Homotopy Colimits - Comparison Lemmas for Combinatorial Applications

J. reine angew. Math. 50 9 (1999), 117-149 Journal für die reine und angewandte Mathematik © W a l t e r d e G r u y t e r Berlin · New York 1999

Homotopy colimits - comparison lemmas for combinatorial applications

B y Volkmar Welker1) and Günter M. Ziegler2) at Berlin, and Rade T. Ž i v a l j e v i ć 3 ) at Belgrade

A b s t r a c t .W e provide a “toolkit” of basic lemmas f o r the comparison o f homotopy types o f homotopy colimits o f diagrams of spaces over small categories. W e show how this toolkit can b e u s e d i n quite d i f f e r e n t fields o f applications. W e demonstrate this with respect to

1 . Björner’s “Generalized Homotopy Complementation Formula” [5],

2. the topology of toric varieties,

3. the study o f homotopy types o f arrangements o f subspaces,

4. the analysis o f homotopy types o f subgroup complexes.

1. Introduction

The aim of this paper i s to advertise homotopy colimit considerations f o r topological investigations i n combinatorics. For this, w e provide a toolkit and demonstrate its u s e f u l n e s s b y a number of applications.

A diagram of spaces is a functor from a small category to the category of topological spaces. In various topological, geometric, algebraic and combinatorial situations one has to deal with structures that can profitably be interpreted as (co)limits or homotopy (co)limits of diagrams over small categories, specially over (finite) posets. In fact, if a space is written as a finite union of (simpler) pieces, then it is the colimit of a corresponding diagram of spaces. While (co)limits do not have good functorial properties in homotopy theory, they can usually be replaced by homotopy (co)limits (Puppe [32] may have been the first to exploit this). Homotopy (co)limits have much better functorial properties. Thus there is

1) Supported by Deutsche Forschungsgemeinschaft (DFG). 2) Supported by a “Gerhard-Hess-Förderpreis” of the DFG, which also provided support for the authors’ joint work at the Konrad-Zuse-Zentrum für Informationstechnik Berlin (ZIB) in 1994. 3) Partially supported by the Serbian Science Foundation, Grant 04 M03.

Bereitgestellt von | Technische Universität Berlin Angemeldet Heruntergeladen am | 09.11.18 17:08 118 Welker, Ziegler and Živaljević, Homotopy colimits a wide variety of techniques to manipulate diagrams of spaces in such a way that the homotopy (co)limit is preserved (up to homotopy type).

Basic work on homotopy (co)limits has been done b y S e g a l [37], B o u s f i e l d and Kan [7], tom Dieck [42], V o g t [44], and Dwyer and Kan [11], [12]. S e e Hollender and V o g t [21] f o r a recent survey.

T w o k e y results i n this setting are the “Projection Lemma” [3], [7], XII.3.1(iv), [15], [37], [49] which sometimes allows one to replace colimits b y homotopy colimits, and the “Homotopy Lemma” [42], [7], XII. 4.2, [44] which compares the homotopy types o f diagrams over the same small category. These tools have found striking applications, f o r example, i n the study o f subspace arrangements [49], [41], [36].

The choice o f contents f o r our “toolkit f o r the manipulation o f homotopy colimits” i s partially motivated b y the usefulness o f corresponding lemmas i n the special case o f order complexes (the discrete case, when the spaces o f the diagram are points). In this case, there i s a solid amount o f theory available, which has proved to be extremely powerful and u s e f u l i n quite diverse situations. The k e y result i s the Quillen Fiber Theorem (Quillen's “Theorem A” [33], [34], s e e below). A l l other basic tools o f the “homotopy theory o f posets”, such as the crosscut theorem, order homotopy theorem, complementation f o r ¬ mulas, etc., can be derived f r o m it. W e r e f e r to Björner [4] f o r an excellent account o f the theory [4], Sect. 1 0 , f o r an extensive survey o f applications [4], Part I, and f o r further references.

Homotopy limits have not found immediate applications i n combinatorics and discrete geometry so far, and this i s the reason w h y w e restrict our attention to the case o f homotopy colimits. Also note that many results about homotopy limits can be derived f r o m the case o f homotopy colimits b y standard duality procedures ( s e e B o u s f i e l d and Kan [7], XII. 4.1, and Hollender and V o g t [21], Sect. 3).

W e provide several applications o f our methods to various areas within mathematics. As a first application, i n the field o f topological combinatorics, w e present a n e w proof o f a result b y Björner on the homotopy type o f complexes [5], which generalizes the Homotopy Complementation Formula o f Björner and Walker [6], a tool which has proved to be v e r y powerful i n combinatorics. S i n c e this proof a f f o r d s the application o f many o f the techniques provided i n this paper, w e g i v e a detailed exposition of it here.

Then w e present a n e w v i e w o f toric varieties. Namely, w e start with the observation that toric varieties are homeomorphic to homotopy colimits over the f a c e poset o f the f a n d e f i n i n g the variety. This immediately leads to a spectral sequence to compute the homology o f toric varieties isomorphic to one already employed b y Danilov [9].

We derive a new “combinatorial formula” for the homotopy types of quite general arrangements (such as “Grassmannian” arrangements) that are associated to linear sub- space arrangements by suitable functorial constructions. More briefly we cover two appli¬ cations for which details are contained in other papers: We describe a new result on the homotopy type of the order complex of the poset Sp(G) of non-trivial p-subgroups of a finite group G [31], and we review results obtained by homotopy limit methods on the topology of subspace arrangements in [49], and provide the equivalence with the results of Vassiliev [43].

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2. F u n d a m e n t a l c o n c e p t s a n d c o n s t r u c t i o n s

2.1. Basic definitions and motivating examples of diagrams. In the following, all cate¬ gories are small, so their objects and morphisms form sets. Any partially ordered set can be considered as a category “with morphisms pointing down”, that is, for $ there is a (unique) arrow x → y if and only if $.

A diagram of spaces over a small category A is a covariant functor $ into the category Top of topological spaces. We denote for an object $ the image under $ by Fa or by F (a), and for a morphism $ the image $ or b y F(g). I f there i s a unique morphism g : a → b i n A between a and b, then w e write fab for $. A morphism $ of diagrams $ and $ is a functor F:A → B together with a natural transformation α from $ to $. Given a diagram $, the homotopy colimit hocolim $ i s a space associated to $ b y a homotopy mixing construction, s e e Section 2.2. B e f o r e w e proceed with a reasonably detailed outline o f the theory, w e introduce several motivating examples o f diagrams o f spaces.

Constant diagram. For a topological space X the constant diagram $ is defined by sending each object of A to the space X and each morphism to the identity id : X → X . Of particular interest is the case when $ is the one-point space. In this case the constant diagram $ leads, via homotopy colimits, to the construction of the classifying space BA of the category A. In the special case when A = P is a partially ordered set—a po s e t for short—the classifying space BP will be seen to coincide with the order complex Δ (P) of P .

Group diagram. Given a discrete group G, let A G be th e ca t e g o r y wh i c h co n s i s t s of one single object and a morphism for each element $. Then the classifying space of this category is the K(G,1)-space $ of the group G. Its universal cover EG is constructed in a similar way from the category whose objects are the elements of the group. Here for each pair of elements $ the unique morphism from g to h is given by h g -1. Finally, any AG-diagram $ can be interpreted as a G-space X and it turns out that $.

Subspace diagrams. Let $ be a collection of subspaces of a topological space X. The intersection poset PU of the family U is the partially ordered set of all non¬ empty intersections $, ordered by reversed inclusion. Regard PU as a small cate- gory. Then the subspace diagram associated to U is the diagram $ sending each element o f PU to the corresponding intersection$,with inclusions as morphisms. This class o f examples can be u s e d to s e e many results f r o m topological combinatorics f r o m a higher perspective (e.g. Borsuk’s Nerve theorem [4]).

Arrangements as diagrams. An especially interesting class of examples arises if X in the previous example is a linear space (affine space, sphere, projective space) and the family U is an (affine, spherical, projective) subspace arrangement. The subspace diagrams that arise this way have been successfully used to deduce both new and old results about the homotopy and homology of these arrangements and their complements (see Ziegler and Živaljević [49], Schaper [36]).

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Orbit diagrams. Let G be a Lie group acting on a space X with finitely many orbit types. From this arises a natural diagram $ over the category OG of all G-orbits defined by $, see [14]. Recall that the orbit category OG is defined as the category with $ while the morphisms are G-equivariant maps $.

