interchanges two points p and q. If A is an arc from p to q where f has no fixed points in A, then h has a fixed point in one of the bounded complementary domains of \(A\cup h(A)''\). The proof uses simple properties of the winding number and that the set consisting of \(A\cup h(A)\) and the union with its bounded complementary domains is contractible.

Reviewer: D.Gonçalves (São Paulo)

MSC:

55M20

Fixed points and coincidences in algebraic topology

Keywords:

index; orientation preserving homeomorphism; plane; winding number

Citations:

Zbl 0494.55001