Comparison lemmas and applications

for diagrams of spaces

Volkmar Welker1

Institute for Experimental MathematicsEllernstr. 29

45326 Essen, Germany

welker@exp-math.uni-essen.de

G�unter M. Ziegler2

Department of Mathematics 6{1Technical University of Berlin

10623 Berlin, Germany

ziegler@math.tu-berlin.de

Rade T. �Zivaljevi�c3

Matemati�cki InstitutKnez Mihailova 35/1, P.P. 367

11001 Beograd, Serbia

ezivalje@ubbg.etf.bg.ac.yu

Abstract

We provide a \toolkit" of basic lemmas for the comparison of homotopy types of (ho-motopy) limits of diagrams of spaces over �nite partially ordered sets, among them severalnew ones. In the setting of this paper, we obtain simple inductive proofs that provide ex-plicit homotopy equivalences. (In an appendix we provide the link to the general setting ofdiagrams of spaces over an arbitrary small category.)

We show how this toolkit of old and new diagram lemmas can be used on quite di�erent�elds of applications. In this paper we demonstrate this with respect to

� the \generalized homotopy-complementation formula" by Bj�orner [4],

� the topology of toric varieties (which turn out to be homeomorphic to homotopy limits,and for which the homotopy limit construction provides a suitable spectral sequence),

� in the study of homotopy types of arrangements of subspaces, where we establish anew, general combinatorial formula for the homotopy types of \Grassmannian" ar-rangements, and

� in the analysis of homotopy types of subgroup complexes.

1Supported by a \Habilitationsstipendium" of the DFG2Supported by a \Gerhard-Hess-F�orderpreis" of the DFG, which also provided support for the authors' joint

work at the Konrad-Zuse-Zentrum f�ur Informationstechnik Berlin (ZIB) in 19943Supported by the Mathematical Institute SANU, Belgrade, grant no. 0401D

1

Contents

1 Introduction 2

2 Set-up and Basic Tools 4

3 Homotopy Comparison Lemmas 5

4 Bj�orner's generalized homotopy-complementation formula 16

5 Toric Varieties 18

6 Subspace Arrangements 20

7 Subgroup Complexes 26

8 Appendix: Comparison Lemmas over Small Categories 26

Acknowledgements 34

References 34

1 Introduction

A diagram of spaces is a functor from a small category to a category of topological spaces. In thefollowing, we will consider the special case where the small category is a �nite partially orderedset P , and a P -diagram is an assignment of topological spaces Dp to the elements p 2 P , andcompatible maps dpp0 : Dp ! Dp0 to the order relations p � p

0 in P .In various topological, geometric, algebraic and combinatorial situations one has to deal with

structures that can pro�tably be interpreted as limits or homotopy limits of diagrams over �niteposets. In fact, if a space is written as a �nite union of (simpler) pieces, then it is the limitof a corresponding diagram of spaces. While limits do not have good functorial properties inhomotopy theory, they can usually be replaced by homotopy limits (Puppe [29] may have beenthe �rst to exploit this). Homotopy limits have much better functorial properties. Thus thereis a wide variety of possible techniques to manipulate diagrams of spaces in such a way that thehomotopy limit is preserved (up to homotopy type).

Basic work on homotopy limits has been done by Segal [32], Bous�eld & Kan [6], tomDieck [37], Vogt [39], and Dwyer & Kan [12, 13]. See Hollender & Vogt [19] for a recent survey.

Two key results in this setting are the \Projection Lemma" [32] [6, XII.3.1(iv)], which allowsone to replace limits by homotopy limits, and the \Homotopy Lemma" [37] [6, XII.4.2] [39] whichcompares the homotopy types of diagrams over the same partially ordered set. These tools havefound striking applications, for example, in the study of subspace arrangements [44] [36].

Our objective here is to provide a \tool kit" for manipulation of limits and homotopy limits:basic lemmas that allow one to compare the homotopy types of diagrams de�ned over di�erentpartial orders. Not all of these results are new: in particular, Dwyer & Kan [13, Sect. 9] providea list of some (apparently) \well-known" results, for which a proof can be found in Hollender &Vogt [19, Prop. 4.4 with Remark 3.2]. However, in our attempt to isolate the \central results"we also describe new tools, in particular the Upper Fiber Lemma, Theorem 3.8 below.