Generally any diagram over the orbit category OG w i l l be referred to as an orbit diagram.

Toric diagrams. A toric diagram $ is a diagram for which each space Dd is the standard torus $, and each morphism dab: Da→Db is a standard algebraic homomorphism, i.e., a homomorphism that arises from an integer matrix $. An important observation is that any compact (complex) toric variety can be interpreted as the homotopy colimit of a toric diagram over the face poset of a complete fan, see Section 5.3.

In order to avoid pathological behavior of the topological spaces involved and because this setting covers all application we can think of—in our combinatorial context — we assume that all spaces are compactly generated. For the same reason we restrict our attention to small categories which are not topologized. Note however that the general theory (see [21]) can be developed for the case of topological categories A for which the map $ is a cofibration.

2.2. L i m i t s p a c e c o n s t r u c t i o n s .W e w i l l now discuss several constructions o f “limit spaces” f r o m a diagram $. For that w e recall the notion o f a simplicial space and its geometric realization. The references provide more information about simplicial s e t s and spaces, their geometric realizations, and other general categorical constructions used i n this paper. Nevertheless, w e try to b e as self-contained as possible.

The category Ord has as objects the finite ordinals $; its mor¬ phisms are the non-decreasing maps f : [n] → [m] (i.e., $). The set of mor¬ phisms of Or d is ge n e r a t e d [2 5 ] , p. 4 , [1 9 ] , II . 2. 2 by th e ma p s $ a n d $ ,where $ is the increasing injection that does not assume the value i (for $), and $ is the non-decreasing surjection that assumes the value i twice (for $).

A simplicial space i s a contravariant functor F : Ord → Top. A simplicial set i s a contravariant functor F : Ord → Set. W e w i l l not distinguish between simplicial sets and simplicial spaces equipped with the discrete topology. SimplicialSets and SimplicialSpaces are categories with the “obvious” morphisms, that is, whose morphisms are the natural transformations between functors Ord → Set resp. Ord→Top.

There i s a simple geometric realization functor R : Ord → Top that associates with [n] the standard n-dimensional simplex

$,

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N o w w e are a b l e to r e v i e w s o m e important “ l i m i t s p a c e ” o r “total s p a c e ” c o n s t r u c ¬ tions a r i s i n g f r o m c a t e g o r i e s a n d diagrams.

Geometric realization. Given a simplicial space F one defines its geometric realiza¬ tion |F|. Denote by $ the image of [ n] under F, by $ the image of $ under F and by $ the image of $ under F. Then | F | is obtained as the quotient of the disjoint union $ modulo the equivalence relation “~” defined by

$ f o r $ a n d $, $ for $ and $.

W i t h this g e o m e t r i c realization, |·| i s a f u n c t o r f r o m SimplicialSpaces to Top.

C l a s s i f y i n g s p a c e s .I f A i s a n y s m a l l category, t h e n there i s a natural s i m p l i c i a l s e t FA : Ord→Set associated to it. Namely, the image of [n ] u n d e r F A consists of all sequences $ of objects $ and maps $ in the category A. The maps $ are given by

$,

$.

T h e maps $ are g i v e n b y

$.

T h e g e o m e t r i c realization o f the s i m p l i c i a l s e t FA i s the classifying space B(A) o f the c a t e ¬ g o r y A.

C o l i m i t . T h e co l i m i t of a diagram $ is the topological space obtained from the direct sum $ modulo the equivalence relation “~” generated by x ~ y fo r $, $ if $ for some g:a → b. We write colim $ for the colimit of the diagram $.

Homotopy colimit. The homotopy colimit of a diagram $ is defined as the quotient of the direct sum $ by an equivalence relation “~.” Here A↓a denotes, for a given category A and $, the category of all arrows a → b emanating from a with commutative triangles as morphisms between objects a → b 1 a n d a → b 2. If A = P i s

Bereitgestellt von | Technische Universität Berlin Angemeldet Heruntergeladen am | 09.11.18 17:08 122 Welker, Ziegler and Živaljević, Homotopy colimits a partially ordered set, then $. The equivalence relation “~” is defined as follows. For all morphisms f : a → b consider the maps:

$, $, $, $.

Then “~” i s the transitive closure o f α(p,x) ~β(p,x).

If it is important to emphasize the indexing category A, then the homotopy colimit of a diagram $ is denoted by $.

Bar construction. A very general construction of a limit space is a construction given by Hollender and Vogt [21]. Given diagrams $ and $ over a category A and its opposite category Aop, there is an associated total space $. Its construction generalizes all limit space constructions presented so far (except for the colimit construc¬ tion). To the data $ and $ one associates a simplicial set $. The space of n -simplices o f $ is the space of all “threads”

$, where $ and $. More precisely, this space can be described as the space $, where the direct sum i s taken over a l l n-chains

$ of morphisms in A. The maps $ are given by

$

$,

$,

$.

The maps $ are given by

$.

Denote by $ the geometric realization of the simplicial space $. The homotopy colimit and the classifying space are special cases of the geometric realization of simplicial spaces of the form $. Namely, let $ be the diagram over the category A that assigns to each object a in A the one-point space $. Then the classifying space of the category A is the geometric realization of $. For a diagram $ the homotopy colimit $ is homeomorphic to the space $ (see [21]).

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We may observe that for every diagram $ over a category that is a poset P there is a canonical partial order on $ defined by setting $ and $, i f and only if $ and dpp'(x) = x'. With this definition there is a canonical bijection $; however, the topology on the two spaces is different: in fact, with the usual topology on simplicial complexes the subspaces $ get a discrete topology. Note however, see Section 4.1, that there exists a convenient geometric model for $ realizing this space as a subspace of the join $. This construction can be s e e n as a natural extension o f the order complex construction to diagrams over posets.

E x a m p l e s 2.1.Here are some simple examples f o r the construction o f homotopy colimits o f diagrams over posets.

( i ) I f D p is a one-point space for all $, then $ is (isomorphic to) the order complex of P.

( i i ) I f P = { p } is a one-element poset, then $.

( i i i ) I f P = { p,p'} has two points and p' >p, then $ is the mapping cylinder o f t h e m a p f = dp'p : Dp' → Dp.

(The mapping cylinder i s homotopy equivalent to the image space Dp, which i s the colimit o f the diagram.)

( i v ) I f P = { p,p'} and p, p' are not comparable, then $ i s t h e disjoint union.

(In this case it agrees with the colimit $ of the diagram.)

( v ) L e t P = { p,p',p"} be the three-element poset with $, $, f o r w h i c h p' and p" are incomparable, and let $ be a diagram over P with $ a one-point space. Then the homotopy colimit $ is homeomorphic to the mapping cone of the map dpp' : Dp → Dp'.

(vi) If $ is a diagram such that the spaces Dp are identical, and the maps are identity maps, then $.

(The colimit o f such a diagram i s Dp, i f P i s connected.)

2.3. Simplicial homotopy lemma a n d the gluing lemma.

D e f i n i t i o n 2.2.A simplicial space F : Ord → Top i s called good i f the inclusion map σ(F[n]) → F([n + 1]) is a cofibration where $ is the space of all degenerated simplices i n F([n + 1]). According to L i l l i g ’ s union theorem f o r cofibrations [23], F is good if forany surjectivemap σ : [n +1] → [n] in Ord the inclusion Im(F((σ)) → F([n + 1]) is a closed cofibration.

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Proposition 2.3 (Simplicial Homotopy Lemma). Let A, B be two good simplicial spaces. Let F : A → B be a map of simplicial spaces. If Fn : An → Bn is a homotopy equivalence for all n, then the induced map |F|:|A| → |B|

is a homotopy equivalence.

Good simplicial spaces w e r e d e f i n e d b y S e g a l [38], Appendix. In [27] and i n [26] the c l o s e l y related, albeit not coinciding, classes o f proper and strictly proper spaces are introduced. Proofs o f Proposition 2 . 3 can b e found i n either o f the references mentioned above.

The Gluing Lemma i s a k e y result. It has repeatedly been used i n inductive proofs o f results establishing that two spaces are homotopy equivalent; a notable example i s the proof o f Proposition 2.3. This lemma i s a basic example f o r an important general principle which informally says that a local homotopy equivalence i s also a global homotopy e q u i ¬ valence - s e e the Homotopy Lemma 3 . 7 .

