We recall a group-theoretic description of the first non-vanishing homotopy group of a certain (n+1)-ad of spaces and show how it yields several formulae for homotopy and homology groups of specific spaces. In particular we obtain an alternative proof of J. Wu’s group-theoretic description of the homotopy groups of a 2-sphere.

The quadric Q2n is the Z-scheme defined by equation ?i=1nxiyi=z(1?z). We show that a homogeneous space for split reductive group scheme SO2n+1 over Z. known to have A1-homotopy type of motivic sphere and identification as allows us give characteristic independent affine representability statement spheres. This last observation comparison results between Chow–Witt groups, stable cohomotopy group...

We bound the volume of homotopy groups 2-local Goodwillie approximations a sphere in terms amount 2-torsion stable stems, providing Goodwillie-theoretic refinement result Burklund and Senger. At 2 k $2^k$ -excisive approximation, this is obtained by ‘multiplying answer polynomial degree $k$ ’. The main tool Behrens' Goodwillie–EHP long exact sequence.

Article history: Received 20 November 2014 Received in revised form 17 February 2015 Communicated by C.A. Weibel MSC: 14F42; 55T15; 55S10 We compute the h1-localized cohomology of the motivic Steenrod algebra over C. This serves as the input to an Adams spectral sequence that computes the motivic stable homotopy groups of the η-local motivic sphere. We compute some of the Adams differentials, a...

We revisit Freyd’s generating hypothesis in stable homotopy theory. We derive new equivalent forms of the generating hypothesis and some new consequences of it. A surprising one is that I, the Brown-Comenetz dual of the sphere and the source of many counterexamples in stable homotopy, is the cofiber of a self map of a wedge of spheres. We also show that a consequence of the generating hypothesi...