In this paper, we study the relation between the Lusternik-Schnirelmann category and the topological complexity of two closed oriented manifolds connected by a degree one map.

The inverse involution on a Lie group $G$ is the periodic $2$ transformation $\gamma $ that sends each element $g\in G$ to its inverse $g^{-1}$. The variety of the fixed point set ${\rm Fix}(\gamma )$ is of importance for the relevances with Morse function on the Lie group $G$, and the $G$-representations of the cyclic group $\mathbb{Z}_{2}$. In this paper we develop an approach to calculate the diffeomorphism types of the fixed point sets ${\rm Fix}(\gamma)$ for the simple Lie groups.

We compute the topological complexity of a polyhedral product $\mathcal{Z}$ defined in terms of an ${\rm LS}$-logarithmic family of locally compact connected ${\rm CW}$ topological groups. The answer is given by a combinatorial formula that involves the ${\rm LS}$ category of the polyhedral-product factors. As a by-product, we show that the Iwase-Sakai conjecture holds true for $\mathcal{Z}$. The proof methodology uses a Fadell-Husseini viewpoint for the monoidal topological complexity $\big(\mathsf{TC}^M\big)$ of a space, which, under mild conditions, recovers Iwase-Sakai's original definition. In the Fadell-Husseini context, the stasis condition - $\mathsf{TC}^M$'s raison d'être - can be encoded at the covering level. Our Fadell-Husseini inspired definition provides an alternative to the $\mathsf{TC}^M$ variant given by Dranishnikov, as well as to the ones provided by Garcia-Calcines, Carrasquel-Vera and Vandembroucq in terms of relative category.

The Reeb graph of a circle-valued function is a topological space obtained by contracting connected components of level sets (preimages of points) to points. For some smooth functions, the Reeb graph has the structure of a finite graph. This notion finds numerous applications in the theory of dynamical systems, as well as in the topological classification of circle-valued functions and the study of their homotopy properties. However, important theoretical facts on the topological properties of the Reeb graphs of circle-valued functions are scattered across numerous papers on different topics, according to the specific needs of the corresponding application. In this paper, we systematize the existing results on the Reeb graphs of circle-valued functions and generalize some of them to wider classes of functions or spaces. We also show how some results can be carried out from real-valued functions. Finally, we adapt some facts from the theory of foliations to the Reeb graphs of circle-valued functions. In particular, we analyze the cycle rank of the Reeb graph and address the problem of realization of a finite graph as a Reeb graph.

The classical approach to envy-free division and equilibrium problems arising in mathematical economics typically relies on Knaster-Kuratowski-Mazurkiewicz theorem, Sperner's lemma or some extension involving mapping degree. We propose a different and relatively novel approach where the emphasis is on configuration spaces and equivariant topology, originally developed for applications in discrete and computational geometry (Tverberg type problems, necklace splitting problem, etc.). We illustrate the method by proving several counterparts and extensions of the classical envy-free division theorem of David Gale, where the emphasis is on preferences allowing the players to choose degenerate pieces of the cake.

In this paper, motivated by recent work of Schnider and Axelrod-Freed and Soberón, we study an extension of the classical Grünbaum-Hadwiger-Ramos mass partition problem to mass assignments. Using the Fadell-Husseini index theory we prove that for a given family of $j$ mass assignments $\mu_1,\dots,\mu_j$ on the Grassmann manifold $G_{\ell}\big(\mathbb{R}^d\big)$ and a given integer $k\geq 1$ there exist a linear subspace $L\in G_{\ell}\big(\mathbb{R}^d\big)$ and $k$ affine hyperplanes in $L$ that equipart the masses $\mu_1^L,\dots,\mu_j^L$ assigned to the subspace $L$, provided that $d\geq j + (2^{k-1}-1)2^{\lfloor\log_2j\rfloor}$.

Let $G$ be a fct Lie group and let $U$ and $V$ be finite-dimensional real $G$-modules with $V^G=0$. A theorem of Marzantowicz, de Mattos and dos Santos estimates the covering dimension of the zero-set of a $G$-map from the unit sphere in $U$ to $V$ when $G$ is an elementary abelian $p$-group for some prime $p$ or a torus. In this note, the classical Borsuk-Ulam theorem will be used to give a refinement of their result estimating the dimension of that part of the zero-set on which an elementary abelian $p$-group $G$ acts freely or a torus $G$ acts with finite isotropy groups. The methods also provide an easy answer to a question raised in [17].

