One of the important theorems in homotopy theory is the Hilton Splitting Theorem which states: there is an isomorphismH = ⊕γ∈ΓHγ from them-th homotopy group of the wedge of a number of spheres to the direct sum of the m-th homotopy groups of some spheres, see [Hi]. In this paper we will construct geometrically all Hilton homomorphisms Hγ and prove a family of sharper symmetry relations between ...

As is well known, it is our manifest destiny as 21st century algebraic topologists to compute homotopy groups of spheres. This noble venture began even before the notion of homotopy was around. In 1931, Hopf was thinking about a map he had encountered in geometry from S to S and wondered whether or not it was essential. He proved that it was by considering the linking of the fibers. After Hurew...

Secondary homotopy groups supplement the structure of classical homotopy groups. They yield a 2-functor on the groupoid-enriched category of pointed spaces compatible with fiber sequences, suspensions and loop spaces. They also yield algebraic models of (n− 1)-connected (n+1)-types for n ≥ 0. Introduction The computation of homotopy groups of spheres in low degrees in [Tod62] uses heavily secon...

Let p be a prime greater than three. In the p–local stable homotopy groups of spheres, R L Cohen constructed the infinite –element n 1 2 2pnC1 2pnC2p 5.S/ of order p . In the stable homotopy group 2pnC1 2pnC2p2 3.V .1// of the Smith– Toda spectrum V .1/ , X Liu constructed an essential element $k for k 3 . Let ˇ s D j0j1ˇ s 2 ŒV .1/;S 2sp2 2s 2p denote the Spanier–Whitehead dual of the generat...

The unstable homotopy groups of spheres can be approached by the EHP spectral sequence. There are computations of the low dimensional portion of the EHP sequence by Toda [1, 2] for the 2,3-primary part, and Behrens [3], Harper [4] for the 5-primary part. There are certain stable phenomenon in the EHP sequence. In fact, there is one portion in the E1-term which are in the stable range, which mea...

We establish a bridge between homotopy groups of spheres and commutator calculus in groups, solve this manner the “dimension problem” by providing converse to Sjogren's theorem: every abelian group bounded exponent can be embedded dimension quotient group. This is proven embedding for arbitrary s , d $s,d$ torsion π ( S ) $\pi _s(S^d)$ into quotient, via result Wu. In particular, invalidates so...

In this paper, some groups Ext A (Zp, Zp) with specialized s and t are first computed by the May spectral sequence. Then we make use of the Adams spectral sequence to prove the existence of a new nontrivial family of filtration s+5 in the stable homotopy groups of spheres πpnq+(s+3)pq+(s+1)q−5S which is represented (up to a nonzero scalar) by β̃s+2b0hn ∈ Ext q+(s+3)pq+(s+1)q+s A (Zp, Zp) in the ...

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...

For a compact Lie group G with maximal torus T, Pittie and Smith showed that the flag variety G/T is always a stably framed boundary. We generalize this to the category of p-compact groups, where the geometric argument is replaced by a homotopy theoretic argument showing that the class in the stable homotopy groups of spheres represented by G/T is trivial, even G-equivariantly. As an applicatio...