Orientation-preserving self-homeomorphisms of the surface of genus two have points of period at most two

Abstract

Abstract. We show that for any orientation-preserving self-homeomorphism α of the double torus Σ 2 there exists a point p of Σ 2 such that α(α(p)) = p. This answers a question raised by Jakob Nielsen in 1942. BackgroundThroughout this article, R will denote a commutative ring, and g a positive integer. We shall write Σ g to denote the closed, connected, orientable surface of genus g, and Z g to denote the ring Z/gZ.Nielsen [10] (cf. [7]) showed that, for any g ≥ 2, there exists an orientationpreserving self-homeomorphism α of Σ g such that α, α 2 , . . . , α 2g−3 are all fixedpoint free, that is, have no fixed-points. He showed further that, for any orientationpreserving self-homeomorphism α of Σ g , at least one of α, α 2 , . . . , α 2g−3 , α 2g−2 has a fixed-point if g ≥ 3, and at least one of α, α 2 , α 3 has a fixed-point if g = 2.Since 2g − 2 = 2 for g = 2, this left open the question of exactly what happens in the case g = 2, and he commented that it seemed difficult to him [10, Section 4] (cf. [7]). The problem has not been forgotten; for example, in a recent paper in which he obtains results analogous to Nielsen's for orientation-reversing selfhomeomorphisms and for non-orientable surfaces, Wang [13] mentions that it is still open. In the next section we solve this problem, rounding off Nielsen's abovementioned results by showing that, for g = 2, at least one of α, α 2 has a fixed-point, or, equivalently, α 2 has a fixed-point.This result has its origins in classical topology, has connections with dynamical systems, and has a proof which is mainly algebraic. Let us review the basic information that will be used.Recall that the fundamental group of Σ g has a one-relator presentationwhere [x, y] denotes xyx −1 y −1 . Recall also that the first homology R-module of Σ g , denoted H 1 (Σ g , R), is the R-module obtained by abelianizing the fundamental group and then tensoring over Z with R. There is a natural group homomorphism

Minimum periods of homeomorphisms of orientable surfaces

“…The problem about the determination of m( H + 2 ), raised by Nielsen in 1942, remained open until 1996 when Dicks and Llibre [8] gave an algebraic proof that m( H + 2 ) = 2, which completes (1.1). The only remaining case in (1.2) is m( H − 0 ).…”

Vortices and a TQFT for Lefschetz fibrations on 4–manifolds

“…For g > 2, Corollary 1.8 can also be deduced via elementary methods: by considering the relationship of (what are now called) the Lefschetz numbers of the iterates of φ to the characteristic polynomial of the action of φ on H 1 (Σ; Z), Nielsen showed in [31] that an orientation-preserving homeomorphism φ must have a periodic point of period at most 2g − 2 and that this estimate is best possible; by examining his argument more carefully one can show that it implies that one of the Lefschetz numbers L(φ), L(S 2 φ), L(S 2g−2 φ) is nonzero, so that in any event S 2g−2 φ has a fixed point. Since Nielsen's argument does not work for the case g = 2, he asked in [31] whether orientation-preserving homeomorphisms of surfaces of genus 2 always have points of period at most 2; this question remained open for decades before eventually being answered affirmatively by Dicks and Llibre [4], using methods quite different from those we use in the special case considered here.…”