## Colimits, Stanley-Reisner Algebras, and Loop Spaces (2003)

Citations: | 4 - 3 self |

### BibTeX

@MISC{Panov03colimits,stanley-reisner,

author = {Taras Panov and Nigel Ray and Rainer Vogt},

title = {Colimits, Stanley-Reisner Algebras, and Loop Spaces},

year = {2003}

}

### OpenURL

### Abstract

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.

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Citation Context ...e during June 2001. They are particularly grateful to the organisers for providing the opportunity to work in such magnicent surroundings. 2. Categorical Prerequisites We refer to the books of Kelly [=-=21]-=- and Borceux [3] for notation and terminology associated with the theory of enriched categories, and to Barr and Wells [1] for background on the theory of monads (otherwise known as triples). For more... |

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Citation Context ...plex K, in various algebraic and topological categories. We are particularly interested in colimits and homotopy colimits of such diagrams. We are motivated by Davis and Januszkiewicz's investigation =-=[12]-=- of toric manifolds, in which Ksrst arises as the boundary of the quotient polytope. In the course of their cohomological computations, Davis and Januszkiewicz construct real and complex versions of a... |

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Citation Context ...t(K)), defined by the vertices v. The facets determine the faces, according to the expression B(σ ↓cat(K)) = ⋂ B(v ↓cat(K)) for each σ ∈ K, and form a panel structure on Bcat(K) as described by Davis =-=[11]-=-. This terminology is motivated by our next example, which lies at the heart of recent developments in the theory of toric manifolds. Example 3.6. The boundary of a simplicial polytope P is a simplici... |

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Citation Context ...hey refer to the second horizontalsbration of the diagram (6.6), which is homotopy equivalent to the third whenever K = L issag, by Proposition 6.1. The second examples also appeal to James's Theorem =-=[19-=-], which identies the loop space S n with the free monoid F + (S n 1 ) for any n > 1. Examples 6.8. If K is the discretesag complex V , then U F (K) is homotopy equivalent to the commutator subgroup o... |

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Citation Context ...). Such groups are sometimes called graph groups, and are special examples of graph products [10]. As explained to us by Dave Benson, neither should be confused with the graphs of groups described in =-=-=-[31]. In the continuous case, we dene the circulation group Cir(K (1) ) as colim tmg T K in tmg. Every element of Cir(K (1) ) may therefore be represented as a word (4.4) t i 1 (1) t i k (k); where... |

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Citation Context ...ired, using the isomorphism D ~ a op a( ; a) = D(a) of (2.15). It is important to establish when the simplicial topological monoids B tmg (; a; D) are proper simplicial spaces, in the sense of [25]=-=-=-, because we are interested in the homotopy type of their realisations. This is achieved in Proposition 7.8, and leads on to the analogue of the Homotopy Lemma for tmg. These are two of the more memor... |

38 |
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Citation Context ...ial algebra. For example, Herzog, Reiner, and Welker [17] discuss combinatorial issues associated with calculating the k-vector spaces Tor SR k (K) (k; k) over an arbitrary groundseld k, and refer to =-=[16]-=- for historical background. Such calculations have applications to diagonal subspace arrangements, as explained by Peeva, Reiner and Welker [28]. Since these Tor spaces also represent the E 2 -term of... |

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Citation Context ...erminology associated with the theory of enriched categories, and to Barr and Wells [1] for background on the theory of monads (otherwise known as triples). For more specic results, we cite [14] and [=-=18]-=-. Unless otherwise stated, we assume that all our categories are enriched in one of the topological senses 4 TARAS PANOV, NIGEL RAY, AND RAINER VOGT described below, and that functors are continuous. ... |

33 |
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Citation Context ...oposition 5.11. The space ZG (K; 2 V ) is homotopy equivalent to U F (K), for any complex K. Proof. Substitute L = 2 V in Corollary 5.3 and apply (5.10). By specialising certain results of [37] and [=-=38-=-], we may also describe S W = 2K F W n 0 as a homotopy colimit. This space is dual to U F (K), and appears to have a more manageable homotopy type in many relevant cases. For G = C 2 and T , a versio... |