L e m m a 2 . 4 (Gluing Lemma [8], [42]).Consider the commutative diagram of spaces

$

$.

Assume that h0,, hl, h2 are homotopy equivalences and that either both f1 and g1 are cofi- brations or both f2 and g2 are cofibrations. Let $ ( r e s p . , $) denote the colimit (or push-out) of the diagram formed by the maps (f 1,f 2) ( r e s p . , (g1,g2)). Then the induced map h : X → Y is also a homotopy equivalence.

3. C o m p a r i s o n r e s u l t s f o r d i a g r a m s o f s p a c e s

In this section we set up the toolkit for applications of diagrams of spaces. In the subsequent section we will show how these very general results specialize to forms applicable in various combinatorial situations where the emphasis is on diagrams over (locally) finite posets. The toolkit consists of a sequence of propositions and lemmas most of which are part of standard tools of an algebraic topologist. These results allows us to recognize a topological space as the homotopy colimit of a diagram (Projection Lemma), to check if a given morphism $ of diagrams induces a homotopy equivalence

$

of the corresponding homotopy colimits (Homotopy Lemma), to change the underlying small category of the diagram without changing the homotopy type of the homotopy colimit, to “compute” a standard, combinatorial form of the space $ ( W e d g e Lemma), etc.

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3.1. Projection, wedge a n d the quasifibration lemma.

Proposition 3.1 (P r o j e c t i o n Le m m a [3 ] , [4 9 ] ) . Let P be a finite poset and let $ a diagram having the property that for each $ the induced map $ is a closed cofibration. Then the natural map

$ is a homotopy equivalence.

R e m a r k .A more general form o f the Projection Lemma has been established f o r the so called free diagrams over arbitrary small categories, s e e [14] and [15], Appendix HL. The f i r s t “Projection Lemma” was established b y S e g a l [37] f o r numerable coverings o f topological spaces.

The next result that w e are heading for, the W e d g e Lemma, has been formulated and u s e d before i n the case o f diagrams over posets [49]. In order to formulate and prove a more general form o f this lemma w e need two preliminary definitions.

Definition 3.2. A diagram $ is called a diagram with constant maps if f o r a l l $ and any nonidentity morphism f : a → b, $, the map X(f): X(a) → X(b) is a constant map. A diagram with coherent constant maps is a diagram with constant maps with the additional property that for any object $ there exists a point $ such that for any morphism f : b → a, $, the one element set Im(X(f)) is precisely { xa}. In addition it is assumed that all spaces X(a) are well pointed [8], which means that (X(a), {xa}) is a cofibration pair. If the category A has an initial object a, (i.e., an object such that for each $ there is a unique morphism from a to b) then a diagram with constant maps over A is called an initial diagram with constant maps.

Note that an initial diagram with constant maps i s automatically a diagram with coherent constant maps.

Recall, see Section 2.2, that for a given category A and $, the undercategory A↓a, o f t e n denoted b y a\A or a↓Id, i s the category with objects

$

and an arrow from $ to $ for each commutative triangle formed by f, g and a morphism $.

Definition 3.3. Given $, the truncated category obtained by deleting c and all incident morphisms is denoted by A\\c.If we delete from A↓a the initial object $ and all incident morphisms we obtain the truncated undercategory A↓a \\ida which is denoted by $.

Specially, if A is a poset category P, then A↓aand$ are the poset categories $ and P

9 Journal für Mathematik. Band 509

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L e m m a 3 . 4 . Let $ be an initial diagram with constant maps over a category A with a as an initial object. If X(b) is a one-element space for each object $,then the homotopy colimit of this diagram has the following join decomposition:

$, where B(A\\a)is the classifying space of A\\a. Moreover, if X(a) is not a singleton then these spaces are homeomorphic.

Proof. For an object $ l e t $ be the unique morphism con¬ necting a and b (e.g., fa = ida). If $ then $ hence $. If $ this is possible only if X (a) consists of a single element. In this case $ and it is well known [37] (see also Section 3.3) that B(A) is contractible i f A ha s an in i t i a l ob j e c t . On th e ot h e r ha n d th e ri g h t ha n d si d e is ob v i o u s l y co n t r a c t i b l e .

I f | X (a) | > 1 then the only non-identity morphisms with a as the domain or codomain are of the form $ f o r s o m e $, $. In this case the formula $ follows by the definition of the homotopy colimit through the bar-construction.

R e m a r k . There are interesting examples of diagrams $, see [49] and Section 5.4, which have the property that for all $ and f : a→b, $, the map X(f): X(a) → X(b) i s homotopically trivial. Typically these diagrams can be t r a n s ¬ formed, usually with the aid o f the Homotopy Lemma (Proposition 3.7), into diagrams with coherent constant maps without a f f e c t i n g the homotopy type o f the homotopy colimit. The following lemma shows that i n these cases w e can actually “compute” these homotopy types.

P r o p o s i t i o n 3.5 ( W e d g e Lemma).Suppose that $ is a diagram with coherent constant maps over A. If the category A is connected then the homotopy colimit of $ has the wedge decomposition $.

(It is assumed that the base point of $ is the base point of its subspace (Xa) while the base point of the connected CW-complex B (A) is arbitrary.)

Proof. For each $ we consider two diagrams:

• The initial diagram with constant maps X [a] : A↓a→Top,and $ i f $ and $. The maps are the obvious constant maps.

• The diagram with coherent constant maps Y[a] : A → Top, and b → {x b} if $ and $. Again the maps are the obvious constant maps.

Since $ is a diagram with coherent constant maps, there exist maps of diagrams $ and $ such that $. This implies that $ is a retract of $. In particular, $ is a closed subspace of $. Obviously, $. Moreover, $ is the colimit of the covering by the $.

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In other words a subset F of $ is closed if and only if $ is closed in $ for all $. This follows from the fact that this is true on the level of simplicial spaces and that the geometric realization functor commutes with colimits.

Each space $ i s homeomorphic to the push-out o f the diagram:

( 1 ) $.

From here we deduce that $ is homeomorphic to the push-out of the diagram

( 2 ) $.

Each o f the spaces B(A↓a) i s contractible to its base point ida. W e conclude f r o m the Gluing Lemma 2 . 4 that the push-out o f ( 2 ) i s homotopy equivalent to the push-out o f the following diagram:

( 3 ) $.

B y assumption B(A) i s connected which implies that α i s homotopic to a constant map. This observation together with the Gluing Lemma 2 . 4 and 3 . 4 complete the proof.□

Note that i f the category A i s not connected, then the formula from Proposition 3 . 5 applies to the diagrams induced on each o f the connected components.

The obvious “collapsing” map of diagrams $ where $ is an arbitrary A-diagram and $ is the constant point-diagram over A,inducesamap $. Under certain strong conditions this map turns out to be a qua- sifibration, a concept introduced by Dold and Thorn [10]. A map f : X → Y between topological spaces is a qu a s i f i b r a t i o n if for any $ and $ the induced maps $ are isomorphisms for all $. The following proposition is essentially due to Quillen [33].

Proposition 3.6 (Quasifibration Lemma [3 3 ] , §1). If $ is an A-diagram such that all the maps X(f): X(a) → X(b), $ are homotopy equivalences, then the natural projection map $ is a quasifibration. In particular, if B(A) is contractible, then for every $ the map $ is a homotopy equivalence.

3.2. F r o m l o c a l t o g l o b a l h o m o t o p y e q u i v a l e n c e s .This i s the first circle o f results about homotopies between homotopy colimits o f diagrams.

Proposition 3.7 (H o m o t o p y Le m m a ) . Suppose that $ and $ are diagrams over C and $ is a morphism such that α(c ) : X (c) → Y (c) is a (w e a k ) ho m o t o p y eq u i ¬ valence for all $. Then the induced map

$ is also a (weak) homotopy equivalence.

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This Homotopy Lemma is a central result about diagrams of spaces. As noted above it has evolved from the Gluing Lemma, Proposition 2.4, in the papers of tom Dieck [42], Bousfleld and Kan [7], XII. 4.2, Vogt [44] and others. Both the Homotopy Lemma and the following more general result from [21] for the bar construction (as discussed in Section 2 . 2 ) can be deduced f r o m the results stated i n Section 2 .

Proposition 3.8. Let $, $ and $, $ be diagrams. Suppose $ and $, are morphisms of diagrams such that α(c ) : X (c) → X' (c) and β(c) : Y(c) → Y'(c) are homotopy equivalences for all $. Then the induced map

$ is also a homotopy equivalence.

