13923104
On the K-Theory Spectrum of a Ring of Algebraic Integers⋆
Suppose that F is a number field (i.e. a finite algebraic extension of the field Q of rational numbers) and that OF is the ring of algebraic integers in F . One of the most fascinating and apparently difficult problems in algebraic K-theory is to compute the groups KiOF . These groups were shown to be finitely generated by Quillen [36] and their ranks were calculated by Borel [7]. Lichtenbaum and Quillen have formulated a very explicit conjecture about what the groups should be [37]. Nevertheless, there is not a single number field F for which KiOF is known for any i ≥ 5. For most fields F , these groups are unknown for i ≥ 3. The groups KiOF are the homotopy groups of a spectrum KOF (“spectrum” in the sense of algebraic topology [1]). In this paper, rather than concentrating on the individual groups KiOF , we study the entire spectrum KOF from the viewpoint of stable homotopy theory. For technical reasons, it is actually more convenient to single out a rational prime number l and study instead the spectrum KRF , where RF = OF [1/l] is the ring of algebraic l-integers in F . “At l”, that is, after localizing or completing at l, the spectra KOF and KRF are almost the same (cf. [34] and [35, p. 113]).
123630598
The bott filtration of a loop group
In [10] it was shown that the "degree" filtration on the homology of OSU(n) has a geometric realization. Indeed, by the theory of loop groups, OSU(n) is homotopy equivalent to an infinite dimensional flag variety, and the spaces of the filtration are in fact Schubert subvarieties, indexed by an explicitly determined family of elements in the affine Weyl group. At first glance the existence of this filtration, and in particular the way it can be obtained from the affine Weyl group, appears to be peculiar to SU(n). In this paper we show that in fact the construction is a very general one: If G is a compact Lie group, a theorem of Bott [3] shows that OoG admits a finite generating complex-e-i.e., a finite complex X, together with an inclusion i : X --. OoG, such that i*(H*(X; 7l)) generates the Pontrjagin ring of OoG. This defines a multiplicative filtration on the homology of OoG, and again one can ask for a geometric realization. (The degree filtration mentioned above corresponds to the usual generating complex (:pn-l --. OSU(n).) In fact the map X --. OoG extends to a map OEX --. !loG, so that the filtration we seek corresponds precisely to the James filtration on OEX. Our main result is that whenever X is the generating complex associated to a "miniscule node" of the Dynkin diagram of G, there is indeed a filtration of OoG by Schubert varieties realizing the filtration on homology (Theorem 2.3). We even give explicit descriptions of the elements of the affine Weyl group indexing the filtrations; that X is a generating complex reduces, from this point of view, to the fact these elements form a cofinal sequence for the Bruhat order (Corollary 2.5). The use of miniscule nodes is very convenient, but unfortunately excludes the groups G2 , F., ana E8 • However, this restriction could probably be avoided.
118505299
The Morava K -theory of algebraic K -theory spectra
3236051
The All-Hex Geode-Template for Conforming a Diced Tetrahedral Mesh to any Diced Hexahedral Mesh
Abstract. Take a hexahedral mesh and an adjoining tetrahedral mesh that splits each boundary quadrilateral into two triangles. Separate the meshes with a buffer layer of hexes. Dice the original hexes into eight, and the tetrahedra into four hexahedra. Then I show that the buffer layer hexes can be filled with the geode-template, creating a conforming all-hex mesh of the entire model. The geode-template is composed of 26 hexahedra. The hexahedra have acceptable quality, depending on the geometry of the buffer layer. The method used to generate the geode-template is general, based on interleaving completed dual surfaces, and might be extended to other transition problems.