2

The choice of contents for the \toolkit for the manipulation of homotopy limits" is partiallymotivated by the usefulness of corresponding lemmas in the special case of order complexes (thediscrete case, when the spaces Dp are points). In this case, there is a solid amount of theoryavailable, which has proved to be extremely powerful and useful in quite diverse situations. Inthis situation the key result is the Quillen Fiber Theorem (Quillen's \Theorem A" [30, 31],see below). All other basic tools of the \homotopy theory of posets," such as the crosscuttheorem, order homotopy theorem, complementation formulas, etc., can be derived from it. Werefer to Bj�orner [3] for an excellent account of the theory [3, Sect. 10], for an extensive surveyof applications [3, Part I], and for further references.

The main part of our exposition is restricted to the case of a diagram over a �nite partiallyordered set. Our primary formulation and proofs are for diagrams over �nite posets; this is avery concrete setting that allows one to build concrete intuition for the behaviour of homotopylimits under natural operations that occur \in practice." In this setting, we are also able toprovide explicit proofs, which in the combinatorial setting we have in mind (where the categoryis a �nite poset, and the spaces are compact CW-complexes) also allow one to understand theexplicit maps, retractions, etc that induce the homotopy equivalences. In an appendix, we showhow they extend to the setting of arbitrary diagrams over a small category.

In the following, we work in the category of spaces having the homotopy type of a �niteCW-complex. Certainly much more generality is possible (see the appendix). One can expectthat techniques such as tom Dieck's partitions of unity [37] are su�cient to establish extensionsof our results in the generality of diagrams admitting a numerable covering (as considered bytom Dieck [37]) for any small category instead of a �nite partial order (the setting of Bous�eld &Kan [6]). Also, the underlying category of topological spaces can be replaced by an (equivalent)category of semisimplicial spaces and maps (as used by Bous�eld & Kan [6], and by Dwyer &Kan [12, 13].

Also, the following does not contain any details on the case of inverse limits. One reason forthis is that such results can be derived from the case of homotopy limits by standard dualityprocedures (see Bous�eld & Kan [6, XII.4.1], and Hollender & Vogt [19, Sect. 3]). Another reasonis that homotopy inverse limits seem to be much less useful for the applications in combinatoricsand discrete geometry that we have in mind.

We provide several applications of our methods to various areas within mathematics. As a�rst application, in the �eld of topological combinatorics, we present a new proof of a result byBj�orner on the homotopy type of complexes [4], which generalizes the Homotopy Comple-mentation Formula of Bj�orner and Walker [5]; a tool which has proved to be very powerfulin combinatorics. Since this proof a�ords the application of many of the techniques provided inthis paper, we give a detailed exposition of it here.

Then we present a new view of toric varieties. Namely, we show that toric varieties arehomeomorphic to homotopy limits over the face poset of the fan de�ning the variety. This leadsto a spectral sequence to compute the homology of toric varieties isomorphic to one alreadyemployed by Danilov [8].

More brie y we cover two applications for which details are contained in other papers. Wedescribe a new result on the homotopy type of the order complex of the poset Sp(G) of non-trivial p-subgroups of a �nite group G [28]. Finally, we review results obtained by homotopylimit methods on the topology of subspace arrangements in [44], and provide the equivalencewith the results of Vassiliev [38]. We also derive a new \combinatorial formula" for the homotopytypes of quite general arrangements (such as \Grassmannian" arrangements) that are associatedto linear subspace arrangements by suitable functorial constructions.

In an appendix, we give complete statements and proofs (or references to proofs) for our main

3

results (homotopy lemmas from our toolkit) in a more general setting of diagrams of spaces overan arbitrary small category.

