arXiv research

We study Cohen-Macaulay actions, a class of torus actions on manifolds, possibly without fixed points, which generalizes and has analogous properties as equivariantly formal actions. Their equivariant cohomology algebras are computable in the sense that a Chang-Skjelbred Lemma, and its stronger version, the exactness o…

Let T be a torus. We present an exact sequence relating the relative equivariant cohomologies of the skeletons of an equivariantly formal T-space. This sequence, which goes back to Atiyah and Bredon, generalizes the so-called Chang-Skjelbred lemma. As coefficients, we allow prime fields and subrings of the rationals, i…

Expands Bredon's trick for applications in geometry and topology.

Compute Bredon homology for a specific type of Artin groups.

New triangulations of octonionic projective plane found with restricted symmetry groups.

In dimension 3 and above, Bredon cohomology gives an acurate purely algebraic description of the minimal dimension of the classifying space for actions of a group with stabilisers in any given family of subgroups. For some Coxeter groups and the family of virtually cyclic subgroups we show that the Bredon cohomological…

Bredon's trick helps extend local properties to global topological spaces.

A remarkable result of Gersten states that the class of hyperbolic groups of cohomological dimension $2$ is closed under taking finitely presented (or more generally $FP_2$) subgroups. We prove the analogous result for relatively hyperbolic groups of Bredon cohomological dimension $2$ with respect to the family of para…

New cohomology theory shows compact Lie group actions are Morita invariant.

Let $G$ be a group that admits a cocompact classifying space for proper actions $X$. We derive a formula for the Bredon cohomological dimension for proper actions of $G$ in terms of the relative cohomology with compact support of certain pairs of subcomplexes of $X$. We use this formula to compute the Bredon cohomologi…

We generalize a class of groups introduced by Herbert Abels to produce examples of virtually torsion free groups that have Bredon-finiteness length m-1 and classical finiteness length n-1 for all 0 < m <= n. The proof illustrates how Bredon-finiteness properties can be verified using geometric methods and a version of …

Let G be the fundamental group of a connected, closed, orientable 3-manifold. We explicitly compute its virtually cyclic geometric dimension. Among the tools we use are the prime and JSJ decompositions of M, several push-out type constructions, as well as some Bredon cohomology computations.

Bredon has constructed a 2-dimensional compact cohomology manifold which is not homologically locally connected, with respect to the singular homology. In the present paper we construct infinitely many such examples (which are in addition metrizable spaces) in all remaining dimensions $n \ge 3$.

Let $G$ be a matrix group. Topological $G$-manifolds with Palais-proper action have the $G$-homotopy type of countable $G$-CW complexes (3.2). This generalizes E Elfving's dissertation theorem for locally linear $G$-manifolds (1996). Also we improve the Bredon--Floyd theorem from compact groups $G$ (1960).

Study fixed-point sets of $S^{1}$-actions on quaternionic manifolds.

We continue the work of [10], studying properties of digital images determined by fixed point invariants. We introduce pointed versions of invariants that were introduced in [10]. We introduce freezing sets and cold sets to show how the existence of a fixed point set for a continuous self-map restricts the map on the c…

We study the fixed point set in the ideal boundary of a parabolic isometry of a proper CAT(0)-space. We show that the radius of the fixed point set is at most pi/2, and study its centers. As a consequence, we prove that the set of fixed points is contractible with respect to the Tits topology.

New tools for constructing fixed point sets in digital topology.

Fixed point sets of certain group actions are contractible.

Study circle actions on unitary manifolds with discrete fixed points.

Paper extends Brouwer Fixed Point Theorem with amiable and almost amiable fixed sets.

In this paper, we study a circle action on a compact oriented manifold with a discrete fixed point set. The fixed point data consists of the weights of the $S^1$-representations at the fixed points. We prove various results and properties of the action, in terms of the fixed point data. We show that the manifold can be…

The paper proves group actions on spheres with odd fixed points.

Formula for fixed points on noncompact spaces.

