A description of homotopy groups of the 2-dimensional sphere in terms of combinatorial group theory was discovered by the second author in 1994 and given in his thesis [25], with a published version in [27]. In this article we give a combinatorial description of the homotopy groups of k-dimensional spheres with k ≥ 3. The description is given by identifying the homotopy groups as the center of ...

‎Let $p$ be a prime with $pgeq 7$ and $q=2(p-1)$‎. ‎In this paper‎ ‎we prove the existence of a nontrivial product of‎ ‎filtration $s+4$ in the stable homotopy groups of spheres‎. ‎This nontrivial‎ ‎product is shown to be represented up to a nonzero scalar by‎ ‎the product element $widetilde{gamma}_{s}b_{n-1}g_{0}in‎ ‎{Ext}_{mathcal{A}}^{s+4,(p^n+sp^2+sp+s)q+s-3}(mathbb{Z}/p,mathbb{Z}/p)$‎ ‎in ...

We describe explicit presentations of all stable and the first nonstable homotopy groups of the unitary groups. In particular, for each n ≥ 2 we supply n homotopic maps that each represent the (n − 1)!-th power of a suitable generator of π2nSU(n) ≈ Zn!. The product of these n commuting maps is the constant map to the identity matrix. Introduction The homotopy groups of compact Lie groups have b...

‎let $p$ be a prime with $pgeq 7$ and $q=2(p-1)$‎. ‎in this paper‎ ‎we prove the existence of a nontrivial product of‎ ‎filtration $s+4$ in the stable homotopy groups of spheres‎. ‎this nontrivial‎ ‎product is shown to be represented up to a nonzero scalar by‎ ‎the product element $widetilde{gamma}_{s}b_{n-1}g_{0}in‎ ‎{ext}_{mathcal{a}}^{s+4,(p^n+sp^2+sp+s)q+s-3}(mathbb{z}/p,mathbb{z}/p)$‎ ‎in ...

We present a new technique for analyzing the v0-Bockstein spectral sequence studied by Shimomura and Yabe. Employing this technique, we derive a conceptually simpler presentation of the homotopy groups of the E(2)-local sphere at primes p ≥ 5. We identify and correct some errors in the original ShimomuraYabe calculation. We deduce the related K(2)-local homotopy groups, and discuss their manife...

The main result of this paper is that the group Diff(S×S) of diffeomorphisms S1×S2→S1×S2 has the homotopy type one would expect, namely the homotopy type of its subgroup of diffeomorphisms that take each sphere {x}×S to a sphere {y}×S by an element of the isometry group O(3) of S , where the function x,y is an isometry of S , an element of O(2) . It is not hard to see that this subgroup is home...

Determining the stable homotopy groups of spheres has been a vexing and fascinating problem in algebraic topology for the past 70 years. Recall • πn+k(S) is defined to be the set of homotopy classes of maps from S (the unit sphere in R) to S. The set has a natural abelian group structure. • πn+k(S) is known to be independent of n for n > k + 1. We denote this group by π k or πk(S ). • Here are ...