On a homotopy converse to the Lefschetz fixed point theorem

Brown, Robert F.

doi:10.2140/pjm.1966.17.407

Cited by 62 publications

(95 citation statements)

References 6 publications

(11 reference statements)

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…the number of solutions of f (z) − z = ε in B (each one with a plus or minus sign depending on whether f −Id preserves or reverses orientation at that point), where ε ∈ M is sufficiently close to zero and a regular value of f −Id. It is important to note that the Poincaré index is constant under small perturbations of the map (see [5,6]). …”

Section: Lefschetz Numbersmentioning

confidence: 99%

“…Knowing the homology class of the map, one can compute its Lefschetz number L(f ) and, if the result is nonzero, conclude the existence of a fixed point. Clearly, the same process, applied to the k th iterate of the function, f k , would give the existence of a periodic orbit of period k, or a divisor of k. We have gone a long way from this theorem, and there is plenty of literature on its generalizations and applications (see [2,5,8,16]). …”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Periodic points of holomorphic maps via Lefschetz numbers

Fagella¹,

Llibre²

2000

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. In this paper we study the set of periods of holomorphic maps on compact manifolds, using the periodic Lefschetz numbers introduced by Dold and Llibre, which can be computed from the homology class of the map. We show that these numbers contain information about the existence of periodic points of a given period; and, if we assume the map to be transversal, then they give us the exact number of such periodic orbits. We apply this result to the complex projective space of dimension n and to some special type of Hopf surfaces, partially characterizing their set of periods. In the first case we also show that any holomorphic map of CP (n) of degree greater than one has infinitely many distinct periodic orbits, hence generalizing a theorem of Fornaess and Sibony. We then characterize the set of periods of a holomorphic map on the Riemann sphere, hence giving an alternative proof of Baker's theorem.

show abstract

Section: Lefschetz Numbersmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Periodic points of holomorphic maps via Lefschetz numbers

Fagella¹,

Llibre²

2000

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…Hence the sum of the indices is 0, which is why the Lefschetz fixed-point theorem cannot detect the existence of any fixedpoints for α 2 . For details on the theory of indices of isolated fixed-points, see, for example, [3].…”

Section: An Examplementioning

confidence: 99%

“…Then det R (α) = 1; for details, see, for example, [11,II.5]. By the Lefschetz fixed-point theorem for compact polyhedra and by Poincaré duality for surfaces, if α is also fixed-point free, then trace Z (α) = 2; for details, see, for example, [12], [3]. Further, it is easy to see that if trace Z (α) = 2, then trace R (α) = 2, and it is this consequence which interests us.…”

mentioning

confidence: 99%

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

Dicks¹,

Llibre²

1996

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

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

show abstract

“…Obviously, if such a map <r exists, then <r(l) is a fixed point free map homotopic to identity. R. F. Brown [8] proved that x(^) = : 0 is necessary and sufficient for the existence of a nonzero path field on M and the result was extended to topological manifolds with boundary in [ll]. It is interesting to note that in the case of differentiable manifold a nonzero vector field implies the existence of a nonzero path field cr such that a(x) is a simple path (no self-intersections) for each x.…”

Section: Theorem Let ƒ: X-+x Denote a Self-map Of A Wecken Space Satmentioning

confidence: 99%

Recent results in the fixed point theory of continuous maps

Fadell¹

1970

Bull. Amer. Math. Soc.

View full text Add to dashboard Cite

On a homotopy converse to the Lefschetz fixed point theorem

Cited by 62 publications

References 6 publications

Periodic points of holomorphic maps via Lefschetz numbers

Periodic points of holomorphic maps via Lefschetz numbers

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

Recent results in the fixed point theory of continuous maps

Contact Info

Product

Resources

About