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.
20093815
On the K-theory spectrum of a smooth curve over a finite field☆
Suppose that X is a nonsingular curve over a finite field F. According to ideas of Quillen and others, there is an algebraic K-theory spectrum KX associated to the category of locally free sheaves on X . The word “spectrum” here designates an object in the stable homotopy category of spaces. If X = SpecO is affine, then KX is equivalent to the usual K-theory spectrum associated to the category of finitely generated projective modules over O. A certain amount is known about the groups KiX = πiKX. They are abelian, and they vanish for i < 0. They are finitely generated and with a few exceptions are actually finite: if X is complete then K0X has rank 2, while if X is affine then K0X has rank 1 and K1X has rank s−1, where s is the number of points at infinity of X. For i ≤ 2, KiX can be described in fairly classical terms (see §2). In general, though, these groups seem very difficult to pin down. In this paper we detour around the groups KiX in favor of studying the entire spectrum KX, or more accurately the `-completion KX of KX, where ` is an odd rational prime different from the characteristic of F. Let K be the `-completion of the periodic topological complex K-theory spectrum and K∗ the associated cohomology theory. We begin by calculating K∗(KX) = K∗(KX) as a module over the ring of cohomology operations in K∗ (5.1, 5.2). The answer is surprisingly simple and leads to a formula for the `-adic topological K-theory K(Ω0 KX) of the basepoint component of the zero space in the associated Ω-spectrum (7.5). If X = SpecO is affine, then K(Ω0 KX) is the same as K∗(BGL(O)), so in this case we obtain a formula for the `-adic topological K-theory of the classifying space of the infinite general linear group of O. Next we compute the homotopy type of a certain localization L(KX) = L(KX) of KX; this is localization in the sense of Bousfield with respect to the cohomology theory K∗ (4.2). The homotopy groups πiL(KX) do not vanish for i < 0, but there is a form of the Lichtenbaum-Quillen conjecture which suggests that the localization map KX −→ L(KX)
6041165
The generalized Burnside ring and theK-theory of a ring with roots of unity
Determining the algebraic K-theory of rings of integers in number fields has been the goal of much research. In [10] D. Quillen showed that the Hurewicz map h : Q0(S ) → BGL(Z) (see 1.1 for the notation) induces an interesting map on homotopy groups from the stable homotopy groups of spheres to the algebraic K-theory of the ring Z of rational integers. Quillen observed that if ` is an odd prime and if p 6= ` is another prime generating the `-adic units, then the composite map
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.