We study the Brunnian subgroups and the boundary Brunnian subgroups of the Artin braid groups. The general higher homotopy groups of the sphere are given by mirror symmetric elements in the quotient groups of the Artin braid groups modulo the boundary Brunnian braids, as well as given as a summand of the center of the quotient groups of Artin pure braid groups modulo boundary Brunnian braids. T...

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 ...

We describe a homotopy-theoretic approach to the theory of moduli realizations Blanc-Dwyer-Goerss, reproducing their obstructions realizing given Π-algebra as homotopy groups pointed space. Our techniques are based on ∞-category PΣ(Sph) product-preserving presheaves finite wedges positive-dimensional spheres, leading more conceptual and streamlined arguments.

This paper proves that, for any generator x ∈ Ext A (Zp, Zp), if (1L ∧ i)∗φ∗(x) ∈ Ext A (H ∗L ∧ M,Zp) is a permanent cycle in the Adams spectral sequence (ASS), then h0x ∈ Ext A (Zp, Zp) also is a permenent cycle in the ASS. As an application, the paper obtains that h0hnhm ∈ Ext3,pnq+pmq+q A (Zp, Zp) is a permanent cycle in the ASS and it converges to elements of order p in the stable homotopy ...

Ever since its introduction by J. F. Adams [8] in 1958, the spectral sequence that bears his name has been a source of fascination to homotopy theorists. By glancing at a table of its structure in low dimensions (such have been published in [7], [i0] and [27]; one can also be found in ~2) one sees not only the values of but the structural relations among the corresponding stable homotopy groups...