The Homotopy Lemma i s a v e r y convenient tool i f w e want to compare homotopy types o f homotopy colimits o f two diagrams over the same category C. For comparison of homotopy colimits o f diagrams over d i f f e r e n t categories w e have at our disposal the following two elegant results. They both describe possible changes o f the indexing category o f a diagram that do not a f f e c t the homotopy type o f the homotopy colimit.

Definition 3.9. Given a functor F : C → D, the pullback functor F* associates a C- diagram with every D -diagram: It is defined by $ f o r a n y D-diagram $. If $ l e t d ↓F be th e ca t e g o r y wh i c h ha s as ob j e c t s al l ar r o w s of th e fo r m d → F(c) f o r s o m e $ and for morphisms between arrows $ and $ commutative triangles $ where $. I f F = Id i s the identity functor w e recover i n d ↓ Id the definition o f undercategory f r o m Definition 3.3. Dually to Definition 3 . 9 one d e f i n e s the category d↑F. Again i n the case F = Id w e obtain the “overcategory” D↑d.

Proposition 3.10 (Cofinality Theorem [7], XL 9.2, [12], [21]). If a functor F:C→D is right cofinal, i.e., if the classifying space B(d↓F)is homotopically trivial for all $, then the induced map

$ is a homotopy equivalence.

The (Segal’s) homotopy pushdown functor $ associates a D-diagram to any C-diagram $, [38], [21].

Definition 3.11. For every category D there is a tautological (Dop×D))-diagram $ o f di s c r e t e spaces defined by $. I f $

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P r o p o s i t i o n 3.12 (Homotopy Pushdown Theorem).Given a functor F : C → D and a C-diagram $, there is a natural homotopy equivalence

$.

R e m a r k . The Homotopy Pushdown Theorem is particularly transparent if C = P and D = Q are poset categories. Suppose that F : P → Q is a functor (a monotone map). Then for each $, $, the “overcategory” q↑F is the poset category $ and the space $ is homeomorphic to the space $ where $ is the associated restriction diagram.

3.3. H o m o t o p i e s a r i s i n g f r o m n a t u r a l t r a n s f o r m a t i o n s .A proposition o f S e g a l [37], Prop. 2 . 1 says that any natural transformation of functors F0, F1 : C → D induces a h o m o ¬ topy on the l e v e l o f geometric realizations. This principle has numerous consequences and applications; some of them are collected in this section.

Recall that a morphism $ of diagrams $ and $ i s a functor F : A → B and a natural transformation α from $ to $. The result of Segal deals with topological categories and not with diagrams. However, the connection with results quoted here becomes obvious if a diagram $ is viewed as a topological category with $ as the space of objects and an arrow $ for each pair of objects x, y and a morphism $ such t h a t $, $ and X(f)(x)= y. We are indebted to the referee for the following proposition and its corollaries. The referee also pointed out that these results can be immediately deduced from Proposition 3.1.7 in [21]; however, in the more explicit form presented here they are closer to our intended com¬ binatorial applications.

Proposition 3.13. Suppose that $ and $ are diagrams of spaces over A and B. Assume that $ and $ are morphisms of these dia¬ grams such that either

1 . th e r e ex i s t s a na t u r a l tr a n s f o r m a t i o n $ such that $ or

2. a natural transformation $ such that $.

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$

$ $

Then (F,α) and (G,β) induce continuous maps

$ such that g is a left homotopy inverse to f,

$.

Moreover, if there exists a morphism of diagrams $ and either a natural trans¬ formation $ such that $ or a natural transformation $ such that $, then (F,α), (G,β) and (H,γ) induce homotopy equivalences

$.

Corollary 3.14. Let i : B→A be an inclusion of B as a full subcategory of A, : A F→B a functor which maps B to B, and $ a natural transformation. Let $ be an A-diagram, and let $denote the restriction of $ to B. Then the canonical inclusion map induces a homotopy equivalence

$.

Proof. The inclusion of diagrams $ induces an inclusion map $. The natural transformation ξdefinesa map $ of diagrams, $, which induces a map

$.

Then h(Φ) is both the right and left homotopy inverse ofh (Ξ). Indeed, this follows from Proposition 3.13 with (F,α) = ( H,γ) = (F,Xξ), (G,β) = (i,id), $, $, and $. □

C o r o l l a r y 3.15.Let F : A→B be a left adjoint to G : B→A. Let $ be an A-diagram. Then ( G , id) is a map of diagrams which induces a homotopy equivalence

$.

Proof. Let $ and $ be the adjunction transformations. Apply Proposition 3.13 with (F,α) = (H,γ) = (F,η), $, (G , id), $ and $. □

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4. D i a g r a m s o v e r p o s e t s

Diagrams over posets play a special role in this paper since the majority of diagrams arising in combinatorics are of this type. The situation resembles the use of diagrams of spaces in the theory of transformation groups where this technique is also of utmost im¬ portance. A topologist primarily interested in group actions may find it convenient to have the key statements formulated in the language of G -spaces although they are consequences of more general results. An example is the result that an equivariant map f : X → Y induces a homotopy equivalence $ if fH : XH → YH is a homotopy equivalence for any subgroup $.

A similar tendency exists i n combinatorics. For example, the reference [34], where Propositions 4.2 and 4.3 are formulated, deals only with posets despite the f a c t that Quillen himself was in possession of much more general statements, his w e l l known Theorems A and B [33]. A l l these results are relatives of theorems formulated i n Sections 3 . 3 and 3 . 2 and they have played an important role i n the development o f the general theory.

This i s the first reason w h y w e collect some o f the corollaries of the general results i n a separate section. A second reason i s that there exist results which hold f o r posets and w h i c h do not have u s e f u l analogs f o r general categories.

4.1. Homotopy colimits revisited. The homotopy colimit construction is supposed to be a direct extension of the order complex construction for (locally) finite posets P. It is shown in Section 5.1 that the join [4], $ of a family $ o f spaces has a simple description as the homotopy colimit of a natural diagram of spaces. Here w e note that conversely, at least i n the case o f posets, the homotopy colimit can b e described as a subspace o f a join o f spaces.

Proposition 4.1. Let $ be a diagram over a (locally) finite poset P. Let $ be the join of all spaces in this diagram and let

$.

Then this space is naturally homeomorphic to the homotopy colimit of $,

$.

This characterization of homotopy colimits generalizes the order complex construc¬ tion, since $, where $ is the constant diagram associated to P, as in Section 2.

4.2. Comparison results for diagrams over posets. In this section we briefly review some of the most useful comparison lemmas which are meaningful only for order complexes of posets. Examples of such results are the Crosscut Theorem 4.4 and the Homotopy Complementation Formula [6], the latter one being a special case of Theorem 5.2. These results have found many interesting combinatorial applications. Also, by showing some of the most useful corollaries of general results from Section 3 we provide a link between these results and the general theory of diagrams.

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Quillen’s Theorem A ([33]) and its specialization for posets the so called Fiber Theorem can be seen as a corollary and a result that motivates both the Cofinality Theorem (Proposition 3.10) and Segal’s Pushdown Theorem (Proposition 3.12). The Order Homo¬ topy Theorem is a corollary and a good illustration of results from Section 3.3.

Proposition 4.2 (Quillen Fiber Theorem: Quillen [33], [34], Walker [45]). If f : P→Q is a map of posets such that the order complex of $ is contractible for all $, then the simplicial map Δf : Δ(P) → Δ(Q) is a homotopy equivalence.

Similarly, if the order complex of $ is contractible for all $, then Δf is a homotopy equivalence.

Proposition 4.3 (Order Homotopy Theorem: Quillen [34], see Bjorner [4], (10)). If f:P→Pis a decreasing poset map (t h a t i s , $ for all $), then P is homotopy equivalent to f(P).

A join semilattice is a poset P such that every finite subset has a unique minimal upper bound. If P is of finite length, then this implies that P has a unique maximal element $. The crosscut complex Γ(P) of a join semilattice is the simplicial complex of all nonempty subsets of min(P) that have an upper bound in $.

Proposition 4.4 (Crosscut Theorem: see Bj ö r n e r [4 ] , (1 0 . 8 ) ) . For every join semilat¬ tice P of finite length, the crosscut complex Γ(P) is homotopy equivalent to the order com¬ plex of $.

