20,041 research outputs found
Indicators of Tambara-Yamagami categories and Gauss sums
We prove that the higher Frobenius-Schur indicators, introduced by Ng and
Schauenburg, give a strong enough invariant to distinguish between any two
Tambara-Yamagami fusion categories. Our proofs are based on computation of the
higher indicators as quadratic Gauss sums for certain quadratic forms on finite
abelian groups and relies on the classification of quadratic forms on finite
abelian groups, due to Wall.
As a corollary to our work, we show that the state-sum invariants of a
Tambara-Yamagami category determine the category as long as we restrict to
Tambara-Yamagami categories coming from groups G whose order is not a power of
2. Turaev and Vainerman proved this result under the assumption that G has odd
order and they conjectured that a similar result should hold for groups of even
order. We also give an example to show that the assumption that G does not have
a power of 2, cannot be completely relaxed.Comment: 29 page
Geometric and homological finiteness in free abelian covers
We describe some of the connections between the Bieri-Neumann-Strebel-Renz
invariants, the Dwyer-Fried invariants, and the cohomology support loci of a
space X. Under suitable hypotheses, the geometric and homological finiteness
properties of regular, free abelian covers of X can be expressed in terms of
the resonance varieties, extracted from the cohomology ring of X. In general,
though, translated components in the characteristic varieties affect the
answer. We illustrate this theory in the setting of toric complexes, as well as
smooth, complex projective and quasi-projective varieties, with special
emphasis on configuration spaces of Riemann surfaces and complements of
hyperplane arrangements.Comment: 30 pages; to appear in Configuration Spaces: Geometry, Combinatorics
and Topology (Centro De Giorgi, 2010), Edizioni della Normale, Pisa, 201
The computation of the cohomology rings of all groups of order 128
We describe the computation of the mod-2 cohomology rings of all 2328 groups
of order 128. One consequence is that all groups of order less than 256 satisfy
the strong form of Benson's Regularity Conjecture.Comment: 15 pages; revised versio
On the Cohomology of Central Frattini Extensions
We use topological methods to compute the mod p cohomology of certain
p-groups. More precisely we look at central Frattini extensions of elementary
abelian by elementary abelian groups such that their defining k-invariants span
the entire image of the Bockstein. We show that if p is sufficiently large,
then the mod p cohomology of the extension can be explicitly computed as an
algebra
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