Some Examples of Free actions on Products of Spheres

Cyclic group actions and embedded spheres in 4-manifolds

In this note we derive an upper bound on the number of 2-spheres in the fixed point set of a smooth and homologically trivial cyclic group action of prime order on a simply-connected 4-manifold. This improves the a priori bound which is given by one half of the Euler characteristic of the 4-manifold. The result also shows that in some cases the 4-manifold does not admit such actions of a certain order at all or that any such action has to be pseudofree.

具有自由 $S^1$ 和 $S^3$ 作用的无穷族非常怪球面

Infinite families of very exotic spheres with free $S^1$- and $S^3$-actions

利用拓扑模形式检测具有非平凡 2-和 3-局部 Kervaire-Milnor 不变量的无穷族非常怪球面上的光滑自由 $S^1$ 和 $S^3$ 作用。

The stable free rank of symmetry of products of spheres

A well known conjecture in the theory of transformation groups states that if p is a prime and (Z/p)^r acts freely on a product of k spheres, then r is less than or equal to k. We prove this assertion if p is large compared to the dimension of the product of spheres. The argument builds on tame homotopy theory for non simply connected spaces.

在秩为一的素数幂等变的情况下，群作用于球面

Group actions on spheres with rank one prime power isotropy

本文研究了秩为二的有限群G是否能构造出同调于球面的有限G-CW复形，其等变秩为一的素数幂，当且仅当G不包含Qd(p)的p'-因子。

On topological actions of finite, non-standard groups on spheres

The standard actions of finite groups on spheres S^d are linear actions, i.e. by finite subgroups of the orthogonal group O(d+1). We prove that, in each dimension d>5, there is a finite group G which admits a faithful, topological action on a sphere S^d but is not isomorphic to a subgroup of O(d+1). The situation remains open for smooth actions.

A Borsuk-Ulam theorem for $(\mathbb Z_p)^k$-actions on products of (mod $p$) homology spheres

It is proved that for a product action of $(\mathbb Z_p)^k$ on a product of (mod p) homology spheres $N^{n_1}\times...\times N^{n_k}$, where all $n_i$'s are assumed to be odd if $p$ is odd, and any continuous map $f\colon N^{n_1}\times...\times N^{n_k}\to \mathbb R^m$ the set $A(f)=\{x\in N^{n_1}\times...\times N^{n_k}| f(x)=f(gx) \forall g\in(\mathbb Z_p)^k\}$ has dimension at least $n_1+...+n_k-m(p^k-1)$, provided $n_i\ge mp^{i-1}(p-1)$ for all $i (1\le i\le k)$. Moreover, if $n_i\ge mp^{k-1}(p-1)$ for all $i(1\le i\le k)$ then the free action $μ$ can be assumed arbitrary.

诱导 exotic smooth 两个固定点作用于球体

Inducing of exotic smooth two fixed point actions on spheres

本文通过群表示诱导技术，证明有限群在球体上光滑作用且有两个固定点时，切空间的群模结构不一定同构，首次指出某些可解非幂零 Oliver 群给出 Smith 问题的否定回答。

A note on actions of the symplectic group Sp(2g,Z) on homology spheres

The symplectic group Sp(2g,Z) is a subgroup of the linear group SL(2g,Z) and admits a faithful action on the sphere S^(2g-1), induced from its linear action on Euclidean space R^(2g). Generalizing corresponding results for linear groups, we show that, if m < 2g-1 and g > 2, any continuous action of Sp(2g,Z) on a homology m-sphere, and in particular on S^m, is trivial.

在秩为一的同源群作用下的球体群作用

Group actions on spheres with rank one isotropy

研究秩为二的有限群G，通过特定群论条件构造具有秩一p子群的G-CW复形X，其固定集为同调球面，提供新的非线性G-CW复形例子。

Actions of SL(n,Z) on homology spheres

Any continuous action of SL(n,Z), where n > 2, on a r-dimensional mod 2 homology sphere factors through a finite group action if r < n - 1. In particular, any continuous action of SL(n+2,Z) on the n-dimensional sphere factors through a finite group action.

Quotients of Spheres By Linear Actions of Tori

We consider quotients of spheres by linear actions of real tori. To each quotient we associate a matroid built out of a diagonalization of the torus action. We find the integral homology groups of the resulting quotient spaces in terms of the Tutte polynomial of the matroid. We also find the homotopy type and homology of the singular space of such an action. Lastly, we consider the circumstances under which the orbit space is a manifold or, more specifically, a (homology) sphere.

Matrix group actions on product of spheres and Zimmer's program

Let SL(n,Z) be the special linear group over integers and $M =S^r_1 \times S^r_2,T^r_1 \times S^r_2$ , or $T^r_0 \times S^r_1 \times S^r_2$, products of spheres and tori. We prove that any group action of SL(n,Z) on $M^r$ by diffeomorphims or piecewise linear homeomorphisms is trivial if $r<n-1$. This confirms a conjecture on Zimmer's program for these manifolds.