2 Set-up and Basic Tools

In the following all posets are �nite partially ordered sets. A basic operation we will use is theconstruction of subposets like P�p := fp

0 2 P : p0 � pg, P<p := fp0 2 P : p0 < pg, etc. Also,

the order complex of P is the (abstract) simplicial complex �(P ) given by the collection ofchains in P . By j�j we denote the geometric realization of the complex �. (See Stanley [34,Chap. 3] for other basic concepts, notation, etc.) Whenever we talk about the topology of aposet, we refer to the order complex. Thus a contractible poset is a poset whose order complexis contractible, and a poset is a cone whenever the order complex is a cone, etc. All topologicalspaces will have the homotopy type of a �nite CW-complex. (See Whitehead [42] for the basicsabout this category.)

The following lemma is extremely useful for our inductive proofs (in the next section, andelsewhere): it can be used for proving that a map is a homotopy equivalence if it is \gluedtogether" from two homotopy equivalences.

Lemma 2.1 (Gluing Lemma: see for example tom Dieck [37])Let X = A [ B and Y = C [D be spaces and f : X ! Y a continuous map with the propertyf(A) � C and f(B) � D. Assume that the inclusions A \ B ,! A and A \ B ,! B areco�brations. Then if f jA : A ! C, f jB : B ! D and f j(A \ B) : A \ B ! C \ D are allhomotopy equivalences, then the map f : X ! Y is a homotopy equivalence.

There various other formulations of a \Gluing Lemma." In particular, the condition thatthe maps A \B ,! A;B are closed co�brations can be replaced by, for example, the conditionthat the pair (A;B) de�nes a numerable covering.

Lemma 2.2 (Contractible Carrier Lemma: Lundell & Weingram [22], Walker [40])Let � be a simplicial complex and let X be a topological space. Assume f; h : j�j ! X are mapsand assume C is a map that assigns to every simplex � 2 � a subspace C(�) � X such that:

(i) C(�) is contractible,

(ii) � � � implies C(�) � C(�),

(iii) f(j�j)[ h(j�j) � C(�).

Then there is a homotopy H : j�j � [0; 1]! X such that:

(a) H(x; 0) = f(x), H(x; 1) = h(x),

(b) H(x; t) 2 C(�) for x 2 j�j and t 2 [0; 1].

Conversely, assume that we are given a map C satisfying (i) and (ii). Then there exists a mapg : j�j ! X such that g(j�j)� C(�).

We call a map C : � ! X satisfying conditions (i) to (iii) of the preceding lemma acontractible carrier for f; h and g.

In the rest of this section we list the main tools to manipulate homotopy types of ordercomplexes. An extensive survey, with all the proofs and references, is in Bj�orner [3, Sect. 10].

4

Thus we give no proofs here. (Most of them are not hard to derive from the ContractibleCarrier Lemma.) In the next section we will state and prove extensions of all these resultsfor the much more general setting of diagrams.

A �rst | trivial | observation is that if P has a minimal element 0, or a maximal element 1,then �(P ) is a cone, and hence contractible.

Proposition 2.3 (Quillen Fiber Theorem: Quillen [30, 31], Walker [40])If f : P ! Q is a map of posets such that the order complex of f�1(Q�q) is contractible for allq 2 Q, then the simplicial map �f : �(P )! �(Q) is a homotopy equivalence.

Similarly, if the order complex of f�1(Q�q) is contractible for all q 2 Q, then �f is ahomotopy equivalence.

Proposition 2.4 (Order Homotopy Theorem: Quillen [31], see Bj�orner [3, (10.12)])If f : P ! P is a decreasing poset map (that is, f(p) � p for all p 2 P ), then P is homotopyequivalent to f(P ).

A join semilattice is a poset P such that every �nite subset has unique minimal upper bound.Such a P has a unique maximal element 1 (note that P is �nite). The crosscut complex �(P )of a join semilattice is the simplicial complex of all nonempty subsets of min(P ) that have anupper bound in Pn1.

Proposition 2.5 (Crosscut Theorem: see Bj�orner [3, (10.8)])For every join semilattice, the crosscut complex �(P ) is homotopy equivalent to the order complexof Pn1.

