Ming Yang
In this paper, by use of techniques associated to Cobordism theory and Morse theory, we give a proof of Space-Form-Conjecture, i.e. a free action of a finite group on 3-manifold is equivalent to a linear action.
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The reader of doi:10.1016/j.topol.2010.08.006 might conjecture that Δ(\barΠ_n)/G is homotopy equivalent to a wedge of spheres for any n>=3 and any subgroup G<S_n. We disprove this by showing that Δ(\barΠ_p)/C_p is not homotopy equivalent to a wedge of spheres for any prime number p>=5.
Some homotopy coherent finite group actions on spheres
研究球面上同伦相干有限群作用，给出非严格作用的例子作为贡献，核心方法未提及。
William Browder in his paper "Surgery and the theory of differentiable transformation groups" developed surgery techniques to study semi-free actions of S1 on homotopy spheres, under the additional assumption that the fixed point set is a homotopy sphere. He used this surgery to show how to construct such actions. In this paper, I discussed a similar theory for semi-free actions of S3 on homotopy spheres. An open problem is raised at the end of the paper.
In this paper, we describe the homotopy type of the homotopy fixed point sets of $S^3$-actions on rational spheres and complex projective spaces, and provide some properties of $S^1$-actions on a general rational complex.
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.
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.
Let p be an odd prime. We prove that every rank three p-group acts freely and smoothly on a product of three spheres. To construct this action, we first prove a generalization of a theorem of L\" uck and Oliver on constructions of G-equivariant vector bundles. We also give some other applications of this generalization.
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.
Free Circle Actions on the Product of Three Spheres
本文将三个球面乘积上自由S^0-作用的轨道空间分类推广到自由S^1-作用，并建立Borsuk-Ulam型定理。
On actions of tori and quaternionic tori on products of spheres
本文研究了环面和四元数环面在球面积上的作用，证明了特定作用的轨道空间与球面同胚，并扩展了相关拓扑群作用的结果。
This paper generalizes the concept of index and co-index and some related results for free actions of G = S0 on a paracompact Hausdorff space which were introduced by Conner and Floyd. We define the index and co-index of a finitistic free G-space X, where G = Sd , d = 1 or 3 and prove that the index of X is not more than the mod 2 cohomology index of X. We observe that the index and co-index of a (2n + 1)-sphere (resp. (4n+3)-sphere) for the action of componentwise multiplication of G = S1 (resp. S3) is n. We also determine the orbit spaces of free actions of G = S3 on a finitistic space X wit…
A classical conjecture in transformation group theory states that if $G=(\bbZ/p)^r$ acts freely on a product of $k$ spheres $S^{n_1} \times ... \times S^{n_k}$, then $r\leq k$. We prove this conjecture in the case where the average of the dimensions ${n_i}$ is large compared to the differences $|n_i-n_j|$ between the dimensions.
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…
Equivariant formality of corank-one isotropy actions and products of rational spheres
本文完全刻画了满足秩差为1且左作用等变形式化的连通李群对，并修正与扩展了具有奇偶维球面乘积有理同伦型的齐性商分类。
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。
We compute the monoid of essential self-maps of of the product of two n-spheres fixing the diagonal. More generally, we consider products S x S, where S is a suspension. Essential self-maps of S x S demonstrate the interplay between the pinching action for a mapping cone and the fundamental action on homotopy classes under a space. We compute examples with non-trivial fundamental actions.
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.
We introduce a new class of poset edge labelings for locally finite lattices which we call $SB$-labelings. We prove for finite lattices which admit an $SB$-labeling that each open interval has the homotopy type of a ball or of a sphere of some dimension. Natural examples include the weak order, the Tamari lattice, and the finite distributive lattices.
