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Register a new userThe homotopy groups of the {eta}-periodic motivic sphere spectrum
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We compute the homotopy groups of the {eta}-periodic motivic sphere spectrum over a finite-dimensional field k with characteristic not 2 and in which -1 a sum of four squares. We also study the general characteristic 0 case and show that the {eta}-periodic slice spectral sequence over Q determines the {eta}-periodic slice spectral sequence over all extensions of Q. This leads to a speculation on the role of a connective Witt-theoretic J-spectrum in {eta}-periodic motivic homotopy theory.
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We survey computations of stable motivic homotopy groups over various fields. The main tools are the motivic Adams spectral sequence, the motivic Adams-Novikov spectral sequence, and the effective slice spectral sequence. We state some projects for future study.
Over any field of characteristic not 2, we establish a 2-term resolution of the $eta$-periodic, 2-local motivic sphere spectrum by shifts of the connective 2-local Witt K-theory spectrum. This is curiously similar to the resolution of the K(1)-local sphere in classical stable homotopy theory. As applications we determine the $eta$-periodized motivic stable stems and the $eta$-periodized algebraic symplectic and SL-cobordism groups. Along the way we construct Adams operations on the motivic spectrum representing Hermitian K-theory and establish new completeness results for certain motivic spectra over fields of finite virtual 2-cohomological dimension. In an appendix, we supply a new proof of the homotopy fixed point theorem for the Hermitian K-theory of fields.
We construct well-behaved extensions of the motivic spectra representing generalized motivic cohomology and connective Balmer--Witt K-theory (among others) to mixed characteristic Dedekind schemes on which 2 is invertible. As a consequence we lift the fundamental fiber sequence of $eta$-periodic motivic stable homotopy theory established in [arxiv:2005.06778] from fields to arbitrary base schemes, and use this to determine (among other things) the $eta$-periodized algebraic symplectic and SL-cobordism groups of mixed characteristic Dedekind schemes containing 1/2.
A C-motivic modular forms spectrum mmf has recently been constructed. This article presents detailed computational information on the Adams spectral sequence for mmf. This information is essential for computing with the C-motivic and classical Adams spectral sequences that compute the C-motivic and classical stable homotopy groups of spheres.
Let $M$ be a topological monoid with homotopy group completion $Omega BM$. Under a strong homotopy commutativity hypothesis on $M$, we show that $pi_k (Omega BM)$ is the quotient of the monoid of free homotopy classes $[S^k, M]$ by its submonoid of nullhomotopic maps. We give two applications. First, this result gives a concrete description of the Lawson homology of a complex projective variety in terms of point-wise addition of spherical families of effective algebraic cycles. Second, we apply this result to monoids built from the unitary, or general linear, representation spaces of discrete groups, leading to results about lifting continuous families of characters to continuous families of representations.
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