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...

Abstract We examine the relation between gauge groups of $$\mathrm {SU}(n)$$ SU ( n ) - and {PU}(n)$$ PU -bundles over $$S^{2i}$$ S 2 i , with $$...

In this paper, using Waldhausen’s standard degree 1-maps between handlebodies, we show that the Casson invariant of oriented integral homology 3-spheres is a homotopy invariant. This provides a map from homotopy classes of oriented integral homology 3-spheres to the integers. It also gives an obstruction to finding orientationreversing self homotopy equivalences of such 3-manifolds.

These are notes for a five lecture series intended to uncover large-scale phenomena in the homotopy groups of spheres using the Adams-Novikov Spectral Sequence. The lectures were given in Strasbourg, May 7–11, 2007.

This paper proves the existence of a new nontrivial family of filtration three in the stable homotopy group of spheres p2ð p nþpÞð p 1Þ 3S which is of order p and is represented by (b0hn þ h1bn 1Þ in the E 3; 2ð p þpÞð p 1Þ 2 -term of the Adams spectral sequence, where pb 5 is a prime and nb 3.

The purpose of this article is to describe connections between the loop space of the 2-sphere, Artin’s braid groups, a choice of simplicial group whose homotopy groups are given by modules called Lie(n), as well as work of Milnor [25], and Habegger-Lin [17, 22] on ”homotopy string links”. The current article exploits Lie algebras associated to Vassiliev invariants in work of T. Kohno [19, 20], ...