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Buildings and their applications in geometry and topology
 ASIAN J. MATH
"... Buildings were first introduced by J.Tits in 1950s to give systematic geometric interpretations of exceptional Lie groups and have been generalized in various ways: Euclidean buildings (BruhatTits buildings), topological buildings, Rbuildings, in particular Rtrees. They are useful for many differ ..."
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Buildings were first introduced by J.Tits in 1950s to give systematic geometric interpretations of exceptional Lie groups and have been generalized in various ways: Euclidean buildings (BruhatTits buildings), topological buildings, Rbuildings, in particular Rtrees. They are useful for many different applications in various subjects: algebraic groups, finite groups, finite geometry, representation theory over local fields, algebraic geometry, Arakelov intersection for arithmetic varieties, algebraic Ktheories, combinatorial group theory, global geometry and algebraic topology, in particular cohomology groups, of arithmetic groups and Sarithmetic groups, rigidity of cofinite subgroups of semisimple Lie groups and nonpositively curved manifolds, classification of isoparametric submanifolds in R n of high codimension, existence of hyperbolic structures on three dimensional manifolds in Thurstonâ€™s geometrization program. In this paper, we survey several applications of buildings in differential geometry and geometric topology. There are four underlying themes in these applications: 1. Buildings often describe the geometry at infinity of symmetric spaces and locally symmetric
Representations of Matroids in Semimodular Lattices
"... We prove equivalence of two definitions of representability of matroids: representation by vector configurations and representation by retraction of buildings of type An . Proofs are given in a more general context of representation of matroids in semimodular lattices and Coxeter matroids in chamber ..."
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We prove equivalence of two definitions of representability of matroids: representation by vector configurations and representation by retraction of buildings of type An . Proofs are given in a more general context of representation of matroids in semimodular lattices and Coxeter matroids in chamber systems with group metric. 3
On the Topology of the Combinatorial Flag Varieties  From a New Point of View
, 2000
"... We prove that the simplicial complex# n of chains of matroids (with respect to the ordering by the quotient relation) on n elements is shellable. This follows from a more general result on shellability of the simplicial complex of Wmatroids for an arbitrary finite Coxeter group W . The paper gener ..."
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We prove that the simplicial complex# n of chains of matroids (with respect to the ordering by the quotient relation) on n elements is shellable. This follows from a more general result on shellability of the simplicial complex of Wmatroids for an arbitrary finite Coxeter group W . The paper generalises the well known results by SolomonTits and Bjorner on spherical buildings, and improves the results of [6]. # Partially supported by PSCCUNY Research Award 669437 and EPSRC Research Grant GR/M24707. 1 1 Definitions and main results For any positive integer n, let [n] denote the set { 1, . . . , n }, and let M n be the set of all matroids on [n] of rank distinct from 0 and n. Define a partial ordering on M n by M # # M if M # is a quotient of M . Let# n be the simplicial complex of chains in M n ; every simplex # ## n can be written as # = #M 1 , . . . , M r # , with all M i # M n and M 1 # . . . # M r . Following [10, 11], we shall call# n the n th combinatorial flag variety. In contexts where the value of n is not important we shall omit it from the notation and write M and## The works [10, 11] contain detailed discussions of many remarkable properties of the combinatorial flag variety# n . It behaves like a universal, characteristicfree finite geometry. In particular, every matroid on n points is representable in# n in the sense specified in [10, 11], which is the direct reformulation in combinatorial terms of the classical concept of representability of matroids by means of vector configurations. The present work will concentrate on topological and cohomological properties of the combinatorial flag varieties. It is easy to prove that# n is a purely (n2)dimensional simplicial complex, that is, it has dimension n  2 and eve...
On the Topology of the Combinatorial Flag Varieties
, 1999
"... this paper all matroids will have ground set [n], and we shall frequently omit the symbol n from our notation. Define a partial ordering on M + n by M # # M if M # is a quotient of M . Let# + n be the simplicial complex of chains in M + n ; every simplex s ## + n can be written as s = ..."
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this paper all matroids will have ground set [n], and we shall frequently omit the symbol n from our notation. Define a partial ordering on M + n by M # # M if M # is a quotient of M . Let# + n be the simplicial complex of chains in M + n ; every simplex s ## + n can be written as s = #M 1 , . . . , M r #, with all M i # M + n and M 1 # . . . # M r . 1.1 Convention. A superscript number on a matroid will denote its rank. Subscripts will be used for general indexing purposes. According to this convention, in a context where all matroids are on [n], M 0 and M n will denote the unique matroids of rank 0 and rank n respectively. For another example of the use of this convention, when we write s =# M i 0 , M i 1 , . . . , M i r # in# + n , the notation will imply that M i 0 # M i 1 # . . . # M i r and that rank(M i 0 ) = i 0 , rank(M i 1 ) = i 1 etc. 1.2 For any subset N # M + n let #(N ) be the subcomplex of# + n generated by N : #(N ) consists of all simplices of# + n all of whose vertices belong to N . In particular let M n = {M # M + n : 1 #rank(M) # n  1}, and let# n = #(M n )