In this paper we describe the equivariant cobordism classification of smooth actions $(M^m,\phi)$ of the group $G=\mathbb{Z}_2^k$ on closed smooth $m$-dimensional manifolds $M^m$, for which the fixed point set of the action is a connected manifold of dimension $n$ and $2^k n - 2^{k-1} \leq m < 2^k n$. Here, $\mathbb{Z}_2^k$ is considered as the group generated by $k$ commuting smooth involutions defined on $M^m$. This generalizes a previous result of 2008 of the second author, who obtained this type of classification for $k=2$ and $m=4n-1$ or $m=4n-2$.

Parametrized motion planning algorithms [1] have high degree of flexibility and universality, they can work under a variety of external conditions, which are viewed as parameters and form part of the input of the algorithm. In this paper we analyse the parameterized motion planning problem in the case of sphere bundles. Our main results provide upper and lower bounds for the parametrized topological complexity; the upper bounds typically involve sectional categories of the associated fibrations and the lower bounds are given in terms of characteristic classes and their properties. We explicitly compute the parametrized topological complexity in many examples and show that it may assume arbitrarily large values.

We present a comparative study of certain invariants defined for group actions and their analogues defined for orbifolds. In particular, we prove that Fadell's equivariant category for $G$-spaces coincides with the Lusternik-Schnirelmann category for orbifolds when the group is finite.

In this paper dedicated to the memory of Edward Fadell and Sufian Husseini we show how the notion of the Lusternik-Schnirelmann relative category can be used to study a multiplicity problem for brake orbits in a potential well which is homeomorphic to the $N$-dimensional unit disk. The estimate of the relative category of the set of chords with endpoints on the $(N-1)$-unit sphere was shown to the third author by Fadell and Husseini while he was visiting the University of Wisconsin at Madison.

We are trying to look over the Lusternik-Schnirelmann theory (L-S theory, for short) and the Topological Complexity (TC, for short) as a natural extension of the L-S theory. In particular, we focus on the impact of the ideas originated from E. Fadell and S. Husseini on both theories. More precisely, we see how their ideas on a category weight and a relative category drive the L-S theory and the TC.

We prove the existence of critical points of vortex type Hamiltonians \[ H(p_1,\ldots, p_N) = \sum_{{i,j=1}\atop{i\ne j}}^N \Gamma_i\Gamma_jG(p_i,p_j)+\Psi(p_1,\dots,p_N) \]on a closed Riemannian surface $(\Sigma,g)$ which is not homeomorphic to the sphere or the projective plane. Here $G$ denotes the Green function of the Laplace-Beltramioperator in $\Sigma$, $\Psi\colon \Sigma^N\to\mathbb{R}$ may be any function of class${\mathcal C}^1$, and $\Gamma_1,\dots,\Gamma_N\in\mathbb{R}\setminus\{0\}$ are the vorticities. The Kirchhoff-Routh Hamiltonian from fluid dynamics corresponds to $\Psi(p) = -\sum\limits_{i=1}^N \Gamma_i^2h(p_i,p_i)$ where $h\colon \Sigma\times\Sigma\to\mathbb{R}$ is the regular part of the Laplace-Beltrami operator. We obtain critical points $p=(p_1,\dots,p_N)$ for arbitrary $N$ and vorticities $(\Gamma_1,\dots,\Gamma_N)$ in $\mathbb{R}^N\setminus V$ where $V$ is an explicitly given algebraic variety of codimension 1.

The index $\mathrm{ind}(\mathbf{F})$ of a fixed point class $\mathbf{F}$ is a classical invariant in the Nielsen fixed point theory. In the recent paper [13], the authors introduced a new invariant $\mathrm{ichr}(\mathbf{F})$ called the improved characteristic, and proved that $\mathrm{ind}(\mathbf{F})\leq \mathrm{ichr}(\mathbf{F})$ for all fixed point classes of $\pi_1$-injective selfmaps of connected finite graphs. In this note, we show that the two homotopy invariants mentioned above are exactly the same. Since the improved characteristic is totally determined by the endomorphism of the fundamental group, we give a group-theoretical approach to compute indices of fixed point classes of graph selfmaps. As a consequence, we give a new criterion of a fixed point, which extends the one due to Gaboriau, Jaeger, Levitt and Lustig.