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Citation Context ...1. They are particularly grateful to the organisers for providing the opportunity to work in such magnicent surroundings. 2. Categorical Prerequisites We refer to the books of Kelly [21] and Borceux [=-=3-=-] for notation and terminology associated with the theory of enriched categories, and to Barr and Wells [1] for background on the theory of monads (otherwise known as triples). For more specic results... |

30 |
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Citation Context ... of theselds R or C . The study of the coordinate subspace arrangements A F (K), together with their complements, is a special case of a well-developed theory whose history is rich and colourful (see =-=[2-=-], for example). In the exterior case, we replace F by the union of a countably innite collection of 1-dimensional cones in R 2 , which 16 TARAS PANOV, NIGEL RAY, AND RAINER VOGT we call a 1-star and ... |

26 |
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Citation Context ...analogues of hK for all complexes K. Our interest in the loop spaces colim + (BG) K has been stimulated by several ongoing programmes in combinatorial algebra. For example, Herzog, Reiner, and Welker =-=[17]-=- discuss combinatorial issues associated with calculating the k-vector spaces Tor SR k (K) (k; k) over an arbitrary groundseld k, and refer to [16] for historical background. Such calculations have ap... |

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Citation Context ...s and their geometrical interpretations. Some of the results appear in [7], but we believe that our approach oers an attractive and ecient alternative, and eases generalisation. We refer to [18] and [=-=36]-=- for the notation and fundamental properties of homotopy colimits. Several of the results we use are also summarised in [37], together with additional information on combinatorial applications. We beg... |

24 |
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Citation Context ...tanley-Reisner Q-algebra of the simplicial complex K, and written SRQ (K). This ring is a fascinating invariant of K, and re ects many of its combinatorial and geometrical properties, as explained in =-=[-=-33]. Its Q-dual is a graded incidence coalgebra [20], which we denote by SR Q (K). We dene a cat(K) op -diagram DK in top+ as follows. The value of DK on each face is the discrete space + , obtained ... |

23 |
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Citation Context ...that the two models are compatible, and homotopy equivalent, when K issag. We take the category top of k-spaces X and continuous functions f : X ! Y as our underlying topological framework, following =-=[3-=-5]. Every function space Y X is endowed with the corresponding k-topology. Many of the spaces we consider have a distinguished basepoint , and we write top+ for the category of pairs (X; ) and basepoi... |

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Citation Context ...nt surroundings. 2. Categorical Prerequisites We refer to the books of Kelly [21] and Borceux [3] for notation and terminology associated with the theory of enriched categories, and to Barr and Wells =-=[1-=-] for background on the theory of monads (otherwise known as triples). For more specic results, we cite [14] and [18]. Unless otherwise stated, we assume that all our categories are enriched in one of... |

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13 |
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Citation Context ...lly described by a complementary construction involving only the 1-skeleton K (1) of K. Whenever K is determined entirely by K (1) it is known as a flag complex, and results such as those of [12] and =-=[22]-=- may be interpreted as showing that the associated Coxeter and Artin groups are homotopy equivalent to the loop spaces ΩDJ(K), in the real and exterior cases respectively. In other words, the groups a... |

11 |
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(Show Context)
Citation Context ...M n Y n ), it suces to assume that Y is discrete; in this case, BM ^ Y ! B y M y COLIMITS, STANLEY-REISNER ALGEBRAS, AND LOOP SPACES 23 is a homotopy equivalence by the same result of Fiedorowicz [15]. We apply Theorem 7.12 to construct our general model for DJ (K), but require a commutative diagram to clarify its relationship with the special case hK of Proposition 6.3. We deal with a op op... |

10 |
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Citation Context ...notes the commutator b i b j b 1 i b 1 j ), and so is isomorphic to the right-angled Artin group Art(K (1) ). Such groups are sometimes called graph groups, and are special examples of graph products =-=[10-=-]. As explained to us by Dave Benson, neither should be confused with the graphs of groups described in [31]. In the continuous case, we dene the circulation group Cir(K (1) ) as colim tmg T K in tmg.... |

10 |
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(Show Context)
Citation Context ... realisation Btmg (∗,a,D) is wellpointed. Proof. By Lemma 7.7, each degeneracy map Btmg n (∗,a,D) → Btmg n+1 (∗,a,D) is a closed cofibration. The first result then follows from Lillig’s Union Theorem =-=[23]-=- for cofibrations. So22 TARAS PANOV, NIGEL RAY, AND RAINER VOGT Btmg 0 (∗,a,D) ⊂ Btmg (∗,a,D) is a closed cofibration and Btmg 0 (∗,a,D) is well-pointed, yielding the second result. As described in E... |

