A classical theorem of H. Hopf asserts that a closed connected smooth manifold admits a nowhere vanishing vector field if and only if its Euler characteristic is zero. R. Brown generalized Hopf's result to topological manifolds, replacing vector fields with path fields. In this note, we give an equivariant analog of Brown's theorem for locally smooth $G$-manifolds where $G$ is a finite group.

We shall present a new lower bound for the number of roots of maps between graphs in any given homotopy class. We also give an example showing that our new lower bound can be arbitrary larger than the number of essential root classes.

We study the existence of periodic solutions for a nonlinear second order system of ordinary differential equations. Assuming suitable Lazer-Leach type conditions, we prove the existence of at least one solution applying topological degree methods.

We give a formula for the coincidence Reidemeister trace of selfmaps on bouquets of circles in terms of the Fox calculus. Our formula reduces the problem of computing the coincidence Reidemeister trace to the problem of distinguishing doubly twisted conjugacy classes in free groups.

Suppose $f\colon M\to M$ on a compact manifold. Let $m$ be a natural number. One of the most important questions in the topological theory of periodic points is whether the Nielsen-Jiang periodic number $NF_m(f)$ is a sharp lower bound on $\# {\rm Fix}(g^m)$ over all $g\sim f$. This question has a positive answer if ${\rm dim}\, M\geq 3$ but in general a negative answer for self maps of compact surfaces. However, we show the answer to be positive when $M=\mathbb K$ is the Klein bottle. As a consequence, we reconfirm a result of Llibre and compute the set ${\rm HPer} (f)$ of homotopy minimal periods on the Klein bottle.

Let $\Delta^{n}$ be the ball $|x|< 1$ in the complex vector space ${\mathbb C}^{n}$, let $f\colon \Delta^{n}\rightarrow {\mathbb C}^{n}$ be a holomorphic mapping and let $M$ be a positive integer. Assume that the origin $0=(0,\ldots ,0)$ is an isolated fixed point of both $f$ and the $M$-th iteration $f^{M}$ of $f$. Then the (local) Dold index $P_{M}(f,0)$ at the origin is well defined, which can be interpreted to be the number of periodic points of period $M$ of $f$ hidden at the origin: any holomorphic mapping $f_{1}\colon \Delta^{n}\rightarrow {\mathbb C}^{n}$ sufficiently close to $f$ has exactly $P_{M}(f,0)$ distinct periodic points of period $M$ near the origin, provided that all the fixed points of $f_{1}^{M}$ near the origin are simple. Therefore, the number ${\mathcal O}_{M}(f,0)=P_{M}(f,0)/M$ can be understood to be the number of periodic orbits of period $M$ hidden at the fixed point.

According to Shub-Sullivan [A remark on the Lefschetz fixed point formula for differentiable maps, Topology 13 (1974), 189–191] and Chow-Mallet-Paret-Yorke [A periodic orbit index which is a bifurcation invariant, Lecture Notes in Math., vol. 1007, Springer, Berlin, 1983, pp. 109–131], a necessary condition so that there exists at least one periodic orbit of period $M$ hidden at the fixed point, say, ${\mathcal O}_{M}(f,0)\geq 1$, is that the linear part of $f$ at the origin has a periodic point of period $M$. It is proved by the author in [Fixed point indices and periodic points of holomorphic mappings, Math. Ann. 337 (2007), 401–433] that the converse holds true.

In this paper, we continue to study the number ${\mathcal O}_{M}(f,0)$. We will give a sufficient condition such that ${\mathcal O}_{M}(f,0)\geq 2$, in the case that all eigenvalues of $Df(0)$ are primitive $m_{1}$-th,$\ldots$, $m_{n}$-th roots of unity, respectively, and $m_{1},\ldots,m_{n}$ are distinct primes with $M=m_{1}\ldots m_{n}$.

Let $M \to B$, $N \to B$ be fibrations and $f_1,f_2\colon M \to N$ be a pair of fibre-preserving maps. Using normal bordism techniques we define an invariant which is an obstruction to deforming the pair $f_1,f_2$ over $B$ to a coincidence free pair of maps. In the special case where the two fibrations are the same and one of the maps is the identity, a weak version of our $\omega$-invariant turns out to equal Dold's fixed point index of fibre-preserving maps. The concepts of Reidemeister classes and Nielsen coincidence classes over $B$ are developed. As an illustration we compute e.g. the minimal number of coincidence components for all homotopy classes of maps between $S^1$-bundles over $S^1$ as well as their Nielsen and Reidemeister numbers.