3 Homotopy Comparison Lemmas

In our setting, a P -diagram D is an assignment of spaces Dp to the elements p 2 P , and ofmaps dpp0 : Dp ! Dp0 to the order relations p � p0, in such a way that dpp is the identity, anddpp0 � dp0p00 = dpp00 for p � p

0 � p00.If D is a P -diagram, and P 0 � P is a subposet of P (that is, a subset with the induced

partial order), then we write D[P 0] for the induced subdiagram: the P 0-diagram whose spacesand maps are given by the spaces and maps in D.

The limit lim�!D of a P -diagram D is constructed from the disjoint union

Up2P Dp by identi-

�cation of x 2 Dp with dpp0(x) 2 Dp0 , for all p � p0 and all x 2 Dp.

For example, \subspace arrangements" give rise to \inclusion diagrams." The followingde�nition provides suitable generality for our purposes.

De�nition 3.1 An arrangement is a �nite collection A of closed subspaces in some ambientspace U , such that any nonempty intersection of subspaces in A is a (�nite) union of subspacesin A.

Then there is a natural associated diagram, whose poset P is in bijection with A, the orderon P is by reverse inclusion, the spaces Ap are the elements of A corresponding to p 2 P , andthe maps are the inclusion maps Ap ,! Ap0 for p � p

0.In general, a subspace diagram will mean the inclusion diagram of an arrangement. With

this, one sees that the limit of a subspace diagram is exactly the unionSA =

Sp2P Ap.

5

(An assumption one needs for the Projection Lemma below is that the inclusion mapsAp ,! Ap0 are closed co�brations, i.e., that these maps possess the homotopy lifting property.Equivalently, the subspace Ap0 � f0g [ Ap � I is a retract of Ap0 � I , or that (Ap0 ; Ap) is anNDR-pair, see [42, I.5]. This assumption is always satis�ed in a situation where everything issmooth, or where one is dealing with inclusions of subcomplexes in a triangulable arrangement.For a discussion of more general situations, where these assumptions hold we refer the reader tothe work of Lillig [21] and Kraus [20].)

The homotopy limit holim�!D of a P -diagram D can be constructed from the disjoint union

]p2P

�(P�p)�Dp;

by \making the obvious identi�cations," as follows. For each order relation p � p0, there is aninjection i : �(P�p0) ,! �(P�p), and a map dpp0 : Dp ! Dp0 . Using the map id � dpp0 , weidentify the subspaces

�(P�p0)�Dp � �(P�p)�Dp

and�(P�p0)�Dp0 � �(P�p0)�Dp0:

If \�" is the equivalence relation generated by these identi�cations, then the homotopy limit ofthe P -diagram D is the space

holim�!D :=

� ]p2P

�(P�p)�Dp

�.� :

An inductive construction of the homotopy limit is as follows: start with a disjoint unionUp2P Dp. Then, for every pair p > p0, glue to it a copy of Dp � I , to create a mapping cylinder

of the map dpp0 . Then, for every triple p > p0 > p00, glue into this a copy of Dp��(p > p0 > p00),and so on. The \usual" spectral sequence for the (co)homology of homotopy limits follows thisinductive construction, i.e., it uses a �ltration induced by the skeleta of �(P ).

We may observe that for every P -diagram D there is a canonical partial order on P :=Up2P Dp de�ned by setting x � x0 for x 2 Dp and x

0 2 Dp0 if and only if p � p0 and dpp0(x) = x0.

With this de�nition there is a canonical bijection �(P) ! holim�!

(D); however, the topology on

the two spaces is di�erent: in fact, with the usual topology on simplicial complexes the subspacesDp � �(P) get a discrete topology.

Examples 3.2 Here are some trivial examples for the construction of homotopy limits.

(i) If Dp is a one-point space for all p 2 P , then holim�!D is (isomorphic to) the order complex

of P .

(ii) If P = fpg is one point, then holim�!D = Dp.

(iii) If P = fp; p0g has two points, and p0 > p, then holim�!D = Dp0[fDp is the mapping cylinder

of the map f = dp0p : Dp0 ! Dp.(The mapping cylinder is homotopy equivalent to the image space Dp, which is the limitof the diagram.)