A root of an $n$-valued map $\varphi \colon X \to D_n(Y)$ at $a \in Y$ is a point $x \in X$ such that $a \in \varphi(x)$. We lift the map $\varphi$ to a split $n$-valued map of finite covering spaces and its single-valued factors are defined to be the lift factors of $\varphi$. We describe the relationship between the root classes at $a$ of the lift factors and those of $\varphi$. We define the Reidemeister root number ${\rm RR} (\varphi)$ in terms of the Reidemeister root numbers of the lift factors. We prove that the Reidemeister root number is a homotopy invariant upper bound for the Nielsen root number $NR(\varphi)$, the number of essential root classes, and we characterize essentiality by means of an equivalence relation called the $\Phi$-relation. A theorem of Brooks states that a single-valued map to a closed connected manifold is root-uniform, that is, its root classes are either all essential or all inessential. It follows that if $Y$ is a closed connected manifold, then the lift factors are root-uniform and we relate this property to the root-uniformity of $\varphi$. If $X$ and $Y$ are closed connected oriented manifolds of the same dimension then, by means of the lift factors, we define an integer-valued index of a root class of $\varphi$ that is invariant under $\Phi$-relation and this implies that if its index is non-zero, then the root class is essential.

Let $G$ be a torsion-free abelian group of rank two and let $\phi$ be an endomorphism of $G$, called a rank-two solenoidal endomorphism. Then it is represented by a $2\times 2$-matrix $M_\phi$ with rational entries. The purpose of this article is to prove the following: The group, $\mathrm{coker}(\phi)$, of the cokernut of $\phi$ is finite if and only if $M_\phi$ is nonsingular, and if it is so, then we give an explicit formula for the order of $\mathrm{coker}(\phi)$, $[G:\mathrm{im}(\phi)]$, in terms of $p$-adic absolute values of the determinant of $M_\phi$. Since $G$ is abelian, the Reidemeister number of $\phi$ is equal to the order of the cokernut of $\mathrm{id}-\phi$ and, when it is finite, it is equal to the number of fixed points of the Pontryagin dual $\widehat\phi$ of $\phi$. Thereby, we solve completely the problem raised in [16] of finding the possible sequences of periodic point counts for all endomorphisms of the rank-two solenoids.

Let $f\colon M\to M$ be a self-map of a compact manifold and $n\in \mathbb{N}$. The least number of $n$-periodic points in the smooth homotopy class of $f$ may be smaller than in the continuous homotopy class. We ask: for which self-maps $f\colon M\to M$ the two minima are the same, for each prescribed multiplicity? In the study of self-maps of tori and compact Lie groups a necessary condition appears. Here we give a criterion which helps to decide whether the necessary condition is also sufficient. We apply this result to show that for self-maps of the product of the lens space $M=L(3)\times L(3)$ the necessary condition is also sufficient.

We prove a Brooks type coincidence minimization result for boundary preserving maps on compact surfaces with boundary. As an application we obtain non-boundary Wecken results for pairs of maps $f,g\colon (X,\partial X) \to (X,\partial X)$ for most surfaces $X$.

Let $\Sigma$ be a $C^3$ compact symmetric convex hypersurface in $\mathbb{R}^{8}$. We prove that when $\Sigma$ carries exactly four geometrically distinct closed characteristics, then there are at least two irrationally elliptic closed characteristics on $\Sigma$.

We study a nonlocal elliptic equation of $p$-Kirchhoff type involving the critical Sobolev exponent. First we give sufficient conditions for the $(\text{PS})$ condition to hold. Then we prove some existence and multiplicity results using tools from Morse theory, in particular, the notion of a cohomological local splitting and eigenvalues based on the Fadell-Rabinowitz cohomological index.

We consider a Neumann boundary value problem driven by the anisotropic $(p,q)$-Laplacian plus a parametric potential term. The reaction is "superlinear". We prove a global (with respect to the parameter) multiplicity result for positive solutions. Also, we show the existence of a minimal positive solution and finally, we produce a nodal solution.

We study a nonlinear, nonlocal eigenvalue problem driven by the fractional $p$-Laplacian with an indefinite, singular weight chosen in an optimal class. We prove the existence of an unbounded sequence of positive variational eigenvalues and alternative characterizations of the first and second eigenvalues. Then, by means of such characterizations, we prove strict decreasing monotonicity of such eigenvalues with respect to the weight function.

