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The definition of the class is actually given in the cited sentence. For the relevant motivic cohomology group we have $${\rm H}^{0,1}({\rm Spec} k,\mathbb{Z}/p\mathbb{Z})\cong {\rm H}^0_{\rm ét}({\rm …

The relation between motivic cohomology and cohomology of the Milnor K-theory sheaf is discussed in Motivic cohomology and cohomology of Milnor K-theory sheaf There is a natural comparison morphism …

Examples of smooth real varieties whose Witt group has odd torsion can be found in a paper of Jacobson: J.A. Jacobson. From the global signature to higher signatures. arXiv:1411.0993, https://arxiv …

The degrees of the attaching maps (and hence the integral chain complex) for the real Grassmannians have been determined in L. Casian and Y. Kodama. On the cohomology of real Grassmann manifolds. a …

The Atiyah-Hirzebruch spectral sequence does have a multiplicative structure, and I think this can be used to determine the multiplication on K-theory. From the paper of Braun, it follows that the spe …

Here are some known computations for infinite discrete groups. Basically, most of these proceed by computing the equivariant K-homology of the classifying space of proper actions and deduce the comput …

First, some general remarks on the situation for $R$ an arbitrary commutative ring. Since ${\rm diag}(-1,-1)$ acts by multiplication by $-1$ on $R^{\oplus 2}$, the homology groups ${\rm H}_i({\rm SL}_ …

Here are some relevant references. First, three papers on different methods to compute the K-theory of compact Lie groups with finite cyclic fundamental group. (The first one also discusses the projec …

To elaborate on my comment, the comparison between the regulators of Beilinson and Borel can be found in the book J.I. Burgos Gil. The regulators of Beilinson and Borel. CRM Monograph Series, 15. Am …

Some alternative references, maybe more classical (and with cycles): Section 12.3.3 of vol I of C. Voisin: Hodge theory and complex algebraic geometry. Cambridge studies in advanced mathematics 76. …

We have a short exact sequence of sheaves of abelian groups (see e.g. Morel's book on $\mathbb{A}^1$-algebraic topology): $$ 0\to\mathbf{I}^{n+1}\to\mathbf{K}^{\rm MW}_n\to \mathbf{K}^{\rm M}_n\to 0, …

This is going to be a slightly extended explanation, I apologize. The short version is basically that little is actually known (and even that is hard to prove), but conjecturally everything permitted …

I would like to add a couple of references to the ones provided in Timo Keller's comment. First of all, the conjecture is not correctly stated. According to the conjectures, the rank of the curve (w …

Yes, the natural multiplication morphisms induce a morphism of Gersten complexes from Milnor to Quillen K-theory. The basic points are made in the paper M. Rost. "Chow groups with coefficients", Do …

There are two papers that could be interesting to you. First, there is a paper of Kevin Hutchinson: K. Hutchinson. On the low-dimensional homology of ${\rm SL}_2(k[t,t^{-1}])$. J. Algebra 425 (2015) …