Let $h : \mathbb{R}^2 \to \mathbb{R}^2$ be an orientation preserving homeomorphism of the plane. For any bounded orbit $\mathcal{O}(x)=\{h^n(x):n\in\mathbb{Z}\}$ there exists a fixed point $p\in\mathbb{R}^2$ of h linked to $\mathcal{O}(x)$ in the sense of Gambaudo: one cannot find a Jordan curve $C\subseteq\mathbb{R}^2$ around $\mathcal{O}(x)$, separating it from p, that is isotopic to h(C) in $\mathbb{R}^2\setminus\left(\mathcal{O}(x)\cup\{p\}\right)$.

Given maps $f_1,\ldots ,f_n:X\to Y$ between (finite and connected) graphs, with $n\geq 3$ (the case $n=2$ is well known), we say that they are loose if they can be deformed by homotopy to coincidence free maps, and totally loose if they can be deformed by homotopy to maps which are two by two coincidence free. We prove that: (i) if Y is not homeomorphic to the circle, then any maps are totally loose; (ii) otherwise, any maps are loose and they are totally loose if and only if they are homotopic.

Let W be a real vector space and let V be an orthogonal representation of a group G such that $V^{G} = \{0\}$ (for the set of fixed points of G). Let $S(V)$ be the sphere of V and suppose that $f: S(V) \to W$ is a continuous map. We estimate the size of the $(H, G)$ -coincidences set if G is a cyclic group of prime power order $\mathbb {Z}_{p^k}$ or a p-torus $\mathbb {Z}_p^k$ .

We show that provided that n is a power of two, any nD measures in ℝn can be bisected by an arrangement of D hyperplanes.

An isotopic to the identity map of the 2-torus, that has zero rotation vector with respect to an invariant ergodic probability measure, has a fixed point by a theorem of Franks. We give a version of this result for nilpotent subgroups of isotopic to the identity diffeomorphisms of the 2-torus. In such a context we guarantee the existence of global fixed points for nilpotent groups of irrotational diffeomorphisms. In particular, we show that the derived group of a nilpotent group of isotopic to the identity diffeomorphisms of the 2-torus has a global fixed point.

In this paper, we investigate free actions of some compact groups on cohomology real and complex Milnor manifolds. More precisely, we compute the mod 2 cohomology algebra of the orbit space of an arbitrary free ℤ2 and $\mathbb{S}^1$-action on a compact Hausdorff space with mod 2 cohomology algebra of a real or a complex Milnor manifold. As applications, we deduce some Borsuk–Ulam type results for equivariant maps between spheres and these spaces. For the complex case, we obtain a lower bound on the Schwarz genus, which further establishes the existence of coincidence points for maps to the Euclidean plane.

We prove that any cyclic quadrilateral can be inscribed in any closed convex $C^{1}$-curve. The smoothness condition is not required if the quadrilateral is a rectangle.

Let G be a finite group acting freely on a finitistic space X having cohomology type (0, b) (for example, $\mathbb S$n × $\mathbb S$2n is a space of type (0, 1) and the one-point union $\mathbb S$n ∨ $\mathbb S$2n ∨ $\mathbb S$3n is a space of type (0, 0)). It is known that a finite group G that contains ℤp ⊕ ℤp ⊕ ℤp, p a prime, cannot act freely on $\mathbb S$n × $\mathbb S$2n. In this paper, we show that if a finite group G acts freely on a space of type (0, 1), where n is odd, then G cannot contain ℤp ⊕ ℤp, p an odd prime. For spaces of cohomology type (0, 0), we show that every p-subgroup of G is either cyclic or a generalized quaternion group. Moreover, for n even, it is shown that ℤ2 is the only group that can act freely on X.

In this paper, we show that for every nonnilpotent hyperbolic map $f$ on an infra-nilmanifold, the set $\operatorname{HPer}(f)$ is cofinite in $\mathbb{N}$. This is a generalization of a similar result for expanding maps in Lee and Zhao (J. Math. Soc. Japan 59(1) (2007), 179–184). Moreover, we prove that for every nilpotent map $f$ on an infra-nilmanifold, $\operatorname{HPer}(f)=\{1\}$.

The aim of this work is to represent the elements of the homotopy classes of maps from S2 to S2 ∨ S1 by maps with some minimality root properties.

Let E → B be a fibre bundle and let Eʹ → B be a vector bundle. Let G be a compact Lie group acting fibre preservingly and freely on both E and Eʹ – 0, where 0 is the zero section of Eʹ → B. Let f : E → Eʹ be a fibre-preserving G-equivariant map and let Zf = {x ∈ E | f(x) = 0} be the zero set of f. In this paper we give a lower bound for the cohomological dimension of the zero set Zf when a fibre of E → B is a real Stiefel manifold with a free ℤ/2-action or a complex Stiefel manifold with a free 𝕊1-action. This generalizes a well-known result of Dold for sphere bundles equipped with free involutions.

First published online by Duke University Press 15 January 2010, subsequently published online by Cambridge University Press 11 January 2016, doi:10.1017/S0027763000009818

For the locally symmetric space X attached to an arithmetic subgroup of an algebraic group G of ℚ-rank r, we construct a compact manifold by gluing together 2r copies of the Borel–Serre compactification of X. We apply the classical Lefschetz fixed point formula to and get formulas for the traces of Hecke operators ℋ acting on the cohomology of X. We allow twistings of ℋ by outer automorphisms η of G. We stabilize this topological trace formula and compare it with the corresponding formula for an endoscopic group of the pair (G,η) . As an application, we deduce a weak lifting theorem for the lifting of automorphic representations from Siegel modular groups to general linear groups.

Bo Ju Jiang introduced an invariant lying in the braid group which is the best lower bound of the number of fixed points in a homotopy class of a given pair of self maps of a surface. Here we modify this construction to get a lower bound of the number of coincidence points of a pair of maps between two closed surfaces.