We establish the existence of positive normalized (in the $L^2$ sense) solutions to non-variational weakly coupled elliptic systems of $\ell$ equations. We consider couplings of both cooperative and competitive type. We show the problem can be formulated as an operator equation on the product of $\ell$ $L^2$-spheres and apply a degree-theoretical argument on this product to obtain existence.

In this paper, we construct multiple normalized solutions of the following from quasi-linear Schrödinger equation:$$-\Delta u-\Delta(|u|^{2})u-\mu u=|u|^{p-2}u, \quad\text{in } \mathbb{R}^N,$$subject to a mass-subcritical constraint. In order to overcome non-smoothness of the associated variational formulation we make use of the dual approach.The constructed solutions possess energies being clustered at $0$ level which makes it difficult to use existing methods for non-smooth variational problems such as the variational perturbation approach.

Assuming that there is a known (trivial) branch of solutions of a parameterized family of equations, topological bifurcation studies the topological invariants of the linearized equations along the trivial branch whose nonvanishing entails the appearance of bifurcation from the trivial branch. We introduce here some refined topological invariants for semilinear elliptic boundary value problems equivariant with respect to the action of the circle $U(1)$ allowing to improve, in this case, some previously obtained bifurcation criteria for general nonlinear elliptic boundary value problems.

Let $\mathcal{G}:= (V,E)$ be a weighted locally finite graph whose finite measure $\mu$ has a positive lower bound. Motivated by a wide interest in the current literature, in this paper we study the existence of classical solutions for a class of elliptic equations involving the $\mu$-Laplacian operator on the graph $\mathcal{G}$, whose analytic expression is given by \begin{equation*} \Delta_{\mu} u(x) := \frac{1}{\mu (x)} \sum_{y\sim x} w(x,y) (u(y)-u(x))\quad (\text{for all } x\in V), \end{equation*} where $w \colon V\times V \rightarrow [0,+\infty)$ is a weight symmetric function and the sum on the right-hand side of the above expression is taken on the neighbours vertices $x,y\in V$, that is $x\sim y$ whenever $w(x,y) > 0$. More precisely, by exploiting direct variational methods, we study problems whose simple prototype has the following form \begin{equation*} \begin{cases} -\Delta_{\mu} u(x)=\lambda f(x,u(x))& \text{for } x \in \mathop D\limits^ \circ,\\ u|_{\partial D}=0, \end{cases} \end{equation*} where $D$ is a bounded domain of $V$ such that $\mathop D\limits^ \circ\neq \emptyset$ and $\partial D\neq \emptyset$, the nonlinear term $f \colon D \times \mathbb R \rightarrow \mathbb R$ satisfy suitable structure conditions and $\lambda$ is a positive real parameter. By applying a critical point result coming out from a classical Pucci-Serrin theorem in addition to a local minimum result for differentiable functionals due to Ricceri, we are able to prove the existence of at least two solutions for the treated problems. We emphasize the crucial role played by the famous Ambrosetti-Rabinowitz growth condition along the proof of the main theorem and its consequences. Our results improve the general results obtained by A. Grigor'yan, Y. Lin, and Y. Yang in [17].

We consider a geodesic problem in a manifold endowed with a Randers-Kropina metric. This is a type of a singular Finsler metric arising both in the description of the lightlike vectors of a spacetime endowed with a causal Killing vector field and in the Zermelo's navigation problem with a wind represented by a vector field having norm not greater than one. By using Lusternik-Schnirelman theory, we prove existence of infinitely many geodesics between two given points when the manifold is not contractible. Due to the type of non-holonomic constraints that the velocity vectors must satisfy, this is achieved thanks to some recent results about the homotopy type of the set of solutions of an affine control system associated with a totally non-integrable distribution.

We investigate the existence of multiple solutions for the $(p,q)$-quasilinear elliptic problem\[\begin{cases}-\Delta_p u -\Delta_q u\ =\ g(x, u) + \varepsilon\ h(x,u)& \mbox{in } \Omega,\\u=0 & \mbox{on } \partial\Omega,\\ \end{cases}\]where $1< p< q< +\infty$, $\Omega$ is an open bounded domain of ${\mathbb R}^N$, the nonlinearity $g(x,u)$ behaves at infinity as $|u|^{q-1}$, $\varepsilon\in{\mathbb R}$ and $h\in C(\overline\Omega\times{\mathbb R},{\mathbb R})$. In spite of the possible lack of a variational structure of this problem, from suitable assumptions on $g(x,u)$ and appropriate procedures and estimates, the existence of multiple solutions can be proved for small perturbations.