Let $X$ be a compact Hausdorff space, $\varphi\colon X\to S^{n}$ a continuous map into the $n$-sphere $S^n$ that induces a nonzero homomorphism $\varphi^{*}\colon H^{n}(S^{n};{\mathbb{Z}}_{p})\to H^{n}(X;{\mathbb{Z}}_{p})$, $Y$ a $k$-dimensional CW-complex and $f\colon X\to Y$ a continuous map. Let $G$ a finite group which acts freely on $S^{n}$. Suppose that $H\subset G$ is a normal cyclic subgroup of a prime order. In this paper, we define and we estimate the cohomological dimension of the set $A_{\varphi}(f,H,G)$ of $(H,G)$-coincidence points of $f$ relative to $\varphi$.

Let $M$ be two-holed $3$-dimensional closed ball, $r$ a given natural number. We consider $f$, a continuous self-map of $M$ with real eigenvalues on the second homology group, and determine the minimal number of $r$-periodic points for all smooth maps homotopic to $f$.

Let $H$ be a real infinite dimensional and separable Hilbert space. With an isolated invariant set ${\rm inv}(N)$ of a flow $\phi^t$ generated by an $\mathcal L\mathcal S$-vector field $f\colon H\supseteq \Omega\to H$, $f(x)=Lx+K(x)$, where $L\colon H\to H$ is strongly indefinite linear operator and $K\colon H\supseteq \Omega\to H$ is completely continuous, one can associate a homotopy invariant $h_{\mathcal L\mathcal S}({\rm inv}(N),\phi^t)$ called the $\mathcal L\mathcal S$-Conley index. In fact, this is a homotopy type of a finite CW-complex. We define the Betti numbers and hence the Euler characteristic of such index and prove the formula relating these numbers to the Leray-Schauder degree ${\rm deg}_{\mathcal L\mathcal S}(\widehat{f},N,0)$, where $\widehat f\colon H\supseteq \Omega\to H$ is defined as $\widehat f(x)=x+L^{-1}K(x)$.

We explain why the Poincaré translation operators along the trajectories of upper-Carathéodory differential inclusions do not satisfy the exceptional cases, described in our earlier counter-examples, for upper semicontinuous maps. Such a discussion was stimulated by a recent paper of F. Obersnel and P. Omari, where they show that, for Carathéodory scalar differential equations, the existence of just one subharmonic solution (e.g. of order $2$) implies the existence of subharmonics of all orders. We reprove this result alternatively just via a multivalued Poincaré translation operator approach. We also establish its randomized version on the basis of a universal randomization scheme developed recently by the first author.

In this work the problem of the existence of almost homoclinic solutions for a Newtonian system $\ddot{q}+V_{q}(t,q)=f(t)$, where $t\in\mathbb{R}$ and $q\in\mathbb{R}^n$, is considered. It is assumed that a potential $V\colon\mathbb{R}\times\mathbb{R}^{n}\to\mathbb{R}$ is $C^{1}$-smooth with respect to all variables and $T$-periodic in a time variable $t$. Moreover, $f\colon\mathbb{R}\to\mathbb{R}^{n}$ is a continuous bounded square integrable function and $f\neq 0$. This system may not have a trivial solution. However, we show that under some additional conditions there exists a solution emanating from $0$ and terminating to $0$. We are to call such a solution almost homoclinic to $0$.

Let $\Omega \subset \mathbb R^n$ be a bounded Lipschitz domain and consider the Dirichlet energy functional $$ {\mathbb F} [u , \Omega] := \frac{1}{2} \int_\Omega |\nabla u (x )|^2 dx, $$ over the space of measure preserving maps $$ {\mathcal A}(\Omega)=\{u \in W^{1,2}(\Omega, \mathbb R^n) : u |_{\partial \Omega} = x , {\rm det}\, \nabla u = 1 \text{ ${\mathcal L}^n$-a.e. in $\Omega$} \}. $$ Motivated by their significance in topology and the study of mapping class groups, in this paper we consider a class of maps, referred to as twists, and examine them in connection with the Euler-Lagrange equations associated with ${\mathbb F}$ over ${\mathcal A}(\Omega)$. We investigate various qualitative properties of the resulting solutions in view of a remarkably simple, yet seemingly unknown explicit formula, when $n=2$.