Motivic stable homotopy and the stable 51 and 52 stems

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...

A Comparison of Motivic and Classical Stable Homotopy Theories

Let k be an algebraically closed field of characteristic zero. Let c : SH → SH(k) be the functor induced by sending a space to the constant presheaf of spaces on Sm/k. We show that c is fully faithful. In particular, c induces an isomorphism c∗ : πn(E)→ Πn,0(c(E)) for all spectra E. Fix an embedding σ : k → C and let ReB : SH(k)→ SH be the associated Betti realization. Let Sk be the motivic sph...

Open Problems in the Motivic Stable Homotopy Theory, I

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...

Equivariant Stable Stems

Let S(r) denote the w-sphere with a linear involution having a fixed point set of codimension r, where O ^ r ^ w . We pick some fixed point as a base point and consider the set [S(r); S(t)] of base point preserving equivariant homotopy classes of maps from S(r) to S(t). This has a natural group structure for n—r^l and is abelian if n—r^2. There is a suspension functor S without action and one 2...