A remark about dihedral group actions on spheres

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 …

结的对称类型

Types of Symmetries of Knots

本文分类了3球面中有限群作用于结的所有情况，通过几何化方法证明所有作用可转化为等距作用，并利用正交表示理论描述了三种循环和七种二面体对称家族。通过构造实例证明循环和四种二面体家族出现在素结中，其余三种仅适用于复合结。

Geometric approach towards stable homotopy groups of spheres. The Steenrod-Hopf invariant I

In this paper a geometric approach toward stable homotopy groups of spheres, based on the Pontrjagin-Thom construction is proposed. From this approach a new proof of Hopf Invariant One Theorem by J.F.Adams for all dimensions except $15,31,63,127$ is obtained. It is proved that for $n>127$ in the stable homotopy group of spheres $Π_n$ there is no elements with Hopf invariant one. The new proof is based on geometric topology methods. The Pontrjagin-Thom Theorem (in the form proposed by R.Wells) about the representation of stable homotopy groups of the real projective infinite-dimensional space (…

关于Moebius-Kantor图的注记

Remarks about the Moebius-Kantor graph

本文研究Moebius-Kantor图MK=G(8,3)的群论和拓扑性质，包括其作为三个非阿贝尔群的Cayley图、与Heawood数7和Tucker群的关系、Lefschetz数的计算、作为3-球面G的2-骨架复形的对偶、在Hopf纤维化中表示平坦Clifford环面，以及度量结构唯一确定Pauli群代数结构。

On Superintegral Kleinian Sphere Packings, Bugs, and Arithmetic Groups

We develop the notion of a Kleinian Sphere Packing, a generalization of "crystallographic" (Apollonian-like) sphere packings defined by Kontorovich-Nakamura [KN19]. Unlike crystallographic packings, Kleinian packings exist in all dimensions, as do "superintegral" such. We extend the Arithmeticity Theorem to Kleinian packings, that is, the superintegral ones come from Q-arithmetic lattices of simplest type. The same holds for more general objects we call Kleinian Bugs, in which the spheres need not be disjoint but can meet with dihedral angles pi/m for finitely many m. We settle two questions f…

关于拓扑特殊一般映射并推广单位球面典型投影的实代数映射的一些注记

Some remark on real algebraic maps which are topologically special generic maps and generalize the canonical projections of the unit spheres

本文针对推广单位球面典型投影的拓扑特殊一般映射，构造了像为光滑嵌入流形的实代数映射，作为先前构造的补充注记。