1,022 research outputs found
Symmetric products and subgroup lattices
Let G be a finite group. We show that the rational homotopy groups of
symmetric products of the G-equivariant sphere spectrum are naturally
isomorphic to the rational homology groups of certain subcomplexes of the
subgroup lattice of G.Comment: final published versio
Finiteness properties of soluble arithmetic groups over global function fields
Let G be a Chevalley group scheme and B<=G a Borel subgroup scheme, both
defined over Z. Let K be a global function field, S be a finite non-empty set
of places over K, and O_S be the corresponding S-arithmetic ring. Then, the
S-arithmetic group B(O_S) is of type F_{|S|-1} but not of type FP_{|S|}.
Moreover one can derive lower and upper bounds for the geometric invariants
\Sigma^m(B(O_S)). These are sharp if G has rank 1. For higher ranks, the
estimates imply that normal subgroups of B(O_S) with abelian quotients,
generically, satisfy strong finiteness conditions.Comment: Published by Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol8/paper15.abs.htm
B^F Theory and Flat Spacetimes
We propose a reduced constrained Hamiltonian formalism for the exactly
soluble theory of flat connections and closed two-forms over
manifolds with topology . The reduced phase space
variables are the holonomies of a flat connection for loops which form a basis
of the first homotopy group , and elements of the second
cohomology group of with value in the Lie algebra . When
, and if the two-form can be expressed as , for some
vierbein field , then the variables represent a flat spacetime. This is not
always possible: We show that the solutions of the theory generally represent
spacetimes with ``global torsion''. We describe the dynamical evolution of
spacetimes with and without global torsion, and classify the flat spacetimes
which admit a locally homogeneous foliation, following Thurston's
classification of geometric structures.Comment: 21 pp., Mexico Preprint ICN-UNAM-93-1
Finite Approximations to Quantum Physics: Quantum Points and their Bundles
There exists a physically well motivated method for approximating manifolds
by certain topological spaces with a finite or a countable set of points. These
spaces, which are partially ordered sets (posets) have the power to effectively
reproduce important topological features of continuum physics like winding
numbers and fractional statistics, and that too often with just a few points.
In this work, we develop the essential tools for doing quantum physics on
posets. The poset approach to covering space quantization, soliton physics,
gauge theories and the Dirac equation are discussed with emphasis on physically
important topological aspects. These ideas are illustrated by simple examples
like the covering space quantization of a particle on a circle, and the
sine-Gordon solitons.Comment: 24 pages, 8 figures on a uuencoded postscript file, DSF-T-29/93,
INFN-NA-IV-29/93 and SU-4240-55
Chains of modular elements and shellability
Let L be a lattice admitting a left-modular chain of length r, not
necessarily maximal. We show that if either L is graded or the chain is
modular, then the (r-2)-skeleton of L is vertex-decomposable (hence shellable).
This proves a conjecture of Hersh. Under certain circumstances, we can find
shellings of higher skeleta. For instance, if the left-modular chain consists
of every other element of some maximum length chain, then L itself is
shellable. We apply these results to give a new characterization of finite
solvable groups in terms of the topology of subgroup lattices.
Our main tool relaxes the conditions for an EL-labeling, allowing multiple
ascending chains as long as they are lexicographically before non-ascending
chains. We extend results from the theory of EL-shellable posets to such
labelings. The shellability of certain skeleta is one such result. Another is
that a poset with such a labeling is homotopy equivalent (by discrete Morse
theory) to a cell complex with cells in correspondence to weakly descending
chains.Comment: 20 pages, 1 figure; v2 has minor fixes; v3 corrects the technical
lemma in Section 4, and improves the exposition throughou
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