Let $G$ be a group acting freely, properly discontinuously and cellularly on a finite dimensional $C$W-complex $Σ(2n)$ which has the homotopy type of the $2n$- sphere $\mathbb{S}^{2n}$. Then, this action induces an action of the group $G$ on the top cohomology of $Σ(2n)$. For the family of virtually cyclic groups, we classify all groups which act on $Σ(2n)$, the homotopy type of all possible orbit spaces and all actions on the top cohomology as well. \par Under the hypothesis that $\mbox{dim}\,Σ(2n)\leq 2n+1$, we study the groups with the virtual cohomological dimension $\mbox{vcd}\,G<\infty$ …
Let $G$ be a connected Lie group of rank one. In this paper the existence of free actions of group $G$ on spheres, real projective spaces and lens spaces has been studied. Most of the results have been obtained for finitistic spaces with cohomology ring isomorphic to the cohomology ring of these spaces. The cohomology algebra of the orbit space has been determined in the case of free $G$-action. The non-existence of $G$-equivariant maps from spheres to cohomology spheres has also been established.
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.
We investigate equivariant and invariant topological complexity of spheres endowed with smooth non-free actions of cyclic groups of prime order. We prove that semilinear $\mathbb{Z}/p$-spheres have both invariants either $2$ or $3$ and calculate exact values in all but two cases for linear actions. On the other hand, we exhibit examples which show that these invariants can be arbitrarily high in the class of smooth $\mathbb{Z}/p$-spheres.
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 $S^n \times S^n$.
This paper gives the cohomology classification of finitistic spaces X equipped with free actions of the group G = S3 and the orbit space X/G is the integral or mod 2 cohomology quaternion projective space HPn. We have proved that X is the integral or mod 2 cohomology (4n+3)-sphere or the product of 3-sphere and quaternion projective space HPn. Similar results for G = S1 actions are also discussed.
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.
Let $X$ be a finite CW-complex having mod $p$ cohomology isomorphic to a wedge of three spheres $\mathbb{S}^n\vee \mathbb{S}^m \vee \mathbb{S}^l,~ 1\leq n \leq m \leq l$. The aim of this paper is to determine the fixed point sets of actions of the cyclic group of prime order on $X.$ We also classify the orbit spaces of free actions of $G=\mathbb{Z}_p, p$ a prime or $G=\mathbb{S}^d,~d=1,3,$ on $X$ and derive the Borsuk-Ulam type results.
Moment-angle manifolds associated to neighbourly triangulations of spheres
研究显示，奇维球体的邻近三角剖分所关联的动量-角流形可同伦等价于两个球面乘积的连通和，解决了Buchstaber和Panov提出的问题。
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。
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.
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.
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。
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.
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…
Revisiting the $β_1$-action on the $3$-primary stable homotopy groups of spheres
本文通过BP合成谱和Adams-Novikov谱序列证明了β₁的5重积非零而6重积为零，并推广至其他144周期家族。
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.
For dimensions n greater than or equal to 3, we show that the space of metrics of positive scalar curvature on the n-sphere is homotopy equivalent to a subspace which takes the form of a H-space with a homotopy commutative, homotopy associative product operation. This product operation is based on the connected sum construction. We then exhibit an action of the little n-disks operad on this subspace which, using results of Boardman, Vogt and May implies that when n=3 or n is at least 5, the space of metrics of positive scalar curvature on the n-sphere is weakly homotopy equivalent to an n-fold…
By studying braid group actions on Milnor's construction of the 1-sphere, we show that the general higher homotopy group of the 3-sphere is the fixed set of the pure braid group action on certain combinatorially described group. We also give certain representation of higher differentials in the Adams spectral sequence for the homotopy groups of the 2-sphere.
We survey some results concerning finite group actions on products of spheres.
We prove that independence complex of a bipartite circle graph is homotopy equivalent to a wedge of spheres, resolving a conjecture posed by Przytycki and Silvero. As a corollary, we obtain that extreme Khovanov spectrum, $\mathcal{X}_{j_{extreme}}$ is homotopy equivalent to a wedge of spheres. In particular, the extreme Khovanov homology has no torsion.
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