Period three actions on the three-sphere

We show that a free period three action on the three-sphere is standard, i.e. the quotient is homeomorphic to a lens space. We use a minimax argument involving sweepouts.

同伦等价于球面乘积的光滑流形

Smooth manifolds homotopy equivalent to products of spheres

分类了同伦等价于特定球面乘积的光滑闭定向流形，通过球面丛、Milnor 管道连接和 Novikov 球面丛等显式构造实现了正常不变映射的像。

Nonexistence of Smooth Effective One Fixed Point Actions of Finite Oliver Groups on Low-dimensional Spheres

According to the work of Laitinen, Morimoto, Oliver and Pawałowski, a finite group $G$ has a smooth effective one fixed point action on some sphere if and only if $G$ is an Oliver group. For some finite Oliver groups $G$ of order up to $216$, and for $G=A_5\times C_n$ for $n=3,5,7$, we present a strategy of excluding of smooth effective one fixed point $G$-actions on low-dimensional spheres.

Cyclic Branched Coverings of Brieskorn Spheres Bounding Acyclic 4-Manifolds

We show that standard cyclic actions on Brieskorn homology 3-spheres with non-empty fixed set do not extend smoothly to any contractible smooth 4-manifold it may bound. The quotient of any such extension would be an acyclic $4$-manifold with boundary a related Brieskorn homology sphere. We briefly discuss well known invariants of homology spheres that obstruct acyclic bounding 4-manifolds, and then use a method based on equivariant Yang-Mills moduli spaces to rule out extensions of the actions.

Pseudofree group actions on spheres

R. S. Kulkarni showed that a finite group acting pseudofreely, but not freely, preserving orientation, on an even-dimensional sphere (or suitable sphere-like space) is either a periodic group acting semifreely with two fixed points, a dihedral group acting with three singular orbits, or one of the polyhedral groups, occurring only in dimension 2. It is shown here that the dihedral group does not act pseudofreely and locally linearly on an actual n-sphere when n is congruent to 0 mod 4. The possibility of such an action when n is congruent to 2 mod 4 and n>2 remains open. Orientation-reversing …

Examples of Free Actions on Products of Spheres

We construct a non-abelian extension $Γ$ of $S^1$ by $\cy 3 \times \cy 3$, and prove that $Γ$ acts freely and smoothly on $S^{5} \times S^{5}$. This gives new actions on $S^{5} \times S^{5}$ for an infinite family $\cP$ of finite 3-groups. We also show that any finite odd order subgroup of the exceptional Lie group $G_2$ admits a free smooth action on $S^{11}\times S^{11}$. This gives new actions on $S^{11}\times S^{11}$ for an infinite family $\cE $ of finite groups. We explain the significance of these families $\cP $, $\cE $ for the general existence problem, and correct some mistakes in th…

Homological Stability For The Moduli Spaces of Products of Spheres

We prove a homological stability theorem for moduli spaces of high-dimensional, highly connected manifolds, with respect to forming the connected sum with the product of spheres $S^{p}\times S^{q}$, for $p < q < 2p - 2$. This result is analogous to recent results of S. Galatius and O. Randal-Williams regarding the homological stability for the moduli spaces of manifolds of dimension $2n > 4$, with respect to forming connected sums with $S^{n}\times S^{n}$.

纯辫群自同构在二维球面同伦群上的作用

Action of automorphisms of pure braid groups on homotopy groups of two-sphere

本文研究了纯辫群相关Delta-群结构的Moore复形，证明循环群与边界群在纯辫群所有自同构下不变，从而推广了前人结果，并得出纯辫群自同构在二维球面同伦群上存在诱导作用且计算了少量股数的作用。

Cyclic Group Actions on Contractible 4-Manifolds

There are known infinite families of Brieskorn homology 3-spheres which can be realized as boundaries of smooth contractible 4-manifolds. In this paper we show that free periodic actions on these Brieskorn spheres do not extend smoothly over a contractible 4-manifold. We give a new infinite family of examples in which the actions extend locally linearly but not smoothly.

SL(n,Z) cannot act on small spheres

The group SL(n,Z) admits a smooth faithful action on the (n-1)-sphere S^(n-1), induced from its linear action on euclidean space R^n. We show that, if m < n-1 and n > 2, any smooth action of SL(n,Z) on a mod 2 homology m-sphere, and in particular on the m-sphere S^m, is trivial.

