Twisted Witt Groups of Flag Varieties

Calm\`es and Fasel have shown that the twisted Witt groups of split flag varieties vanish in a large number of cases. For flag varieties over algebraically closed fields, we sharpen their result to an if-and-only-if statement. In particular, we show that the twisted Witt groups vanish in many previously unknown cases. In the non-zero cases, we find that the twisted total Witt group forms a free module of rank one over the untwisted total Witt group, up to a difference in grading. Our proof relies on an identification of the Witt groups of flag varieties with the Tate cohomology groups of their K-groups, whereby the verification of all assertions is eventually reduced to the computation of the (twisted) Tate cohomology of the representation ring of a parabolic subgroup.Comment: inverse Cartan coefficients for E_7 in Figures 1, 2 and 3 corrected; related mistake in the marking scheme for diagrams of type E_n corrected; many minor corrections and clarifications; more example

Smith-Knight/Rayleigh-Knight Essay

On a complex variety X, two different approaches to K-theory are available: the algebraic K-theory of the variety, and the topological K-theory of the underlying topological space. In this context, the algebraic variant known as Hermitian K-theory corresponds to topological KO-theory. Our aim is to compare the two approaches. We start by constructing a comparison map from certain Hermitian K-groups of X to the KO-groups of X. It is clear what this map must be on groups in degree zero, but the definitions of relative and higher groups differ widely in the algebraic and the topological setting. This difficulty can be overcome by viewing relative and higher groups as subgroups of degree zero groups of certain auxiliary spaces. Once the definition of our comparison map is in place, we prove a number of fundamental properties, in particular compatibility with pushforwards along closed embeddings. We also show how we can use it to compare an exact sequence relating usual algebraic K-theory to Hermitian K-theory with a portion of the Bott sequence in topology. This finally allows us to deduce that the map is an isomorphism on smooth cellular varieties. We conclude with some details concerning projective spaces, for which independent computations of the algebraic and the topological groups exist.This work was supported by the Engineering and Physical Sciences Research Council [LFAG/067]

Witt groups of complex cellular varieties

We show that the Grothendieck-Witt and Witt groups of smooth complex cellular varieties can be identified with their topological KO-groups. As an application, we deduce the values of the Witt groups of all irreducible hermitian symmetric spaces, including smooth complex quadrics, spinor varieties and symplectic Grassmannians.Comment: 43 pages. This is the final version, identical in content to the paper published in Documenta Mathematic

KO-Rings of Full Flag Varieties

We present type-independent computations of the KO-groups of full flag varieties, i.e. of quotient spaces G/T of compact Lie groups by their maximal tori. Our main tool is the identification of the Witt ring, a quotient of the KO-ring, of these varieties with the Tate cohomology of their complex K-ring. The computations show that the Witt ring is an exterior algebra whose generators are determined by representations of G.Comment: 22 pages; updated references and other minor change

On the preparation and NMR spectroscopic characterization of potassium aluminium tetrahydride KAlH4

Potassium aluminium tetrahydride KAlH4 of high phase purity (space group Pnma (62)) was synthesized via a mechanochemical route. The thus obtained material was studied by 27Al and 39K MAS NMR spectroscopy. For both nuclei precise data for the isotropic chemical shift and the quadrupole coupling at T=295 K were derived (27Al: delta_iso=(107.6+-0.2) ppm, C_Q = (1.29+-0.02) MHz and eta = 0.64+-0.02; 39K: delta_iso=(6.1+-0.2) ppm, C_Q = (0.562+-0.005) MHz and eta = 0.74+-0.02). The straightforward NMR spectroscopic approach applied here should also work for other complex aluminium hydrides and for many other materials containing half-integer nuclei experiencing small to medium-sized quadrupole couplings.Comment: 27 pages, 5 figures, final version as published in PCCP, with reviewer reports attached to the ES

Vector bundles of non-negative curvature over cohomogeneity one manifolds

We provide several results on the existence of metrics of non-negative sectional curvature on vector bundles over certain cohomogeneity one manifolds and homogeneous spaces up to suitable stabilization. Beside explicit constructions of the metrics, this is achieved by identifying equivariant structures upon these vector bundles via a comparison of their equivariant and non-equivariant K-theory. For this, in particular, we transcribe equivariant K-theory to equivariant rational cohomology and investigate surjectivity properties of induced maps in the Borel fibration via rational homotopy theory.Comment: some upgrades and modification