65 research outputs found
Face enumeration on simplicial complexes
Let be a closed triangulable manifold, and let be a
triangulation of . What is the smallest number of vertices that can
have? How big or small can the number of edges of be as a function of
the number of vertices? More generally, what are the possible face numbers
(-numbers, for short) that can have? In other words, what
restrictions does the topology of place on the possible -numbers of
triangulations of ?
To make things even more interesting, we can add some combinatorial
conditions on the triangulations we are considering (e.g., flagness,
balancedness, etc.) and ask what additional restrictions these combinatorial
conditions impose. While only a few theorems in this area of combinatorics were
known a couple of decades ago, in the last ten years or so, the field simply
exploded with new results and ideas. Thus we feel that a survey paper is long
overdue. As new theorems are being proved while we are typing this chapter, and
as we have only a limited number of pages, we apologize in advance to our
friends and colleagues, some of whose results will not get mentioned here.Comment: Chapter for upcoming IMA volume Recent Trends in Combinatoric
Integrality in the Steinberg module and the top-dimensional cohomology of SL_n(O_K)
We prove a new structural result for the spherical Tits building attached to
SL_n(K) for many number fields K, and more generally for the fraction fields of
many Dedekind domains O: the Steinberg module St_n(K) is generated by integral
apartments if and only if the ideal class group cl(O) is trivial. We deduce
this integrality by proving that the complex of partial bases of O^n is
Cohen-Macaulay. We apply this to prove new vanishing and nonvanishing results
for H^{vcd}(SL_n(O_K); Q), where O_K is the ring of integers in a number field
and vcd is the virtual cohomological dimension of SL_n(O_K). The (non)vanishing
depends on the (non)triviality of the class group of O_K. We also obtain a
vanishing theorem for the cohomology H^{vcd}(SL_n(O_K); V) with twisted
coefficients V.Comment: 36 pages; final version; to appear in Amer. J. Mat
Simplicial bounded cohomology and stability
We introduce a set of combinatorial techniques for studying the simplicial
bounded cohomology of semi-simplicial sets, simplicial complexes and posets. We
apply these methods to prove several new bounded acyclicity results for
semi-simplicial sets appearing in the homological stability literature. Our
strategy is to recast classical arguments (due to Bestvina, Maazen, van der
Kallen, Vogtmann, Charney and, recently, Galatius--Randal-Williams) in the
setting of bounded cohomology using uniformly bounded refinements of well-known
simplicial tools. Combined with ideas developed by Monod and De la Cruz
Mengual--Hartnick, we deduce slope- stability results for the bounded
cohomology of two large classes of linear groups: general linear groups over
any ring with finite Bass stable rank and certain automorphism groups of
quadratic modules over the integers or any field of characteristic zero. We
expect that many other results in the literature on homological stability admit
bounded cohomological analogues by applying the blueprint provided in this
work.Comment: 53 pages. Comments welcome
The homology of the Higman-Thompson groups
We prove that Thompson's group is acyclic, answering a 1992 question of
Brown in the positive. More generally, we identify the homology of the
Higman-Thompson groups with the homology of the zeroth component of
the infinite loop space of the mod Moore spectrum. As , we
can deduce that this group is acyclic. Our proof involves establishing
homological stability with respect to , as well as a computation of the
algebraic K-theory of the category of finitely generated free Cantor algebras
of any type .Comment: 49 page
Reflect-Push Methods Part I: Two Dimensional Techniques
We determine all maximum weight downsets in the product of two chains, where
the weight function is a strictly increasing function of the rank. Many
discrete isoperimetric problems can be reduced to the maximum weight downset
problem. Our results generalize Lindsay's edge-isoperimetric theorem in two
dimensions in several directions. They also imply and strengthen (in several
directions) a result of Ahlswede and Katona concerning graphs with maximal
number of adjacent pairs of edges. We find all optimal shifted graphs in the
Ahlswede-Katona problem. Furthermore, the results of Ahlswede-Katona are
extended to posets with a rank increasing and rank constant weight function.
Our results also strengthen a special case of a recent result by Keough and
Radcliffe concerning graphs with the fewest matchings. All of these results are
achieved by applications of a key lemma that we call the reflect-push method.
This method is geometric and combinatorial. Most of the literature on
edge-isoperimetric inequalities focuses on finding a solution, and there are no
general methods for finding all possible solutions. Our results give a general
approach for finding all compressed solutions for the above edge-isoperimetric
problems.
By using the Ahlswede-Cai local-global principle, one can conclude that
lexicographic solutions are optimal for many cases of higher dimensional
isoperimetric problems. With this and our two dimensional results we can prove
Lindsay's edge-isoperimetric inequality in any dimension. Furthermore, our
results show that lexicographic solutions are the unique solutions for which
compression techniques can be applied in this general setting
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