Extending Smooth Cyclic Group Actions on the Poincare Homology Sphere

Let $X_0$ denote a compact, simply-connected smooth $4$-manifold with boundary the Poincaré homology $3$-sphere $Σ(2,3,5)$ and with even negative definite intersection form $Q_{X_0}=E_8$. We show that free $\mathbb{Z}/p$ actions on $Σ(2,3,5)$ do not extend to smooth actions on $X_0$ with isolated fixed points for any prime $p>7$. The approach is to study the equivariant version of the Yang-Mills instanton-one moduli space for $4$-manifolds with cylindrical ends. As an application we show that for $p>7$ a smooth $\mathbb{Z}/p$ action on $\#^8 S^2 \times S^2$ with isolated fixed points does not …

On finite simple groups acting on homology spheres with small fixed point sets

A finite nonabelian simple group does not admit a free action on a homology sphere, and the only finite simple group which acts on a homology sphere with at most 0-dimensional fixed point sets ("pseudofree action") is the alternating group A_5 acting on the 2-sphere. Our first main theorem is the finiteness result that there are only finitely many finite simple groups which admit a smooth action on a homology sphere with at most d-dimensional fixed points sets, for a fixed d. We then go on proving that the finite simple groups acting on a homology sphere with at most 1-dimensional fixed point …

On minimal actions of linear fractional and finite simple groups on homology spheres

This paper has been withdrawn since it is superated by the latest version of arXiv:0812.1139 (this is the version which will appear in Rend. Circ. Mat. Palermo; it is a strengthening and elaboration of a paper published in Math. Proc. Camb. Phil. Soc. 144, 397-401 (2008))

不动点集具有奇数欧拉示性数的同调六维球面上的光滑有限群作用

Smooth finite group actions on homology six-spheres with odd euler chracteristic fixed point sets

该研究针对整同调六维球面上光滑有效的有限群作用，在不动点集欧拉示性数为奇数的条件下，确定了作用群的三类同构形式，并证明不动点集恰含一个点，推进了变换群相关领域的研究。

映射类群$\boldsymbol{\rm Mod}(S_g)$在$(2g-1)$维球面上的忠实作用

A Faithful Action of the Mapping Class Group $\mathrm{Mod}(S_g)$ on the $(2g-1)$-Sphere

该研究证明当$g\boldsymbol{\rm Mod}(S_g)$时，映射类群$\boldsymbol{\rm Mod}(S_g)$可通过等变爆破与托雷利群双序性，在$(2g-1)$维球面上实现保定向同胚的忠实作用。

On finite simple groups acting on homology spheres

It is a consequence of the classical Jordan bound for finite subgroups of linear groups that in each dimension n there are only finitely many finite simple groups which admit a faithful, linear action on the n-sphere. In the present paper we prove an analogue for smooth actions on arbitrary homology n-spheres: in each dimension n there are only finitely many finite simple groups which admit a faithful, smooth action on some homology sphere of dimension n, and in particular on the n-sphere. We discuss also the finite simple groups which admit an action on a homology sphere of dimension 3, 4 or …

Exclusions of smooth actions on spheres of the non-split extension of $C_2$ by $SL(2,5)$

There are four groups $G$ fitting into a short exact sequence $ 1\rightarrow SL(2,5)\rightarrow G\rightarrow C_2\rightarrow 1, $ where $SL(2,5)$ is the special linear group of $(2\times 2)$-matrices with entries in the field of five elements. Except for the direct product of $SL(2,5)$ and $C_2$, there are two other semidirect products of these two groups and just one non-semidirect product $SL(2,5).C_2$, considered in this paper. It is known that each finite nonsolvable group can act on spheres with arbitrary positive number of fixed points. Clearly, $SL(2,5).C_2$ is a nonsolvable group. Moreo…

Actions of arithmetic groups on homology spheres and acyclic homology manifolds

We establish lower bounds on the dimensions in which arithmetic groups with torsion can act on acyclic manifolds and homology spheres. The bounds rely on the existence of elementary p-groups in the groups concerned. In some cases, including Sp(2n,Z), the bounds we obtain are sharp: if X is a generalized Z/3-homology sphere of dimension less than 2n-1 or a Z/3-acyclic Z/3-homology manifold of dimension less than 2n, and if n \geq 3, then any action of Sp(2n,Z) by homeomorphisms on X is trivial; if n = 2, then every action of Sp(2n,Z) on X factors through the abelianization of Sp(4,Z), which is …

Actions of automorphism groups of free groups on homology spheres and acyclic manifolds

For n at least 3, let SAut(F_n) denote the unique subgroup of index two in the automorphism group of a free group. The standard linear action of SL(n,Z) on R^n induces non-trivial actions of SAut(F_n) on R^n and on S^{n-1}. We prove that SAut(F_n) admits no non-trivial actions by homeomorphisms on acyclic manifolds or spheres of smaller dimension. Indeed, SAut(F_n) cannot act non-trivially on any generalized Z_2-homology sphere of dimension less than n-1, nor on any Z_2-acyclic Z_2-homology manifold of dimension less than n. It follows that SL(n,Z) cannot act non-trivially on such spaces eithe…

Sphere systems, standard form, and cores of products of trees

We introduce the concept of a standard form for two embedded maximal sphere systems in the doubled handlebody, and we prove an existence and uniqueness result. In particular, we show that pairs of maximal sphere systems in the doubled handlebody (up to homeomorphism) bijectively correspond to square complexes satisfying a set of properties. This work is a variant on Hatcher's normal form.

