The actions of a circumference on smooth odd-dimensional spheres

The J-invariant of a smooth structure on an odd-dimensional sphere is constructed, which allows us to obtain information on smooth structures on a sphere with free action of a circumference and, in particular, yields complete information when the sphere is of dimension seven.Translated from Matematicheskie Zametki, Vol. 5, No. 4, pp. 401–408, 1969.In conclusion the author considers it to be his duty to express thanks to his scientific mentor S. P. Novikov for posing the problem and helping with the work, and also to V. M. Bykhshtaber for valuable discussions.     

Ricci solitons and odd-dimensional spheres

As a real hypersurface in a complex space, we prove two criterion inequalities for an odd-dimensional sphere in terms of the shape operator, the Reeb vector field and its associated 1-form. Also, we determine a real hypersurface in a complex space which admits a Ricci soliton with the Reeb vector field the potential vector field.     

Finite group actions on products of spheres

In this work we make some contributions to the theory of actions of finite groups on products of spheres. Suppose that the groupZ _q ^r acts freely on the product of k copies of spheres. Question: Isr≤k? We solve this question for several values ofr andk.     

Cyclic group actions on polynomial rings

We consider a cyclic group of order pⁿ, acting on a module incharacteristic p, and show how to reduce the calculation ofthe symmetric algebra to that of the exterior algebra.     

Fixity and free group actions on products of spheres

A representation G U(n) of degree n has fixity equal to the smallest integer f such that the induced action of G on U(n) /U(n-f-1) is free. Using bundle theory we show that if G admits a representation of fixity one, then it acts freely and smoothly on We use this to prove that a finite p-group (for p > 3) acts freely and smoothly on a product of two spheres if and only if it does not contain ( /p)³ as a subgroup. We use propagation methods from surgery theory to show that a representation of fixity f < n - 1 gives rise to a free action of G on a product of f + 1 spheres provided the order of G is relatively prime to (n - 1)!. We give an infinite collection of new examples of finite p-groups of rank r which act freely on a product of r spheres, hence verifying a strong form of a well-known conjecture for these groups. In addition we show that groups of fixity two act freely on a finite complex with the homotopy type of a product of three spheres. A number of examples are explicitly described.     

Automorphic forms and the distribution of points on odd-dimensional spheres

We discuss standard pairs of generators of cyclic divisionp-algebras of degreep, and prove forp=3 that any two Artin-Schreier elements are connected by a chain of standard pairs. This result has immediate applications to the presentations of such algebras.     

A sphere theorem for odd-dimensional submanifolds of spheres

We establish a topological sphere theorem from the point of view of submanifold geometry for odd-dimensional submanifolds of a unit sphere. We give examples which show that our result is optimal. Moreover, we note the assumption that the dimension is odd is essential.

     

Semifree actions on spheres

The article presents the main results concerning smooth semifree actions of finite groups on spheres. One shows that for every holomorphic function having an isolated singularity, there exists a smooth semifree action on a (possibly homotopic) sphere where the fixed-point set is the boundary for the singularity of the given function. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 5, pp. 197–207, 2005.     

p-Group actions on homology spheres

When an arbitraryp-groupG acts on a _n -homologyn-sphereX, it is proved here that the dimension functionn:S(G)(S(G) is the set of subgroups ofG), defined byn(H)=dimX ^H, (dim here is cohomological dimension) is realised by a real representation ofG, and that there is an equivariant map fromX to the sphere of this representation. A converse is also established.