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Register a new userTopological comparison theorems for Bredon motivic cohomology
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In this paper we study the topological invariant ${sf {TC}}(X)$ reflecting the complexity of algorithms for autonomous robot motion. Here, $X$ stands for the configuration space of a system and ${sf {TC}}(X)$ is, roughly, the minimal number of continuous rules which are needed to construct a motion planning algorithm in $X$. We focus on the case when the space $X$ is aspherical; then the number ${sf TC}(X)$ depends only on the fundamental group $pi=pi_1(X)$ and we denote it ${sf TC}(pi)$. We prove that ${sf TC}(pi)$ can be characterised as the smallest integer $k$ such that the canonical $pitimespi$-equivariant map of classifying spaces $$E(pitimespi) to E_{mathcal D}(pitimespi)$$ can be equivariantly deformed into the $k$-dimensional skeleton of $E_{mathcal D}(pitimespi)$. The symbol $E(pitimespi)$ denotes the classifying space for free actions and $E_{mathcal D}(pitimespi)$ denotes the classifying space for actions with isotropy in a certain family $mathcal D$ of subgroups of $pitimespi$. Using this result we show how one can estimate ${sf TC}(pi)$ in terms of the equivariant Bredon cohomology theory. We prove that ${sf TC}(pi) le max{3, {rm cd}_{mathcal D}(pitimespi)},$ where ${rm cd}_{mathcal D}(pitimespi)$ denotes the cohomological dimension of $pitimespi$ with respect to the family of subgroups $mathcal D$. We also introduce a Bredon cohomology refinement of the canonical class and prove its universality. Finally we show that for a large class of principal groups (which includes all torsion free hyperbolic groups as well as all torsion free nilpotent groups) the essential cohomology classes in the sense of Farber and Mescher are exactly the classes having Bredon cohomology extensions with respect to the family $mathcal D$.
We show that the analogue of the Peterson conjecture on the action of Steenrod squares does not hold in motivic cohomology.
We provide a complete analysis of the motivic Adams spectral sequences converging to the bigraded coefficients of the 2-complete algebraic Johnson-Wilson spectra BPGL<n> over p-adic fields. These spectra interpolate between integral motivic cohomology (n=0), a connective version of algebraic K-theory (n=1), and the algebraic Brown-Peterson spectrum. We deduce that, over p-adic fields, the 2-complete BPGL<n> split over 2-complete BPGL<0>, implying that the slice spectral sequence for BPGL collapses. This is the first in a series of two papers investigating motivic invariants of p-adic fields, and it lays the groundwork for an understanding of the motivic Adams-Novikov spectral sequence over such base fields.
We determine systematic regions in which the bigraded homotopy sheaves of the motivic sphere spectrum vanish.
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 eta (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 and faithful embedding after eta-completion if a motivic version of Serres finiteness theorem is valid. We produce strong necessary conditions on the field extension L/k for this functor to be full and faithful. Along the way, we produce several results on the stable C_2-equivariant Betti realization functor and prove convergence theorems for the p-primary C_2-equivariant Adams spectral sequence.
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