Next we come to the combinatorial version of the Projection Lemma 3.1. The natural combinatorial setting for its application (see Sections 5.2 and 5.4) is the following: A topo- logical space X is covered by a (finite) set $ of closed subspaces. The intersection poset P of the covering $ is defined on the set of all spaces $, that occur as an intersection of spaces $ with the reversed inclusion as its order relation. There is a natural diagram $ associated to the covering. Namely, to each element $ one assigns the corresponding intersection Dp and d e f i n e s the maps as the natural i n c l u ¬ sions.

L e m m a 4 . 5 (P r o j e c t i o n Le m m a : [3 ] , [4 9 ] ) . Let $ be a covering of the space X by a finite set of closed subspaces. Let P be the intersection poset P and let $ be the corresponding P-diagram. Assume that all maps in this diagram are closed cofibrations. Then the natural map $ induces a homotopy equivalence $.

I n general a ‘standard’ application o f the homotopy colimit tools proceeds f r o m a diagram $ associated to a covering to another ‘more s i m p l e ’ diagram b y suitably m o d i f y ¬ i n g $ using a poset version o f the Homotopy Lemma 3 . 7 , the Cofinality Theorem 3 . 1 0 or the Upper Fiber Lemma.

L e m m a 4 . 6 (H o m o t o p y Le m m a ) . Let $ and $ both be P-diagrams, and assume that there is a map of diagrams $ such that αp : Dp→Ep is a homotopy equivalence for each $, then a induces a homotopy equivalence $.

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I f one wants to change the poset underlying a diagram one needs an analog o f the Cofinality Theorem 3 . 1 0 .

L e m m a 4 . 7 (Inverse Image Lemma). Let f : P→Qposet morphism and let $ be a Q-diagram. Define the P-diagram $by $ and f * epp' =ef(p)f(p').

If for all $ is contractible, then f induces a homotopy equivalence

$.

The following Upper Fiber Lemma i s a u s e f u l result which also permits us to compare homotopy colimits o f diagrams over d i f f e r e n t posets. For comparison w e g i v e two proofs o f this lemma. The first and elementary proof i s based on the basic principles w h i l e the second proof illustrates the u s e o f general tools developed i n earlier sections.

L e m m a 4 . 8 (Upper Fiber Lemma). Let $ be a P-diagram and let $ be a Q-diagram. Assume $ is a map of diagrams. If (F, α ) in d u c e s a ho m o t o p y eq u i v a l e n c e of th e restrictions

$ for all $,

then (F, α ) induces a homotopy equivalence $.

Proof. (1st proof) We proceed by induction on the cardinality |Q| of Q. Le t $ be a minimal element of Q. Th e n $ f o r $ and Q2 = Q\{q}. The poset $ equals Q>q. We set $ for i = 1,2,3, which implies $ and $.

I f q is th e un i q u e mi n i m a l el e m e n t of Q (i . e . , Q1 = Q), then the assertion follows trivially from the assumptions. Otherwise a induces a homotopy equivalence $ f o r i = 1, 2, 3, by induction. Since

$ and $,

it follows from the Gluing Lemma 2.4 that α induces a homotopy equivalence between $ and $.

(2nd proof) By the Homotopy Pushdown Theorem, $. By the remark after the Proposition 3.12, $ is the diagram defined by $. Let $ be thediagramdefinedby $. Then the natural map $ is a homotopy equivalence by Corollary 3.14. Finally, since by assumption the natural map $ i s a homotopy equivalence, the proposition follows from the Homotopy Lemma, Proposition 3 . 7 . □

If one is lucky—and surprisingly often one is indeed lucky—one is able to transform a diagram $ into a diagram that allows an application of the Wedge Lemma 3.5 whose poset version we state next.

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L e m m a 4 . 9 (W e d g e Le m m a ) . Let P be a poset with maximal element $. Let $ be a P-diagram so that there exist points $ for all $such that dp p ' is th e co n s t a n t ma p $, for p > p'. Then

$ where the wedge is formed by identifying $ with $, for $.

W e close this toolkit o f lemmas b y a result that i s a simple consequence o f the definition of a homotopy colimit.

L e m m a 4 . 1 0 (E m b e d d i n g Le m m a ) . Let $ be a map of P-diagrams. If $ is a closed embedding for every $, then α induces a closed embedding

$.

In particular, if $ is a P-diagram and $ is a subposet, then $ is a closed embedding.

5. A p p l i c a t i o n s

5.1. C o m b i n a t o r i a l o b j e c t s a s h o m o t o p y c o l i m i t s .In this section w e show how some w e l l known combinatorial constructions can b e u s e f u l l y related to diagrams and their homotopy colimits.

Proposition 5.1. Let $ be a collection of spaces. Denote by $ the poset of all nonempty faces of an n-simplex, i.e., the poset $ ordered by inclusion. Let $ be the following diagram associated to the collection $. For $ let $ and for $ let $ be the canonical projection. Then the join$of all spaces in the family $ is naturally homeomorphic to the homotopy colimit of this diagram $.

Figure 1. The join of S 0 an d S1 as a homotopy colimit

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The formula g i v e n i n Proposition 5 . 1 i s a v e r y simple and u s e f u l representation o f the join o f spaces exposing much o f the underlying combinatorial structure.

An important variation of the diagram above arises if X i = S 1 fo r al l i . This is a toric diagram in the sense of Section 5.3. There is a natural diagonal action of the circle group U(1) on all tori $. Dividing by this action leads to a new diagram $ and a well known fact is that this leads to a combinatorial description of the complex projective space $: $.

A very useful concept, which has found numerous applications in combinatorics, is the deleted join operation [4], [35], [50]. Given a simplicial complex K, the pth deleted join $ is the subcomplex of the join $ o f p co p i e s of K which consists of all simplices $ which have the form $ where $, i = 1,...,p, are pairwise vertex disjoint simplices of K . Some very important complexes can be constructed by iterating the join and the deleted join operation, for example the well known chessboard complexes Δp,q [4], [50], arise this way. The simplest complex of this form is $ where ΔN i s the standard N-dimensional simplex. This space, v i e w e d as a space with the obvious c y c l i c $-action, has been applied i n combinatorics to problems o f Tverberg type, to the ‘necklace’ partition problem, etc., s e e [4]. Motivated b y these and other similar examples, it i s natural to look f o r homotopy colimit representations for deleted joins as well.

There are two obvious ways to associate a diagram to the space of the form $. One can view $ as a subspace of $ which has the homotopy colimit representation described in Proposition 5.1. The subdiagram which arises this way is similar except that the products Kj are replaced by deleted products $, j = 1, . . . , p. Recall that the deleted product $ is a cell subcomplex of the cell complex Kj consisting of all cells of the form $, where $, k = 1,..., j, are pairwise vertex disjoint simplices K . Another in possibility is to link $ with a diagram over the face poset of K itself via the ‘collapsing’ map $. The diagram arising this way is a finite space diagram. For example if K= ΔN, the space $ is the homotopy colimit of a diagram of the type described in Proposition 5.1 where $ f o r a l l i = 0,1,..., N. These and other examples indicate that it may be useful to study discrete analogs of toric diagrams with the discrete ‘tori’ $ in place of T j = S 1 × .. . × S 1. Specially it is an interesting question which convex polytopes admit natural discrete toric diagrams.

5.2. B j ö r n e r ’ s g e n e r a l i z e d h o m o t o p y c o m p l e m e n t a t i o n f o r m u l a .Björner’s generalized homotopy complementation formula [5] i s an e f f e c t i v e tool to compute the homotopy type o f a simplicial complex Δ i n the case when a large, contractible induced subcomplex $ i s known, whose connections to the rest o f the complex are not too complicated.

In the following, w e provide a “diagrammatic” proof o f Bjorner’s result, thereby demonstrating the applicability o f some o f our lemmas.

Let $ be a finite (abstract) simplicial complex with vertex set S , le t $ b e a subset of its vertex set, and denote by $ the complement S\A of A .

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L e t $ be the induced subcomplex on A, and similarly $ the induced complex on $. In the following, the key assumption we will make is that $ is contractible.

T h e o r e m 5 . 2 (Generalized homotopy complementation formula: Björner [5]).For any simplicial complex $ and $, define a new simplicial complex TA by taking the union of all the simplicial complexes

$ for $, where p is an additional point$, and $ denotes the join of two complexes.