Explicit parallelizations on products of spheres

The aim of this paper is to write an explicit orthonormal parallelization for all parallelizable products of spheres, using an explicit isomorphism with a trivial vector bundle.

Anosov diffeomorphisms of products I. Negative curvature and rational homology spheres

We show that various classes of products of manifolds do not support transitive Anosov diffeomorphisms. Exploiting the Ruelle-Sullivan cohomology class, we prove that the product of a negatively curved manifold with a rational homology sphere does not support transitive Anosov diffeomorphisms. We extend this result to products of finitely many negatively curved manifolds of dimensions at least three with a rational homology sphere that has vanishing simplicial volume. As an application of this study, we obtain new examples of manifolds that do not support transitive Anosov diffeomorphisms, inc…

Any finite group acts freely and homologically trivially on a product of spheres

The main theorem is that if K is a finite CW complex with finite fundamental group G and universal cover homotopy equivalent to a product of spheres X, then G acts smoothly and freely on X x S^n for any n greater than or equal to the dimension of X. If the G-action on the universal cover of K is homologically trivial then so is the action on X x S^n. Unlu and Yalcin recently showed that for every finite group G, there is a finite CW complex K with fundamental group G which acts homologicially trivially on the universal cover of K. Thus every finite group acts smoothly, freely, and homologicall…

On the character variety of the four-holed sphere

We study the (relative) SL(2,C) character varieties of the four-holed sphere and the action of the mapping class group on it. We describe a domain of discontinuity for this action, and, in the case of real characters, show that this domain of discontinuity may be non-empty on the components where the relative euler class is non-maximal.

作用于边界球面的双曲群的稳定性

Stability for hyperbolic groups acting on boundary spheres

本文证明，当双曲群的 Gromov 边界为拓扑 n-球面时，其边界同胚作用具有动力学拓扑稳定性，任意邻近作用均与标准作用半共轭。

Almost linear actions by finite groups on S^{2n-1}

A free action of a finite group on an odd-dimensional sphere is said to be almost linear if the action restricted to each cyclic or 2-hyperelementary subgroup is conjugate to a free linear action. We begin this survey paper by reviewing the status of almost linear actions on the 3-sphere. We then discuss almost linear actions on higher-dimensional spheres, paying special attention to the groups SL_2(p), and relate such actions to surgery invariants. Finally, we discuss geometric structures on space forms or, more generally, on manifolds whose fundamental group has periodic cohomology. The geom…

The minimal dimension of a sphere with an equivariant embedding of the bouquet of $g$ circles is $2g-1$

To embed the bouquet of $g$ circles $B_g$ into the $n$-sphere $S^n$ so that its full symmetry group action extends to an orthogonal actions on $S^n$, the minimal $n$ is $2g-1$. This answers a question raised by B. Zimmermann.

Bridge number and tangle products

We show that essential punctured spheres in the complement of links with distance three bridge spheres have bounded complexity. We define the operation of tangle product, a generalization of both connected sum and Conway product. Finally, we use the bounded complexity of essential punctured spheres to show that the bridge number of a tangle product is at least the sum of the bridge numbers of the two factor links up to a constant error.

Actions of homeomorphism groups of manifolds admitting a nontrivial finite free action

In this paper, we study the action of $\text{Homeo}_0(M)$, the identity component of the group of homeomorphisms of an $n$-dimensional manifold $M$ with an $\mathbb{F}_p$-free action, on another manifold $N$ of dimension $n+k<2n$. We prove that if $M$ is not an $\mathbb{F}_p$-homology sphere, then $N\cong M\times K$ for a homology manifold $K$ such that the action of $\text{Homeo}_0(M)$ on $M$ is standard and on $K$ is trivial. In particular, for $M=S^n$ a sphere, any nontrivial action is a generalization of the "coning-off" construction.

Surfaces, braids, Stokes matrices, and points on spheres

Moduli spaces of points on $n$-spheres carry natural actions of braid groups. For $n=0$, $1$, and $3$, we prove that these symmetries extend to actions of mapping class groups of positive genus surfaces, by establishing exceptional isomorphisms with certain moduli of local systems. This relies on the existence of group structure for spheres in these dimensions. We also use the connection to demonstrate that the space of rank 4 Stokes matrices with fixed Coxeter invariant of nonzero discriminant contains only finitely many integral braid group orbits.