Based on Morse theory for the energy functional path spaces we develop a deformation mapping of spheres into orthogonal groups. This is used to show that these are weakly homotopy equivalent, in stable range, associated Clifford representations. Given an oriented Euclidean bundle $V \to X$ rank divisible by four over finite complex $X$ derive decomposition result vector bundles sphere $\mathord...

In this paper we define new homotopy groups for topological spaces. These groups generalize the homotopy groups of Hurewicz. By the use of these groups and by improved methods we obtain new results about the ordinary homotopy groups, and also easier proofs of known results. Among other things, we can show that 76(S3) is non-trivial. 1. One of the principal problems of modern topology is to devi...

For integers n ≥ 1, k ≥ 0, the stable Kneser graph SGn,k (also called the Schrijver graph) has as vertex set the stable n-subsets of [2n + k] and as edges disjoint pairs of n-subsets, where a stable n-subset is one that does not contain any 2-subset of the form {i, i + 1} or {1, 2n + k}. The stable Kneser graphs have been an interesting object of study since the late 1970’s when A. Schrijver de...

In this paper, we investigate the simplicial groups obtained from the link groups of naive cablings on any given framed link. Our main result states that the resulting simplicial groups have the homotopy type of the loop space of a wedge of 3-spheres. This gives simplicial group models for some loop spaces using link groups.

Introduction. It is shown in [4] that there is an infinite family of semifree Zm actions on odd dimensional homotopy spheres. There is also an infinite family of semifree S actions on odd dimensional homotopy spheres (see [2], [5]). On the other hand, it is announced in [2] that there are only finitely many inequivalent semifree S actions on even dimensional homotopy spheres. Hence it is intere...

Secondary homotopy groups supplement the structure of classical homotopy groups. They yield a track functor on the track 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 secondary ...

We prove that if a simplicial complex ∆ is (nonpure) shellable, then the intersection lattice for the corresponding real coordinate subspace arrangement A∆ is homotopy equivalent to the link of the intersection of all facets of ∆. As a consequence, we show that the singularity link of A∆ is homotopy equivalent to a wedge of spheres. We also show that the complement of A∆ is homotopy equivalent ...