If $ is contractible, then Δ and TA are homotopy equivalent.

Proof Let P b e the poset o f a l l nonempty f a c e s o f Δ, ordered b y inclusion, and l e t

$, partially ordered b y the condition

$.

(Thus Q is isomorphic to the poset of all intervals in P , ordered by inclusion: $.)

For $ we define $.

Then we clearly have inclusion maps $ whenever $. This defines a P- diagram $, with $, where the equality holds b y definition o f the colimit (this i s a subspace diagram!), and the homotopy equivalence i s an application of the Projection Lemma 4.5.

Similarly, for $ w e d e f i n e

$.

Then we get inclusion maps $ whenever $. This defines a Q-diagram $, with

$, b y definition o f the colimit and the Projection Lemma. Thus, the claim of the theorem is reduced to proving that the homotopy colimits of the P-diagram $ and the Q-diagram $ are homotopy equivalent. This demands use of our new homotopy lemmas, since the posets P and Q are quite different.

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The canonical poset map to use i s

f : Q → P, (Σ,Τ)→Τ.

This f induces a homotopy equivalence between the posets P and Q (Walker [46]). To see this, we observe that the lower fibers $ are canonically isomorphic to $. On the poset $ we have an increasing map $ g i v e n b y $. Thus, by the Order Homotopy Theorem 4.3 the fiber $ is homo¬ topy equivalent to the image $, which is a cone.

Now w e modify $ and $ a little. W e d e f i n e a n e w Q-diagram $, whose spaces are

$, and whose maps are the obvious inclusions. Furthermore, there is a map $, which i s the identity map on Q, and between the spaces uses the inclusion maps

$ w h i c h are clearly homotopy equivalences, s i n c e $ i s contractible. Thus, b y the Homotopy Lemma 4.6, ψ1 induces a homotopy equivalence

$.

Similarly, w e d e f i n e a n e w P-diagram $, whose spaces are

$, and whose maps are the obvious inclusions. Furthermore, there is a map $, which i s the identity map on P, and between the spaces uses the inclusion maps

$ which are clearly homotopy equivalences. Thus, b y the Homotopy Lemma, ψ2 induces a homotopy equivalence $.

Finally, at this stage we see that $ is an inverse image diagram, $, where f i s a poset map whose lower fibers w e have already checked to be contractible. Hence the Inverse Image Lemma 4.7 implies a homotopy equivalence $ w h i c h completes the proof.

Alternatively, one could derive $ also from the Upper Fiber Lemma 4.8, together with the Projection Lemma 4.5, since $ and $ are subspace dia¬ grams as well. □

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We note that there are alternative ways to describe the construction of T A. For example, one can (as Björner does in [5]) start from a wedge (or a disjoint union) of the spaces $, and then check that all the identifications of the colimit $ are generated by identifying, for $, the identical subcomplexes $.

Also, there are countless variations possible, corresponding to d i f f e r e n t coverings o f the complex Δ. The beauty o f Björner’s set-up i s that his transformations o f Δ end up with a subspace diagram, and thus with a colimit instead o f a homotopy colimit, which leads to an e f f e c t i v e model f o r $. It s e e m s to us that the diagram techniques y i e l d an extremely natural and convenient setting f o r the proof o f the generalized homotopy c o m ¬ plementation formula and similar results.

5.3. T o r i c v a r i e t i e s .In this section w e g i v e a representation o f the topological space underlying a toric variety ( s e e Danilov [9], Fulton [18], and Ewald [13] for general b a c k ¬ ground on toric varieties) as the homotopy colimit o f a diagram. For this w e recall a description, due to MacPherson ( s e e Y a v i n and Fischli [48], [17]), o f a toric variety. A decomposition o f $ into a complex Σ o f closed, convex, polyhedral cones with apex 0 i s called a complete fan. I f a l l cones i n Σ are generated b y lattice points i n $, then Σ i s called rational. Assume that Σ i s a complete and rational f a n i n $. Then l e t $ be the c e l l d e c o m ¬ position o f the unit ball i n $ that i s dual to the one induced b y Σ. For$w e denote b y $ the c e l l i n $ that corresponds to $. Thus $ i s a c e l l o f dimension n — dim(σ).

We identify the n-torus $ with the image of the projection map $. For all cones $ the image of σ un d e r th i s pr o j e c t i o n is a su b t o r u s $ of $. Since σ is ra t i o n a l , th i s is a cl o s e d su b t o r u s of di m e n s i o n di m ( σ). Thus the quotient $ is a real torus of dimension n — dim(σ).

The toric variety XΣ is obtained from $ by taking the quotient of $ b y the action of $ on $ for each $. This leads to a nice (compact, Hausdorff) quotient space since we take quotients by larger tori on $. In particular, we see that the toric variety XΣ has a well-defined map $, for which the fiber over any interior point of $ is isomorphic to $.

Let PΣ b e the poset whose elements are i n bijection with the cones i n Σ and whose order relation i s d e f i n e d b y reversed inclusion o f the cones i n Σ. Thus PΣ i s the poset o f non-empty f a c e s o f $, ordered b y inclusion. In particular, PΣ has a largest element $ c o r ¬ responding to the 0-dimensional cone { 0 } . W e construct a diagram DΣ over the poset PΣ as follows. For $, set $. Topologically, $ is an (n — dim((σ))-torus. The map dτ,σ for $ is the map induced by the projection $.

P r o p o s i t i o n 5.3.Let Σ be a complete and rational fan in $. Then the toric variety XΣ is homeomorphic to the homotopy colimit of the diagram DΣ associated with Σ:

$.

Proof Let $ be the map that sends $ to its image in X Σ. By construction o f D Σ th e ma p $ is co m p a t i b l e wi t h th e eq u i v a l e n c e re l a t i o n $. H e n c e

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$ induces a map $: hocolim DΣ → XΣ. It i s routine to check that $ i s indeed a homeo- morphism.□

The resolution of singularities of a toric variety also fits our homotopy colimit framework. Namely, let Σ' , Σ be tw o co m p l e t e , ra t i o n a l fa n s in $ s u c h th a t Σ' is a refinement of Σ (i . e . , fo r ev e r y op e n co n e τ'° in Σ' th e r e is an op e n co n e Τ° in Σ su c h th a t $). Thus there is an induced map f : PΣ' → PΣ. Also assume that τ' is a cone in Σ' whose interior is contained in the interior of the cone τ of Σ. Th e n th e in c l u s i o n $ induces a surjective map $. It is easily seen that $ induces a map of diagrams. Hence there is an induced map $ : hocolim $. The map $ i s surjective s i n c e f and a l l$are surjective.

Proposition 5.4. Let Σ' be a complete rational fan which is the refinement of the com¬ plete rational fan Σ. Then there is a surjective map $.

It is well known that XΣ is non-singular if and only if Σ is si m p l i c i a l (i . e . , al l co n e s are simplicial) and unimodular (i.e., all full-dimensional cones are equivalent to $ under unimodular transformations from $). For an example that shows that $ is not a combinatorial invariant of Σ in ge n e r a l se e [2 8 ] . It is al s o well known that for any complete rational fan Σ th e r e is a si m p l i c i a l an d un i m o d u l a r complete rational fan Σ' wh i c h re f i n e s Σ. In th i s ca s e $ is a re s o l u t i o n of si n g u l a r i t i e s .

One can also use our results to investigate the (co)homology o f a toric variety. For this w e s e t up a spectral sequence introduced b y S e g a l [37], which uses the filtration o f hocolim D b y the s-skeleta o f the order complexes. For a simplicial complex Δ w e denote b y Δ5 its s-skeleton.

Assume D is a P -diagram for a poset P. Then we denote by hocolim D5 the image o f $ i n h o c o l i m D. The filtration $

defines a spectral sequence with termination at $ in the E2-term and $. Following Segal’s arguments one finds that $ is g i v e n b y $. Now assume (σ0<··· < σ5) × c is a (s + t) - c e l l i n

hocolim D5. Then the differential o f the c e l l complex hocolim D i s g i v e n b y

$

$.

Thus the differential $ applied to the cell (σ0< ··· < σ5) × c equals $, w h e r e c is a cycle in $.

From this it is easily seen that our spectral sequence is isomorphic to the deRahm- Hodge spectral sequence applied by Danilov [9], Chap. 3, § 12, to compute the cohomology of a toric variety.

