The naive Milnor–Witt K-theory relations in the stable motivic homotopy groups over a base

A new family in the stable homotopy groups of spheres

Let $p$ be a prime number greater than three. In this paper, we prove the existence of a new family of homotopy elements in the stable homotopy groups of spheres $pi_{ast}(S)$ which is represented by $h_nh_mtilde{beta}_{s+2}in {rm Ext}_A^{s+4, q[p^n+p^m+(s+2)p+(s+1)]+s}(mathbb{Z}_p,mathbb{Z}_p)$ up to nonzero scalar in the Adams spectral sequence, where $ngeq m+2>5$, $0leq sExt}_A^{s+2,q[(s+2)p...

Galois Equivariance and Stable Motivic Homotopy Theory

For a finite Galois extension of fields L/k with Galois group G, we study a functor from the G-equivariant stable homotopy category to the stable motivic homotopy category over k induced by the classical Galois correspondence. We show that after completing at a prime and η (the motivic Hopf map) this results in a full and faithful embedding whenever k is real closed and L = k[i]. It is a full a...

Open Problems in the Motivic Stable Homotopy Theory, I

Stable Homotopy Theory over a Fixed Base Space

A spectral sequence which may be regarded as an 'Adams spectral sequence over a fixed space 2?' has been constructed by J. F. McClendon [6] and J.-P. Meyer [8]. This note describes a generalization in which there is no need for any orientability assumptions. The construction is carried out in a suitable stable category, which may be of independent interest. An application to the enumeration of ...

Algebraic Homotopy Theory, Groups, and K-Theory

A thesis submitted in partial fulfilment of the requirements for the degree of Doctor of Philosophy in The Faculty of Graduate Studies Department of Mathematics. LetMk be the category of algebras over a unique factorization domain k, and let ind−Affk denote the category of pro-representable functions from Mk to the category E of sets. It is shown that ind−Affk is a closed model category in such...