ar X iv : m at h / 02 02 08 1 v 1 [ m at h . A T ] 9 F eb 2 00 2 COLIMITS , STANLEY - REISNER ALGEBRAS , AND LOOP SPACES
نویسندگان
چکیده
We study diagrams associated with a finite simplicial complex K, in various algebraic and topological categories. We relate their colimits to familiar structures in algebra, combinatorics, geometry and topology. These include: right-angled Artin and Coxeter groups (and their complex analogues, which we call circulation groups); Stanley-Reisner algebras and coalgebras; Davis and Januszkiewicz's spaces DJ(K) associated with toric manifolds and their generalisations; and coordinate subspace arrangements. When K is a flag complex, we extend well-known results on Artin and Coxeter groups by confirming that the relevant circulation group is homotopy equivalent to the space of loops ΩDJ(K). We define homotopy colimits for diagrams of topological monoids and topological groups, and show they commute with the formation of classifying spaces in a suitably generalised sense. We deduce that the homotopy colimit of the appropriate diagram of topological groups is a model for ΩDJ(K) for an arbitrary complex K, and that the natural projection onto the original colimit is a homotopy equivalence when K is flag. In this case, the two models are compatible.
منابع مشابه
ar X iv : m at h / 04 10 62 1 v 1 [ m at h . Q A ] 2 9 O ct 2 00 4 HOMOTOPY ALGEBRAS AND NONCOMMUTATIVE GEOMETRY
We study cohomology theories of strongly homotopy algebras, namely A∞, C∞ and L∞-algebras and establish the Hodge decomposition of Hochschild and cyclic cohomology of C∞-algebras thus generalising previous work by Loday and Gerstenhaber-Schack. These results are then used to show that a C∞-algebra with an invariant inner product on its cohomology can be uniquely extended to a symplectic C∞-alge...
متن کاملar X iv : 0 70 5 . 26 81 v 1 [ m at h - ph ] 1 8 M ay 2 00 7 On Z - graded loop Lie algebras , loop groups , and Toda equations
Toda equations associated with twisted loop groups are considered. Such equations are specified by Z-gradations of the corresponding twisted loop Lie algebras. The classification of Toda equations related to twisted loop Lie algebras with integrable Z-gradations is discussed.
متن کاملar X iv : 0 70 5 . 08 66 v 2 [ he p - th ] 1 0 M ay 2 00 7 Infinite loop superalgebras of the Dirac theory on the Euclidean Taub - NUT space
The Dirac theory in the Euclidean Taub-NUT space gives rise to a large collection of conserved operators associated to genuine or hidden symmetries. They are involved in interesting algebraic structures as dynamical algebras or even infinite-dimensional algebras or superalgebras. One presents here the infinite-dimensional superalgebra specific to the Dirac theory in manifolds carrying the Gross...
متن کاملar X iv : 0 70 5 . 08 66 v 1 [ he p - th ] 7 M ay 2 00 7 Infinite loop superalgebras of the Dirac theory on the Euclidean Taub - NUT space
The Dirac theory in the Euclidean Taub-NUT space gives rise to a large collection of conserved operators associated to genuine or hidden symmetries. They are involved in interesting algebraic structures as dynamical algebras or even infinite-dimensional algebras or superalgebras. One presents here the infinite-dimensional superalgebra specific to the Dirac theory in manifolds carrying the Gross...
متن کاملar X iv : 0 70 6 . 03 22 v 1 [ m at h . R T ] 3 J un 2 00 7 LOOP SPACES AND LANGLANDS PARAMETERS
We apply the technique of S-equivariant localization to sheaves on loop spaces in derived algebraic geometry, and obtain a fundamental link between two families of categories at the heart of geometric representation theory. Namely, we categorify the well known relationship between free loop spaces, cyclic homology and de Rham cohomology to recover the category of D-modules on a smooth stack X a...
متن کامل