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5.4. S u b s p a c e a r r a n g e m e n t s .Arrangements o f a f f i n e subspaces i n $ also allow an application o f the homotopy colimit method. Let A b e a f i n i t e s e t o f a f f i n e subspaces i n $. Let us denote b y $ the corresponding arrangement o f spheres i n the one-point com- pactification Sn o f $. Under our assumptions intersections o f spheres i n $ are again spheres (or the compactification point). The following result can b e deduced f r o m the Projection Lemma 4.5, the Homotopy Lemma 4.6, and the W e d g e Lemma 4.9.

T h e o r e m 5 . 5 (Ziegler and Ž i v a l j e v i ć [49]).Let A be a finite set of affine subspaces in $. Let $ be the one-point compactification of the set-theoretic union of the subspaces in A and let P be the intersection poset of A. Then

$.

An equivalent result can b e found in V a s s i l i e v [43], III. §6, Thm. 1 . In V a s s i l i e v ’ s formulation the spaces Δ(P

$ and $ are homotopic. If $ is empty, then we have to “interpret” $ as the sus¬ pension o f the empty space, which i s i n our definition the join with a two point space. Then the homotopy equivalence also f o l l o w s in this case.

B y Alexander duality on Sn w e i n f e r f r o m Theorem 5 . 5 the following formula o f Goresky and MacPherson [20].

C o r o l l a r y 5 . 6 (Goresky and MacPherson [20]).Let A be a finite set of affine sub¬ spaces in $. Let MA be the complement $ and let P be the intersection poset of A. Then

$, where codim(p) denotes the real codimension of the subspace corresponding to p.

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Analogous results for arrangements of spheres and projective spaces can be found in Goresky and MacPherson [20] and in [49]. In the following, we describe a simpler, more general, and more powerful approach that provides combinatorial formulas for quite general “Grassmannian arrangements”. Let A be a central arrangement in $ (or $, $) with intersection poset P and a dimension function $. Let $ b e the corresponding P-diagram o f linear spaces. In case each o f the linear subspaces Ap, $, is invariant under the action of a finite (or just closed) subgroup $ (resp. U(n), Sp(n)) of the orthogonal (unitary, symplectic) group, a natural step to make is to define the associated orbit diagram. More generally, if T is an op e r a t i o n (a fu n c t o r ) as s o ¬ ciating a space T(V) to a linear subspace $, where $, then T(A) denotes the diagram $ associated to the corresponding arrangement of subspaces in $. There are several examples that come up very naturally in the mathematical practice. For example, if $ is the operation of associating the unit sphere to the linear subspace $, then $ is the associated spherical diagram. Similarly, functors $, $ or simply P( V ) in bo t h ca s e s (a n d in th e ca s e of quaternionic spaces) lead to the corresponding projective arrangements $, $ or $, denoted simply by P( A ) . Pr o j e c t i v e di a g r a m s ar e sp e c i a l ca s e s of th e as s o c i a t e d Grassmann diagrams obtained with the aid of the functor

$.

Lens space arrangements Lm(A) are defined similarly, where $ i s the l e n s space associated to the unit sphere in a complex linear space V.

W e illustrate how the Homotopy Lemma 4 . 6 can b e applied to produce a c o m b i ¬ natorial description of the link of the related arrangement in all the special cases above. W e obtain i n particular simpler, stronger, and more natural proofs of some results f r o m [49], Thms. 2.11 and 2.14 about projective arrangements. The advantage of the proof below is that it uses the Homotopy Lemma in its simplest form and allows a uniform treatment of all the special cases above. It is clear that this method may be useful for other applications, say for other group actions, since the argument no longer requires the arrangement to be shifted to a more special position by a dilatation $, i = 1,...,n, f o r some ε > 0 .

Our objective is to show that the homotopy type of the link has a purely combinatorial description in terms of the poset P and the associated rank function $. The link o f a (spherical, a f f i n e , projective, lens, Grassmann etc.) arrangement i s the union o f all spaces in the arrangement. It follows f r o m the Projection Lemma that the link has the same homotopy type as the homotopy colimit o f the corresponding diagram.

We uniformly construct a combinatorially defined diagram which serves for compa¬ rison with the original one. Choose a flag $, $. L e t $ be one of the functors described above. Then T[F] = T[F](A), the flag diagram associated with T(A), is defined by $ where the morphisms T[F]p → T[F]q are the obvious inclusion maps. Every two flag diagrams T[F]and T[F'] are naturally isomorphic, thus the isomorphism type o f T[F] depends only on P and d.

We want to compare our diagram T(A) with the combinatorially defined diagram T[F](A). There does not seem to exist a natural map of diagrams between T(A) and

10 Journal für Mathematik. Band 509

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T[F](A)because e.g. the projection map is not natural. This difficulty was overcome in [49], i n the case o f projective diagrams, b y shifting the diagram T[F](A) to a more special position and b y applying a more general version o f the Homotopy Lemma b y V o g t [44] that allows noncommutative diagrams i f they commute up to coherent homotopies. The proof o f Theorem 5 . 8 shows that i n practical problems a v e r y natural idea to use i s the comparison with the third, so called “ample space” diagram that contains both T(A) and T[F](A)as subdiagrams. W e need a lemma which explains what i s meant b y “ample” i n a l l the interesting cases above.

L e m m a 5 . 7 . Let T: $, $, be one of the functors defined above which to every vector space $ associates the corresponding projective space P(V), Grassmann manifold Gk(V), the lens space Lm(V)(f o r $), or the unit sphere S(V). Then in each of these cases there exists a functor $ which associates with each subspace $ an “a m p l e ” su b s p a c e $ and to each inclusion V → W an inclusion of spaces RT(V) → RT(W)so that the following condition is satisfied.

Let $, dim(V) = k. Then for any W of dimension k with $, there is an inclusion $ so that the inclusion map

iw:T(W)→RT(V)

is a homotopy equivalence (actually, an inverse to a deformation retraction).

Proof. The space RT(V)is defined in a very similar way for all the examples above. For example in the case of the functor $ w e p u t $. The construction in the case of a projective functor $ is analogous, one removes the projective space $ f r o m $. The inclusion map $ i s a homotopy equivalence since $ is the total space of a vector space bundle (a tubular neighborhood) over P(V). Obviously $ f o r a n y W with the property $. Finally, iw : P(W) → RT(V)is a homotopy equivalence since there exists a linear map $ which maps V to W and leaves $ invariant. Something similar is done in the case of Grassmannians. Here, $ is defined by

$.

In this case $ is also the total space of a vector bundle over Gk(V) which can be seen as follows. If M k(V) is the Stiefel manifold of all 1-1 linear maps (matrices) $ then Gk(V) = Mk(V)/G(k) where G(k) = GL(k,K)is the appropriate group of linear automorphisms. Let $ be the projection map. Note that

$,

where $is the space of all linear maps, and that the group G(k) acts diagonally on the product $. Now it is enough to recall that if X an d Y are two G- spaces and if the action on Y is free, then the orbit space (X × Y ) / G of the diagonal action is represented as a fiber bundle

X → (X × Y)/G → Y/G

Bereitgestellt von | Technische Universität Berlin Angemeldet Heruntergeladen am | 09.11.18 17:08 Welker, Ziegler and Živaljević, Homotopy colimits 143 which in our case means that $ is fibered over Gk(V) with the fiber $. Hence, iv i s a homotopy equivalence. It i s shown that iw i s also a homotopy equivalence analogously to the case o f projective diagrams.

Finally, in the case of the “lens space” functor, let $. The proof that $is “ample” in the sense above is analogous and can rely on the fact that the sphere $ is a join of spheres S(V)and $, and that the action of the cyclic group $ respects this decomposition. □

Theorem 5.8 (H o m o t o p y ty p e s of ar r a n g e m e n t s ) . Let $ be a linear sub- space arrangement of $, where $ is one of the (skew) fields $, $ or $. Let P be the intersection poset of A, with the dimension function $, d(p) = dim(Ap), defined above, and set $ for $.

Let T : Vect → Top, $, be the projective, sphere, Grassmann or the lens space functor defined above and let T(A) be the corresponding arrangement of subspaces of $. Then there is a homotopy equivalence

$

P r o o f .We start by choosing the flag F, $, in sufficiently general position with respect to the arrangement A. This requirement means that for any Fi and $, i f $ the $. Let us define the associated “ample” space diagram $ b y $, $, where $ i s t h e “ample” space functor associated with T described i n Lemma 5 . 7 . Hence, there e x i s t two naturally d e f i n e d maps of diagrams α and β,

$ induced by the inclusion maps αp : T(Ap) → Rp and βp : T[F]p → Rp. By Lemma 5.7 these inclusion maps are homotopy equivalences. From here and the Homotopy Lemma it follows that $ and $, which implies

$.