(iv) If P = fp; p0g and p; p0 are not comparable, then holim�!D = Dp ]Dp0 is the disjoint union.

6

(v) Let P = fp; p0; p00g be the three-element poset with p � p0, p � p00, for which p0 and p00 areincomparable, and let D be a P -diagram with Dp00 = f�g is a one-point space. Then thehomotopy limit holim

�!(D) is homeomorphic to the mapping cone of the map dpp0 : Dp !

Dp0.

(vi) If D is a diagram such that all the spaces Dp are identical, and the maps are identity maps,then holim

�!D �= �(P )�Dp.

(The limit of such a diagram is Dp, if P is connected.)

If D is a P -diagram and E is a Q-diagram, then a map of diagrams � : D ! E consists of anorder preserving map of posets f : P ! Q, together with a collection of maps �p : Dp ! Ef(p),which have to be compatible in the sense that for p > p0, the maps ef(p)f(p0) � �p and �p0 � dpp0 :Dp ! Ef(p0) have to coincide, that is, the square

Dp�p�! Ef(p)

dpp0??y ??yef(p)f(p0)

Dp0�p0�! Ef(p0)

has to commute for all p � p0. (Formally, this amounts to the requirement that � be a naturaltransformation between the functors D and E .)

We now start our presentation of \tools" with the observation that every map of diagramsinduces a map of their homotopy limits in a natural way.

Lemma 3.3 (Maps)Let � : D ! E be a map of diagrams. Then � induces a natural map

b� : holim�!D ! holim

�!E :

Proof. We can write the elements of �(P�p)�Dp in the form��1p1 + � � �+ �kpk; x

�with p1 � p2 � � � � � pk = p, �i � 0,

Pi �i = 1, and x 2 Dp, and de�ne

b���1p1 + � � �+ �kpk; x�

:=��1f(p1) + � � �+ �kf(pk); �p(x)

�:

These maps are clearly well-de�ned on the pieces, and compatible with the equivalence relation

�. Thus they induce a map b� as required. Naturality (i.e., bid is the identity, and [� � = b� � b )is trivial to check.

Lemma 3.4 (Embedding Lemma)Let � : D ! E be a map of P -diagrams associated with id : P ! P . If �p : Dp ,! Ep is a closedembedding for every p 2 P , then � induces a closed embedding

b� : holim�!D ,! holim

�!E :

In particular, if D is a P -diagram and Q � P is a subposet, then holim�!D[Q] ,! holim

�!D is a

closed embedding.

7

Proof. The �rst part is immediate from the construction of the homotopy limit.For the second part we use that if Dp = ;, then the element p can be deleted from the poset

and the diagram without a�ecting the homotopy limit.

Lemma 3.5 (Cone Lemma: Bous�eld & Kan [6, XII.3.1(iii)])Let P be a poset with least element 0. If D is a P -diagram, then the inclusion f0g � P inducesa homotopy equivalence

D0 ' holim�!D:

Proof. Let D[0] be the restricted diagram over the subposet f0g, with holim�!D[0] = D0. The

inclusion i : f0g ,! P induces a diagram map I : D[0] ,! D, and thus an embedding of homotopylimits D0 ,! holim

�!D. In the other direction, we have a diagram map � : D ! D[0] given by

f(p) = 0 and �p = dp0 for all p 2 P , which induces a map of homotopy limits b� : holim�!D ! D0.

We clearly have b� � bI = idD0, while bI � b� is homotopic to the identity. In fact, for t 2 [0; 1]

we can de�ne

Ht��1p1 + � � �+ �kpk; x

�:=

�t0 + (1� t)(�1p1 + � � �+ �kpk); x

�

with 0 � p1 � p2 � � � � � pk = p, �i � 0,P

i �i = 1, and x 2 Dp. This Ht induces a map

holim�!D ! holim

�!D that depends continuously on t, where H0 = idholim

�!D , and H1 = bI � b�.

Lemma 3.6 (Projection Lemma: Segal [32], Bous�eld & Kan [6, XII.3.1(iv)])Let A be an arrangement with intersection poset P , let D be the corresponding P -diagram. Thenthe natural collapsing map � : holim

�!D ! lim

�!D is a homotopy equivalence.

