A classic result of Swan states that a finite group G acts freely on a finite homotopy sphere if and only if every abelian subgroup of G is cyclic. Following this result, Benson and Carlson conjectured that a finite group G acts freely on a finite complex with the homotopy type of n spheres if the rank of G is less than or equal to n. Recently, Adem and Smith have shown that every rank two fini...

X a finite complex. This has brought us into contact with many new and interesting questions concerning the readabi l i ty of certain cyclic S2*-modules as complex bordism modules of finite complexes. Needless to say it has also involved us in the stable homotopy groups of spheres, particularly with the occurrence of certain types of infinite families of elements in the ^-component (see the dis...

The $2$-primary homotopy $\beta$-family, defined as the collection of Mahowald invariants $2^i$, $i \geq 1$, is an infinite periodic elements in stable groups spheres. In this paper, we calculate $\mathit{tmf}$-based approximations to family. Our calculations combine analysis Atiyah-Hirzebruch spectral sequence for Tate construction $\mathit{tmf}$ with trivial $C_2$-action and Behrens' filtered...

We will give a combinatorial description of the homotopy groups for the suspension ofK(π, 1) and wedges of 2-spheres. In particular, all of the homotopy groups of the 2-sphere are given as the centers of certain combinatorially described groups.

Bott’s periodicity theorem is applied to calculate higher-order differentials of the Adams spectral sequence of homotopy groups π∗(SO). The resulting formulas are used to find higher-order differentials of the Adams spectral sequence of homotopy groups of spheres. DOI: 10.1134/S0001434608110126

We discuss the existence of Gribov ambiguities in SU(m)×U(1) gauge theories over the n−spheres. We achieve our goal by showing that there is exactly one conjugacy class of groups of gauge transformations for the theories given above. This implies that these transformation groups are conjugate to the ones of the trivial SU(m)×U(1) fiber bundles over the n−spheres. By using properties of the spac...