In this article, we classify all involutions on S^6 with 3-dimensional fixed point set. In particular, we discuss the relation between the classification of involutions with fixed point set a knotted 3-sphere and the classification of free involutions on homotopy CP^3's.

We prove the equivariant holomorphic Morse inequalities for a holomorphic torus action on a holomorphic vector bundle over a compact Kahler manifold when the fixed-point set is not necessarily discrete. Such inequalities bound the twisted Dolbeault cohomologies of the Kahler manifold in terms of those of the fixed-poin…

The paper classifies involutions on S^4, proving linearities under certain conditions.

New proof for 6D symplectic manifold with 4 fixed points.

The paper classifies circle actions on 6D manifolds with isolated fixed points.

A fixed point theorem is proved for inverse transducers, leading to an automata-theoretic proof of the fixed point subgroup of an endomorphism of a finitely generated virtually free group being finitely generated. If the endomorphism is uniformly continuous for the hyperbolic metric, it is proved that the set of regula…

Macbeath gave a formula for the number of fixed points for each non-identity element of a cyclic group of automorphisms of a compact Riemann surface in terms of the universal covering transformation group of the cyclic group. We observe that this formula generalizes to determine the fixed-point set of each non-identity…

Let $G$ be a compact Lie group acting isometrically on a compact Riemannian manifold $M$ with nonempty fixed point set $M^G$. We say that $M$ is fixed-point homogeneous if $G$ acts transitively on a normal sphere to some component of $M^G$. Fixed-point homogeneous manifolds with positive sectional curvature have been c…

Study finds critical points in perimeter functional for fixed volume sets.

New method produces reflections with nonseparating fixed points.

Study genus-three Torelli maps and their fixed point sets in representation varieties.

Let $G$ be a compact Lie group acting effectively by isometries on a compact Riemannian manifold $M$ with nonempty fixed point set $Fix(M,G)$. We say that the action is \emph{fixed point homogeneous} if $G$ acts transitively on a normal sphere to some component of $Fix(M,G)$, equivalently, if $Fix(M,G)$ has codimension…

Study circle actions on manifolds with 3 fixed points, finding dimension constraints and unique structures.

Improved bounds on $\Z_2$-torus actions on positively curved manifolds.

Klein bottle embeds into specific lens spaces.

Characterizes the Legendre involution on generic frontals.

In this paper, we investigate the fixed-point set of an element of a CAT(0) group in its boundary. Suppose that a group $G$ acts geometrically on a CAT(0) space $X$. Let $g\in G$ and let $\mathcal{F}_g$ be the fixed-point set of $g$ in the boundary $\partial X$. Then we show that $\mathcal{F}_g=L(Z_g)$, where $Z_g$ is …

Belief propagation (BP) is an iterative method to perform approximate inference on arbitrary graphical models. Whether BP converges and if the solution is a unique fixed point depends on both the structure and the parametrization of the model. To understand this dependence it is interesting to find \emph{all} fixed poi…

Fixed points of nonnegative neural networks are analyzed using fixed point theory.

The study proves fixed-point theorems for groups acting on CAT(0) spaces.

Study on cold and freezing sets in digital images.

We obtain a general lower bound for the number of fixed points of a circle action on a compact almost complex manifold $M$ of dimension $2n$ with nonempty fixed point set, provided the Chern number $c_1c_{n-1}[M]$ vanishes. The proof combines techniques originating in equivariant K-theory with celebrated number theory …

We present proofs of basic results, including those developed by Harold Bell, for the plane fixed point problem: does every map of a non-separating plane continuum have a fixed point? Some of these results had been announced much earlier by Bell but without accessible proofs. We define the concept of the variation of a…

In this thesis we study the geometry of the fixed point set $Σ$ of a smooth mapping $Φ: M\to M$ on a smooth compact Riemannian manifold $M$ without boundary by computing the asymptotic expansion of the deformed heat trace $\Trace Φ\exp(tΔ)$ of the Laplace operator $Δ$ on $M$. We assume that the fixed point set $Σ$ is a…