(See [44, Lemma 1.6] for a simple inductive argument.)

We have discussed after De�nition 2.1 under which conditions on arrangements the Pro-jection Lemma is valid. Care is needed for generalizations of the Projection Lemma tocategories that are not posets. Here simple examples show that a \Projection Lemma" does nothold in general if there are more than two arrows between two objects.

Lemma 3.7 (Wedge Lemma: Ziegler & �Zivaljevi�c [44, Lemma 1.8])Let P be a poset with maximal element 1. Let D be a P -diagram so that there exist pointscp0 2 Dp0 for all p0 < 1 such that dpp0 is the constant map dpp0 : x 7! cp0, for p > p0. Then

holim�!D '

_p2P

��(P<p) �Dp

�;

where the wedge is formed by identifying cp 2 �(P<p) �Dp with p 2 �(P<1) �D1, for p < 1.

Theorem 3.8 (Upper Fiber Lemma)Let D be a P -diagram and let E be a Q-diagram. Assume � : D ! E is a map of diagrams. If �induces a homotopy equivalence

b� : holim�!

(D[p 2 P : f(p) � q]) ' holim�!E [Q�q] for all q 2 Q;

then � induces a homotopy equivalence b� : holim�!D ! holim

�!E.

8

Proof. We proceed by induction on the cardinality jQj of Q. Let q 2 Q be a minimal elementof Q. Then Q = Q1[Q2 for Q1 = Q�q and Q2 = Qnfqg. The poset Q3 := Q1\Q2 equals Q>q.We set Pi := f�1(Qi) for i = 1; 2; 3, which implies P1 \ P2 = P3 and P1 [ P2 = P .

If q is the unique minimal element of Q (i.e., Q1 = Q), then the assertion follows triv-ially from the assumptions. Thus we may assume that Q1 6= Q. Then by induction � in-duces a homotopy equivalence b� : holim

�!D[Pi] ! holim

�!E [Qi] for i = 1; 2; 3. Since holim

�!D[P1] \

holim�!D[P2] = holim

�!D[P3] and holim

�!D[Q1]\holim

�!D[Q2] = holim

�!D[Q3], it follows from the Glu-

ing Lemma that � induces a homotopy equivalence between holim�!D = holim

�!D[P1][holim

�!D[P2]

and holim�!E = holim

�!E [Q1][ holim

�!E [Q2].

Note that by the Cone Lemma the condition of the Upper Fiber Lemma implies thatholim�!

(D[p 2 P : f(p) � q]) is homotopy equivalent to Eq for all p 2 P . Thus, in particular, the

Upper Fiber Lemma implies the following Homotopy Lemma, for which a simple inductiveproof was described in [44, Lemma 1.7].

As a special case, when all the spaces Dp and Eq are one-point spaces, the Upper FiberLemma contains the Quillen Fiber Theorem (in the case of upper ideals f�1(P�p)).

Corollary 3.9 (Homotopy Lemma: tom Dieck [37], Bous�eld & Kan [6, XII.4.2], Vogt [39])Let D and E both be P -diagrams, and assume that there is a map of diagrams � : D ! E thatcorresponds to the identity map id : P ! P on the posets. If �p : Dp ! Ep is a homotopy

equivalence for each p 2 P , then � induces a homotopy equivalence b� : holim�!D ! holim

�!E.

Proof. This follows from the Upper Fiber Lemma 3.8, using the Cone Lemma 3.5.

(The fact that a similar Homotopy Lemma is not available for ordinary limits lim�!D is one

of the main reasons to work with homotopy limits instead.)

Corollary 3.10 (Mapping Cylinder Lemma)Let D be a P -diagram and let E be a Q-diagram. Assume � : D ! E is a map of diagrams, withf : P ! Q the accompanying map of posets. We de�ne a partial order P �f Q on the disjointunion P ]Q by

s � s0 :()

8><>:s; s0 2 P and s � s0 in P; ors; s0 2 Q and s � s0 in Q; ors 2 P; s0 2 Q and f(s) � s0 in Q:

Let D �� E be the P �f Q-diagram de�ned by

(D �� E)s :=

�Es if s 2 Q;

Ds if s 2 P:

The maps of D �� E are induced by the maps of the diagram D and E and the map �. Thenthe natural inclusion holim

�!E ,! holim

�!D �� E given by the Embedding Lemma is a homotopy

equivalence.