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(Show Context)
Citation Context ...(K) (k; k) over an arbitrary groundseld k, and refer to [16] for historical background. Such calculations have applications to diagonal subspace arrangements, as explained by Peeva, Reiner and Welker =-=[28-=-]. Since these Tor spaces also represent the E 2 -term of the Eilenberg-Moore spectral sequence for H ( DJ (K); k), it seems well worth pursuing geometrical connections. We consider the algebraic imp... |

8 | Principal quasifibrations and fibre homotopy equivalence of bundles - Dold, R - 1959 |

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(Show Context)
Citation Context ...v [7] in the real and complex cases, and to Kim and Roush [22] in the exterior case (at least when K is 1-dimensional). In homology, they may be made in the context of incidence coalgebras, following =-=[29]-=-. In both cases, the maps of (4.1) induce the homomorphisms (4.9). Such calculations do not themselves identify colim + (BC 2 ) K and colim + (BT ) K with Davis and Januszkiewicz's constructions. Neve... |

5 |
E∞-spaces, group completions, and permutative categories, New Developments in Topology
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Citation Context ...(a) of (2.15).COLIMITS, STANLEY-REISNER ALGEBRAS, AND LOOP SPACES 21 It is important to establish when the simplicial topological monoids Btmg • (∗,a,D) are proper simplicial spaces, in the sense of =-=[25]-=-, because we are interested in the homotopy type of their realisations. This is achieved in Proposition 7.8, and leads on to the analogue of the Homotopy Lemma for tmg. These are two of the more memor... |

4 |
Groups generated by re and aspherical manifolds not covered by Euclidean space
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Citation Context ...(K)), dened by the vertices v. The facets determine the faces, according to the expression B(#cat(K)) = \ v2 B(v#cat(K)) for each 2 K, and form a panel structure on Bcat(K) as described by Davis [11]=-=-=-. This terminology is motivated by our next example, which lies at the heart of recent developments in the theory of toric manifolds. Example 3.6. The boundary of a simplicial polytope P is a simplici... |

4 |
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Citation Context ...LEY-REISNER ALGEBRAS, AND LOOP SPACES 5 topologically parametrized diagrams in the enriched setting; in other words, t is t-complete and t-cocomplete. A summary of the details for top can be found in =-=[26-=-]. Amongst indexed limits and colimits, the enriched analogues of products and coproducts are particularly important. Denitions 2.4. An s-enriched category r is tensored and cotensored over s if there... |

4 | Torus actions, combinatorial topology and homological algebra - Buchstaber, Panov |

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3 |
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Citation Context ..., there is a homotopy equivalence for any flag complex K. colim + (BG) K ≃ B colim tmg G K Since both cases are discrete, B colim tmg G K is, of course, an Eilenberg-Mac Lane space; Charney and Davis =-=[9]-=- have since identified good models for BA, given any Artin group A. Proposition 4.10 fails for arbitrary complexes K, as our next examples show. Examples 4.11. Proposition 4.10 applies when K = V , be... |

3 | On the homotopy-commutativity of groups and loop spaces, Mem - Sugawara |

3 |
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(Show Context)
Citation Context ...ralisation. We refer to [18] and [36] for the notation and fundamental properties of homotopy colimits. Several14 TARAS PANOV, NIGEL RAY, AND RAINER VOGT of the results we use are also summarised in =-=[37]-=-, together with additional information on combinatorial applications. We begin with a general construction, based on a well-pointed topological group Γ and a diagram H : a → tmg of closed subgroups an... |

2 |
Strongly homotopy-commutative monoids revisited
- Brinkmeier
(Show Context)
Citation Context ... and we emphasise this requirement as it arises. Several other important categories are related to top+ . These include tmonh, consisting of associative topological monoids and homotopy homomorphisms =-=[5]-=- (essentially equivalent to Sugawara's strongly homotopy multiplicative maps [34]), and its subcategory tmon, in which the homorphisms are strict. Again, the forgetful functor tmon ! top+ is faithful.... |

