330 research outputs found
Five-Torsion in the Homology of the Matching Complex on 14 Vertices
J. L. Andersen proved that there is 5-torsion in the bottom nonvanishing
homology group of the simplicial complex of graphs of degree at most two on
seven vertices. We use this result to demonstrate that there is 5-torsion also
in the bottom nonvanishing homology group of the matching complex on
14 vertices. Combining our observation with results due to Bouc and to
Shareshian and Wachs, we conclude that the case is exceptional; for all
other , the torsion subgroup of the bottom nonvanishing homology group has
exponent three or is zero. The possibility remains that there is other torsion
than 3-torsion in higher-degree homology groups of when and .Comment: 11 page
The pre-WDVV ring of physics and its topology
We show how a simplicial complex arising from the WDVV
(Witten-Dijkgraaf-Verlinde-Verlinde) equations of string theory is the
Whitehouse complex. Using discrete Morse theory, we give an elementary proof
that the Whitehouse complex is homotopy equivalent to a wedge of
spheres of dimension . We also verify the Cohen-Macaulay
property. Additionally, recurrences are given for the face enumeration of the
complex and the Hilbert series of the associated pre-WDVV ring.Comment: 13 pages, 4 figures, 2 table
Cosmic Neutrinos and the Energy Budget of Galactic and Extragalactic Cosmic Rays
Although kilometer-scale neutrino detectors such as IceCube are discovery
instruments, their conceptual design is very much anchored to the observational
fact that Nature produces protons and photons with energies in excess of
10^{20} eV and 10^{13} eV, respectively. The puzzle of where and how Nature
accelerates the highest energy cosmic particles is unresolved almost a century
after their discovery. We will discuss how the cosmic ray connection sets the
scale of the anticipated cosmic neutrino fluxes. In this context, we discuss
the first results of the completed AMANDA detector and the science reach of its
extension, IceCube.Comment: 13 pages, Latex2e, 3 postscript figures included. Talk presented at
the International Workshop on Energy Budget in the High Energy Universe,
Kashiwa, Japan, February 200
Random geometric complexes
We study the expected topological properties of Cech and Vietoris-Rips
complexes built on i.i.d. random points in R^d. We find higher dimensional
analogues of known results for connectivity and component counts for random
geometric graphs. However, higher homology H_k is not monotone when k > 0. In
particular for every k > 0 we exhibit two thresholds, one where homology passes
from vanishing to nonvanishing, and another where it passes back to vanishing.
We give asymptotic formulas for the expectation of the Betti numbers in the
sparser regimes, and bounds in the denser regimes. The main technical
contribution of the article is in the application of discrete Morse theory in
geometric probability.Comment: 26 pages, 3 figures, final revisions, to appear in Discrete &
Computational Geometr
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