We will give a combinatorial description of homotopy groups of ΣK(π, 1). In particular, all of the homotopy groups of the 3-sphere are combinatorially given.

This question is rather old [3]: Which groups having periodic cohomology can act freely on some homotopy sphere? The first nontrivial restriction was supplied by Milnor in [3]. Every element of order two must lie in the center. Swan [ô] showed that any group with periodic cohomology acts freely on a c.w. complex of the homotopy type of sphere. If the group has odd order then it is metacyclic. T...

We give an explicit algorithm to compute a projective resolution of a module over the noncommutative ring based on the noncommutative Gröbner bases theory. Introduction LetK be a field and Γ be a ring overK. We generally assume that Γ has a unit, ǫ : K → Γ, as well as an augmentation η : Γ → K. For graded Γ-module M and N , Ext Γ (M,N) is defined with a projective resolution of M . This Ext-fun...

In this paper we explore the isotropic stable motivic homotopy category constructed from usual category, following work of Vishik on motives (see [28]), by killing anisotropic varieties. particular, focus cohomology operations in realm and study structure Steenrod algebra. Then, construct an version Adams spectral sequence apply it to find a complete description groups sphere spectrum, which ha...