2 |
Topological groupoids 1: universal constructions
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(Show Context)
Citation Context ... , even when the diagram consists entirely of topological groups. Pioneering results on the completeness and cocompleteness of categories of topological monoids and topological groups may be found in =-=[6]-=-. Our main deduction from Proposition 2.10 is that tmon and tgrp are tensored over top+ . By studying the isomorphisms (2.5), we may construct the tensors explicitly; they are described as pushouts in... |

2 |
Finite K(; 1) for Artin groups
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(Show Context)
Citation Context ... C, there is a homotopy equivalence colim + (BG) K ' B colim tmg G K for anysag complex K. Since both cases are discrete, B colim tmg G K is, of course, an Eilenberg-Mac Lane space; Charney and Davis =-=[9-=-] have since identied good models for BA, given any Artin group A. Proposition 4.10 fails for arbitrary complexes K, as our next examples show. Examples 4.11. Proposition 4.10 applies when K = V , bec... |

2 |
Homology of certain algebras de by graphs
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- 1980
(Show Context)
Citation Context ...ually described by a complementary construction involving only the 1-skeleton K (1) of K. Whenever K is determined entirely by K (1) it is known as asag complex, and results such as those of [12] and =-=[22]-=- may be interpreted as showing that the associated Coxeter and Artin groups are homotopy equivalent to the loop spaces DJ (K), in the real and exterior cases respectively. In other words, the groups a... |

2 |
On the homotopy-commutativity of groups and loop spaces
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(Show Context)
Citation Context ...ies are related to top+ . These include tmonh, consisting of associative topological monoids and homotopy homomorphisms [5] (essentially equivalent to Sugawara's strongly homotopy multiplicative maps =-=[34-=-]), and its subcategory tmon, in which the homorphisms are strict. Again, the forgetful functor tmon ! top+ is faithful. Limiting the objects to topological groups denes a further subcategory tgrp, wh... |

2 |
Živaljević, Homotopy colimits – comparison lemmas for combinatorial applications
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(Show Context)
Citation Context ...tive and ecient alternative, and eases generalisation. We refer to [18] and [36] for the notation and fundamental properties of homotopy colimits. Several of the results we use are also summarised in =-=[37]-=-, together with additional information on combinatorial applications. We begin with a general construction, based on a well-pointed topological group and a diagram H : a ! tmg of closed subgroups and ... |

1 |
Buchstaber and Taras E Panov. Torus actions, combinatorial topology, and homological algebra
- Victor
(Show Context)
Citation Context ...of a space whose cohomology ring is isomorphic to the Stanley-Reisner algebra of K, over Z=2 and Z respectively. We denote the homotopy type of these spaces by DJ (K), and follow Buchstaber and Panov =-=[7]-=- by describing them as colimits of diagrams of classifying spaces. In this context, an exterior version arises naturally as an alternative. Suggestively, the cohomology algebras and homology coalgebra... |

1 |
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(Show Context)
Citation Context ...tion B tmg (; a; D) is wellpointed. Proof. By Lemma 7.7, each degeneracy map B tmg n (; a; D) ! B tmg n+1 (; a; D) is a closed co- bration. Thesrst result then follows from Lillig's Union Theorem [23] for cobrations. So B tmg 0 (; a; D) B tmg (; a; D) is a closed cobration and B tmg 0 (; a; D) is well-pointed, yielding the second result. As described in Examples 2.13, every simplicial obj... |

1 |
The homology and homotopy theory of certain loop spaces
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(Show Context)
Citation Context ...implications for traditional homotopy theoretic invariants such as Whitehead products, Samelson products, and their higher analogues and iterates. We hope to deal with these issues in subsequent work =-=[27]-=-. We now summarise the contents of each section. It is particularly convenient to use the language of enriched category theory, so we devote Section 2 to establishing the notation, conventions and res... |

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for E∞ ring spectra
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Citation Context ...LEY-REISNER ALGEBRAS, AND LOOP SPACES 5 topologically parametrized diagrams in the enriched setting; in other words, t is t-complete and t-cocomplete. A summary of the details for top can be found in =-=[26]-=-. Amongst indexed limits and colimits, the enriched analogues of products and coproducts are particularly important. Definitions 2.4. An s-enriched category r is tensored and cotensored over s if ther... |

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