2,490 research outputs found
James bundles
We study cubical sets without degeneracies, which we call {square}-sets. These sets arise naturally in a number of settings and they have a beautiful intrinsic geometry; in particular a {square}-set C has an infinite family of associated {square}-sets Ji(C), for i = 1, 2, ..., which we call James complexes. There are mock bundle projections pi: |Ji(C)| -> |C| (which we call James bundles) defining classes in unstable cohomotopy which generalise the classical James–Hopf invariants of {Omega}(S2). The algebra of these classes mimics the algebra of the cohomotopy of {Omega}(S2) and the reduction to cohomology defines a sequence of natural characteristic classes for a {square}-set. An associated map to BO leads to a generalised cohomology theory with geometric interpretation similar to that for Mahowald orientation
and word problems for infinite type Artin-Tits groups, and applications to virtual braid groups
Let be a Coxeter graph, let be its associated Coxeter
system, and let ) be its associated Artin-Tits system. We regard
as a reflection group acting on a real vector space . Let be the Tits
cone, and let be the complement in of the reflecting
hyperplanes. Recall that Charney, Davis, and Salvetti have constructed a
simplicial complex having the same homotopy type as
. We observe that, if , then
naturally embeds into . We prove that this embedding admits a
retraction , and we deduce several
topological and combinatorial results on parabolic subgroups of . From a
family \SS of subsets of having certain properties, we construct a cube
complex , we show that has the same homotopy type as the universal
cover of , and we prove that is CAT(0) if and only if \SS is
a flag complex. We say that is free of infinity if has
no edge labeled by . We show that, if is aspherical and
has a solution to the word problem for all free of
infinity, then is aspherical and has a solution to the word
problem. We apply these results to the virtual braid group . In
particular, we give a solution to the word problem in , and we prove that
the virtual cohomological dimension of is
Size of Sets with Small Sensitivity: a Generalization of Simon's Lemma
We study the structure of sets with small sensitivity.
The well-known Simon's lemma says that any of
sensitivity must be of size at least . This result has been useful
for proving lower bounds on sensitivity of Boolean functions, with applications
to the theory of parallel computing and the "sensitivity vs. block sensitivity"
conjecture.
In this paper, we take a deeper look at the size of such sets and their
structure. We show an unexpected "gap theorem": if has
sensitivity , then we either have or . This is shown via classifying such sets into sets that can be
constructed from low-sensitivity subsets of for and
irreducible sets which cannot be constructed in such a way and then proving a
lower bound on the size of irreducible sets.
This provides new insights into the structure of low sensitivity subsets of
the Boolean hypercube
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