Finally, we notice that the embeddings $ induce a diagram map

$, where $ denotes the “constant” P-diagram which has $ for all $, and iden¬ tity maps fpp, for $. From the Embedding Lemma 4.10 we get that $ is an embedding

$, and we easily identify the image of $ with the space $. □

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Note that for the homotopy formula for Grassmann arrangements in Theorem 6.4 only the truncated poset $ is relevant: This corresponds to the fact that spaces with d(p) < k do not have k-dimensional subspaces, so A an d

$ have the same associated k-Grassmann arrangement.

The following proposition shows that there i s a general decomposition formula f o r the homology o f f l a g diagrams and, a posteriori, o f the T- links o f $-arrangements f o r $.

Proposition 5.9 (H o m o l o g y of fl a g di a g r a m s ) . Let A be an arrangement of linear subspaces in $ and let T be one of the functors described above. Let T[F](A)be the com- binatorially defined flag diagram associated to the arrangement A and the functor T. We set $ and $. Assume that for the coefficient ring R and for all $

• the exact sequence of the pair $ splits in homology and

• the homology groups $ are free R-modules.

Then, $

$.

P r o o f . For this proof we fix the coefficient ring R used for homology computations and we write $ for $. For an arbitrary poset Q and a natural number k we introduce “constant” diagrams $ over Q, defined by $, $. If it is clear from the context which poset Q is used, we write $ for the diagram $ Note that our diagram T[F](A)can be squeezed in between two constant diagrams $ and $. Clearly, hocolim $. The Künneth theorem and the freeness o f R-modules $ imply

$.

From this observation and from the fact that the map $ i s a m o n o - morphism for $, we conclude that the map $, f o r $, induces a monomorphism o f homology groups. Then the composition o f homomorphisms

$ is a monomorphism too. Thus the first of them is also a monomorphism. It follows that the long exact sequence of the pair (hocolimT[F](A), $) splits and

$.

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Informally speaking, we peel from the homotopy colimit of the diagram T[F](A)the part of its homology coming from the constant subdiagram $. Let $ be the restriction of the diagram T[F](A)on the poset $. By excision, recalling the definition of the homotopy colimit hocolimT[F](A), we have

$.

By induction on i we may assume that for some $

$

$

$

The long exact sequence o f the triple

$ splits by the same argument as above. This means that we can peel from the homology $ the part isomorphic to

$ $.

The part that remains is isomorphic to the group $. The last group is by excision isomorphic to $. S o the Künneth formula, the process described above and induction on i lead to the desired formula. □

If the arrangement A is essential (i.e., if s = 0 a n d $, then both $ and $ are interpreted as empty spaces and the formula given in Proposition 5 . 9 can b e rewritten as follows.

Corollary 5.10.Let A be an essential arrangement satisfying the assumptions of Pro¬ position 5 . 9 . Then

$.

For e x a m p l e i f T = P i s t h e f u n c t o r w h i c h a s s o c i a t e s t h e c o m p l e x p r o j e c t i v e s p a c e P(V) t o e v e r y c o m p l e x l i n e a r s p a c e V, t h e n t h e f o r m u l a g i v e n i n C o r o l l a r y 5 . 1 0 h a s t h e f o l l o w i n g f o r m

$,

Bereitgestellt von | Technische Universität Berlin Angemeldet Heruntergeladen am | 09.11.18 17:08 146 Welker, Ziegler and Živaljević, Homotopy colimits where $ and $ f o r a l l k > 0. This formula together with its counterpart for the real projective arrangements was formulated and proved in [49]. Unfortunately, as it was kindly pointed to us by Anders Björner and Karanbir Sarkaria, the formulation there suffers from some misprints.

The following example, which w e also owe to A. Björner and K. Sarkaria, shows how a formula o f the type above arises i n connection with the Stanley-Reisner ring o f a simplicial complex Δ.

Corollary 5.11. Let Δ be a simplicial complex on the vertex set {1,...,n }. Let AΔ be the arrangement of complex linear subspaces in $defined by $, where $, ei the ith unit coordinate vector in $. Let $ be the associated projective arrangement. Then the homology of the union $ of the arrange¬ ment P(A)is given by

$,

where $ is a subposet of $ and Δ(Δ[k]) its ordercomplex.

Note that the union o f the arrangement P(AΔ) i s a projective variety whose h o m o ¬ geneous coordinate ring i s the Stanley-Reisner ring o f Δ.

Note that Proposition 5.9 deals with the case when $ i s i n - jective in contrast to the Goresky-MacPherson formula (Corollary 5.6). We already men¬ tioned that the Goresky-MacPherson formula for the cohomology of the complement of an arrangement A of (affine) subspaces can be proved by Alexander duality from the homo¬ logy of an associated arrangement S(A) of spheres. In the case when T = S the map $ is trivial. Although it does not completely fit in the setting of this section one may regard toric varieties—seen from the point of view of Section 5.3 — as an interesting third case, when the map in homology induced by the diagram maps are surjective.

5.5. S u b g r o u p c o m p l e x e s .The order complex o f the poset

$ o f non-trivial p-subgroups o f a f i n i t e group G has r e c e i v e d considerable interest over that past f e w years ( s e e f o r example [1]). It was already observed b y Quillen [34] that Sp(G) i s homotopy equivalent to the poset Ap(G) of non-trivial elementary abelian p-subgroups o f G. In [31] the authors consider the covering o f Δ(Ap(G)) b y the subcomplexes Δ(Ap(NA)) f o r a fixed solvable normal p′-subgroup N and maximal elementary abelian p-subgroups A o f G. Then they use the following facts:

(a) Intersections of the spaces of type Δ(Ap(NA)) are again of the type Δ(Ap(ND)) for some elementary abelian p-subgroup D of G .

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(b) For a solvable normal p'-group N and an elementary abelian p-subgroup A the complex Δ(Ap(NA))i s homotopy equivalent to a wedge o f spheres o f dimension rank(D)—l.

Observation (a) f o l l o w s b y basic group theoretical argumentation. Assertion (b) i s much l e s s obvious. It was established b y Quillen [34], Theorem 11.2, but also follows b y applications o f the homotopy colimit methods ( s e e [31], Theorem (A)). Using facts (a) and (b), the Projection, the Homotopy and the W e d g e Lemma the following wedge d e ¬ composition o f Δ(Ap(G)) f o r f i n i t e solvable groups G with non-trivial normal p'-groupi s proved in [31].

Theorem 5.12 (P u l k u s an d We l k e r [3 1 ] , Th e o r e m (B ) ) . Let G be a finite group and let p be a prime. Let N be a solvable normal p'-subgroup. Let CN/Nbe the intersection of all maximal elementary abelian p-subgroups of G/N. For $ let cAN/N be an arbitrary but fixed point in Δ(Ap(G/N)>AN/N). Then Δ(Sp(G)) is homotopy equivalent to

$

where the wedge is formed by identifying, for AN / N > N / N,

the point $ with the point $.

In particular, if A is a maximal elementary abelian group of rank r in G, then $ is homotopyc to a wedge of (r — 1) - s p h e r e s .

A c k n o w l e d g m e n t s .Thanks to Eva-Maria Feichtner for helpful comments on the manuscript. W e are grateful to Emmanuel Dror Farjoun and Rainer V o g t f o r some c o r ¬ rections and very valuable remarks and references. The r e f e r e e ’ s remarks, corrections and additions ( s e e in particular Section 3 . 3 ) were extremely helpful, and l e d to a substantial improvement o f the paper and its exposition.

W e all thank the Konrad-Zuse-Zentrum i n Berlin f o r its hospitality.

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Fachbereich Mathematik, MA 7-1, T e c h n i s c h e Universitat B e r l i n , D-10623 B e r l i n e-mail: [email protected] e-mail: [email protected]

Mathematics Institute, Knez Mihailova 35/1, P.F. 3 6 7 , YU-11001 Belgrade e-mail: rade@ turing.mi.sanu.ac.yu

Eingegangen 5. April 1995, in revidierter Fassung 1. Juli 1998

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