Bruno P. Zimmermann
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.
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There is a topological embedding $ι:\mathbb{S}^1\to\mathbb{R}^5$ such that $π_3(\mathbb{R}^5\setminusι(\mathbb{S}^1))=0$. Therefore, no $3$-sphere can be linked with $ι(\mathbb{S}^1)$.
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.
In 2016 Levine showed that there exists a knot in a homology 3-sphere which is not smoothly concordant to any knot in the 3-sphere where one allows concordances in any smooth homology cobordism. Whether the same is true if one allows topological concordances is not known. One might hope that such an example might be detected by the powerful filtration of knot concordance introduced by Cochran-Orr-Teichner. We prove that this is not the case, demonstrating that for any knot in any homology sphere there is a knot in the 3-sphere equivalent to the original knot modulo any term of this filtration.…
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.
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.
Topologically isotopic and smoothly inequivalent 2-spheres in simply connected 4-manifolds whose complement has a prescribed fundamental group
本文提出构造单连通4-流形中拓扑同痕但光滑不等价的2-球面无限族的方法，其余集基本群可预设，涵盖有限循环群等，为首批此类纽结现象。\n
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.
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 …
The goal of this article is to establish estimates involving the Yamabe minimal volume, mixed minimal volume and some topological invariants on compact 4-manifolds. In addition, we provide topological sphere theorems for compact submanifolds of spheres and Euclidean spaces, provided that the full norm of the second fundamental form is suitably bounded.
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.
We construct a large class of pathological $n$-dimensional topological spheres in ${\mathbb R}^{n+1}$ by showing that for any Cantor set $C\subset {\mathbb R}^{n+1}$ there is a topological embedding $f:{\mathbb S}^n\to{\mathbb R}^{n+1}$ of the Sobolev class $W^{1,n}$ whose image contains the Cantor set $C$.
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 …
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 …
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))
We construct infinitely many smooth oriented 4-manifolds containing pairs of homotopic, smoothly embedded 2-spheres that are not topologically isotopic, but that are equivalent by an ambient diffeomorphism inducing the identity on homology. These examples show that Gabai's recent "Generalized" 4D Lightbulb Theorem does not generalize to arbitrary 4-manifolds. In contrast, we also show that there are smoothly embedded 2-spheres that are both equivalent and topologically isotopic, but not smoothly isotopic.
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。
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.
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。
In the paper of Montgomery, D. and Yang, C.T. [5], they discuss the de-suspension of smooth free actions of S1 on (2n+1)-dimensional homotopy spheres. In this paper we discuss the de-suspension of smooth free actions of S3 on (4n + 3)-dimensional homotopy spheres.
Classical work of Lee, Schultz, and Stolz relates the smooth transformation groups of exotic spheres to the stable homotopy groups of spheres. In this note, we apply recent progress on the latter to produce smooth circle and cyclic group actions on certain exotic spheres.
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…
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.
We use methods from the cohomology of groups to describe the finite groups which can act freely and homologically trivially on closed 3-manifolds which are rational homology spheres.
We consider the following problem: for which classes of finite groups, and in particular finite simple groups, does the minimal dimension of a faithful, smooth action on a homology sphere coincide with the minimal dimension of a faithful, linear action on a sphere? We prove that the two minimal dimensions coincide for the linear fractional groups PSL(2,p) as well as for various classes of alternating and symmetric groups. We prove analogous results also for actions on Euclidean spaces.
We use the notion of fixity for representations of finite groups to construct free and smooth actions on products of spheres. In particular we show that a finite p-group (for p>3) will act freely and smoothly on a product of two spheres if and only if it does not contain a rank 3 elementary abelian subgroup. We show that if G is a finite subgroup of U(n), acting freely on U(n)/U(k) for some k>0 and if (|G|,(n-1)!)=1, then the action propagates to a free and smooth action on a product of n-k spheres. A number of explicit examples are discussed.
We show that a finite dihedral group does not act pseudofreely and locally linearly on a 2k-dimensional sphere, if k > 1. This answers a question of R. S. Kulkarni from 1982.
If $G_1$ and $G_2$ are finite groups with periodic Tate cohomology, then $G_1\times G_2$ acts freely and smoothly on some product $S^n \times S^n$.
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.
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 …
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 …
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。
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…
We provide a simple topological derivation of a formula for the Reidemeister and the analityc torsion of spheres.
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.
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。
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。
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.
We present a new direct proof of a topological representation theorem for oriented matroids in the general rank case. Our proof is based on an earlier rank 3 version. It uses hyperline sequences and the generalized Sch{ö}nflies theorem. As an application, we show that one can read off oriented matroids from arrangements of embedded spheres of codimension one, even if wild spheres are involved.
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。
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。
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.
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。
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.
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