We prove that, for p an odd prime, every finite p-group of rank 3 acts freely on a finite complex X homotopy equivalent to a product of three spheres.

2010-11-04math.AT

Constructing homologically trivial actions on products of spheres

We prove that if a finite group $G$ has a representation with fixity $f$, then it acts freely and homologically trivially on a finite CW-complex homotopy equivalent to a product of $f+1$ spheres. This shows, in particular, that every finite group acts freely and homologically trivially on some finite CW-complex homotopy equivalent to a product of spheres.

2012-03-27math.AT

Fusion systems and constructing free actions on products of spheres

We show that every rank two $p$-group acts freely and smoothly on a product of two spheres. This follows from a more general construction: given a smooth action of a finite group $G$ on a manifold $M$, we construct a smooth free action on $M \times \bbS ^{n_1} \times \dots \times \bbS ^{n_k}$ when the set of isotropy subgroups of the $G$-action on $M$ can be associated to a fusion system satisfying certain properties. Another consequence of this construction is that if $G$ is an (almost) extra-special $p$-group of rank $r$, then it acts freely and smoothly on a product of $r$ spheres.

2010-06-30math.AT

Constructing the determinant sphere using a Tate twist

Following an idea of Hopkins, we construct a model of the determinant sphere $S\langle det \rangle$ in the category of $K(n)$-local spectra. To do this, we build a spectrum which we call the Tate sphere $S(1)$. This is a $p$-complete sphere with a natural continuous action of $\mathbb{Z}_p^\times$. The Tate sphere inherits an action of $\mathbb{G}_n$ via the determinant and smashing Morava $E$-theory with $S(1)$ has the effect of twisting the action of $\mathbb{G}_n$. A large part of this paper consists of analyzing continuous $\mathbb{G}_n$-actions and their homotopy fixed points in the setup…

2021-09-13math.AT

On the constructions of free and locally standard Z_2-torus actions on Manifolds

We introduce an elementary way of constructing principal (Z_2)^m-bundles over compact smooth manifolds. In addition, we will define a general notion of locally standard (Z_2)^m-actions on closed manifolds for all m>0, and then give a general way to construct all such (Z_2)^m-actions from the orbit space. Some related topology problems are also studied.

2010-05-09math.ATmath.GT

Nonlinearity, Proper Actions and Equivariant Stable Cohomotopy

In this article we extend the classical definitions of equivariant cohomotopy theory to the setting of proper actions of Lie groups. We combine methods originally developed in the analysis of nonlinear differential equations, mainly in connection with Leray-Schauder theory, and on the other hand from developments of equivariant $K$-Theory by N.C. Phillips. We prove the correspondence with a previous construction of W. Lück by constructing an index. As an illustration of these methods, we introduce a Burnside ring defined in analytical terms. With this definition, we extend a weak version of th…

2013-02-07math.ATmath.GT

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Topologically unique maximal elementary Abelian group actions on compact oriented surfaces

We determine all finite maximal elementary abelian group actions on compact oriented surfaces of genus $σ\geq 2$ which are unique up to topological equivalence. For certain special classes of such actions, we determine group extensions which also define unique actions. In addition, we explore in detail one of the families of such surfaces considered as compact Riemann surfaces and tackle the classical problem of constructing defining equations.

2007-12-04math.AGmath.AT

Constructing homotopy equivalences of chain complexes of free ZG-modules

We describe a general method for algorithmic construction of G-equivariant chain homotopy equivalences from non-equivariant ones. As a consequence, we obtain an algorithm for computing equivariant (co)homology of Eilenberg-MacLane spaces K(pi,n), where pi is a finitely generated ZG-module. This result will be used in a forthcoming paper to construct equivariant Postnikov towers of simply connected spaces with free actions of a finite group G and further to compute stable equivariant homotopy classes of maps between such spaces. The methods of this paper work for modules over any non-negatively…

2013-04-24math.AT

Obstructions for constructing equivariant fibrations

Let $G$ be a finite group and $\mathcal{H}$ be a family of subgroups of $G$ which is closed under conjugation and taking subgroups. Let $B$ be a $G$-$CW$-complex whose isotropy subgroups are in $\mathcal{H}$ and let $\mathcal{F}= \{F_H\}_{H \in \mathcal{H}}$ be a compatible family of $H$-spaces. A $G$-fibration over $B$ with fiber $\mathcal{F}= \{F_H\}_{H \in \mathcal{H}}$ is a $G$-equivariant fibration $p:E \rightarrow B$ where $p^{-1}(b)$ is $G_b$-homotopy equivalent to $F_{G_b}$ for each $b \in B$. In this paper, we develop an obstruction theory for constructing $G$-fibrations with fiber $\…

2012-05-02math.AT

Realizing homotopy group actions

For any finite group $G$, we define the notion of a Bredon homotopy action of $G$, modelled on the diagram of fixed point sets $(X_H)_{H\leq G}$ for a $G$-space $X$, together with a pointed homotopy action of the group $N_{G}H/H$ on $X^{H}/(\bigcup_{H<K} X^{K})$. We then describe a procedure for constructing a suitable diagram $\underline{X}:O_G^{op}\to Top$ from this data, by solving a sequence of elementary lifting problems. If successful, we obtain a $G$-space $X'$ realizing the given homotopy information, determined up to Bredon $G$-homotopy type. Such lifting methods may also be used to u…

2014-02-13math.AT

Equivariant K-theory for proper actions of non-compact Lie groups

Generalizing a construction of Wolfgang Lück and Bob Oliver, we define a good equivariant cohomology theory on the category of proper G-CW complexes when G is an arbitrary Lie group (possibly non-compact). This is done by constructing an appropriate classifying space that arises from a Gamma-G-space. It is proven that this theory effectively generalizes Segal's equivariant K-theory when G is compact.

2010-10-30math.ATmath.CTmath.KT

The Homology Groups of a Partial Trace Monoid Action

The aim of this paper is to investigate the homology groups of mathematical models of concurrency. We study the Baues-Wirsching homology groups of a small category associated with a partial monoid action on a set. We prove that these groups can be reduced to the Leech homology groups of the monoid. For a trace monoid with an action on a set, we will build a cubical complex of free Abelian groups with homology groups isomorphic to the integral homology groups of the action category. It allows us to solve the problem posed by the author in 2004 of the constructing an algorithm for computing homo…

2011-11-03cs.MAmath.AT

Lectures on the Cohomology of Finite Groups

In this paper we first survey some basic results in the cohomology of finite groups, and then discuss recent work on constructing free actions of finite groups on products of spheres.

2006-09-28math.AT

克隆系统与作用operad

Cloning systems and action operads

本文针对公理化构造类Thompson群的两类核心工具，证明作用operad等价于带特定结构的双侧克隆系统，并阐明其与交叉区间群、积范畴的关联。

2025-11-17math.ATmath.GR

Equivariant homotopy and deformations of diffeomorphisms

We present a way of constructing and deforming diffeomorphisms of manifolds endowed with a Lie group action. This is applied to the study of exotic diffeomorphisms and involutions of spheres and to the equivariant homotopy of Lie groups.

2007-08-12math.ATmath.DG