Proof. We construct a diagram map � : D �� E ! E by setting

k : P �f Q! Q; p 7! f(p) for p 2 P; q 7! q for q 2 Q:

and�p := �p for p 2 P; �q := idq for q 2 Q:

9

For this map, we getk�1(Q�q) = Q�q [ fp 2 P : f(p) � qg;

which has a unique minimal element q. Thus, by the Cone Lemma, � induces a homotopyequivalence between D �� E [k

�1(Q�q)] ' Eq and E [Q�q] ' Eq. Hence by the Upper FiberLemma b� is a homotopy equivalence

b� : holim�!D �� E ! holim

�!E :

Now i : Q ,! P �f Q induces an inclusion of diagrams � : E ,! D �� E , with � � � = idE .

This implies b� � b� = bid by naturality, and thus b� is a homotopy equivalence as well.Corollary 3.11 (Direct Image Lemma: Dwyer & Kan [13, 9.8])Let D be a P -diagram, and let f : P ! Q be a poset morphism. De�ne a Q-diagram f]D by set-ting f]Dq := holim

�!D[f�1(Q�q)], and taking for f]dqq0 the inclusion maps holim

�!D[f�1(Q�q)] ,!

holim�!D[f�1(Q�q0)].

Then there is a homotopy equivalence holim�!

f]D ' holim�!D.

Proof. The Q-diagram f]D is the subspace diagram of an arrangement (De�nition 3.1): in factwe have

Q�q \ Q�r =[

s�q; s�rQ�s;

and thusholim�!D[Q�q] \ holim

�!D[Q�r ] =

[s�q; s�r

holim�!D[Q�s]

by the Embedding Lemma 3.4, which rewrites every intersection of subspaces of the formholim�!D[Q�q] as a �nite union of such subspaces.

With this observation, we get a map

holim�!

f]D ! lim�!

f]D = holim�!D;

which is a homotopy equivalence according to the Projection Lemma.

The following, quite innocent-looking, lemma contains the key idea to the combinatorialformulas for \Grassmannian" arrangements in Section 6. It uses the notion of a homotopybetween P -diagrams D and E : a P -diagram H with diagram maps � : D ! H and : E ! Hsuch that the maps �p and p are homotopy equivalences. This is motivated by the special casewhere we would have compatible maps fp : Hp ! [0; 1] such that f�1p (0) = Dp and f

�1p (1) = Ep

for all p 2 P , and such that the inclusions Dp; Ep ,! Hp are homotopy equivalences.

Corollary 3.12 (Homotopy between diagrams)If there is a homotopy between P -diagrams D and E, then holim

�!D ' holim

�!E.

Proof. By Homotopy Lemma, the diagram maps D; E ! H induce homotopy equivalenceson the homotopy limits.

From this result, we can see e.g. that mapping cones are well de�ned (for homotopy classesof maps, up to homotopy!). In particular, in Example 3.2(v) one can insert any contractiblespace for Dp00 and still obtain the homotopy type of the mapping cone.

10

Theorem 3.13 (Inverse Image Lemma: Dwyer & Kan [13, 9.4])Let f : P ! Q be a poset morphism and let E be a Q-diagram. De�ne the P -diagram f ]E by(f ]E)p := Ef(p) and f

]epp0 = ef(p)f(p0).If for all q 2 Q, f�1(Q�q) is contractible, then f induces a homotopy equivalence

holim�!

f ]E ! holim�!E :

Proof. In the �rst step of the proof we explicitly construct the maps that show that f induces ahomotopy equivalence between �(P ) and �(Q). This constructions follows the argumentationprovided by Bj�orner [3, (10.5)] for an elementary proof of the Quillen Fiber Lemma.

By the Contractible Carrier Lemma there exists a map g : j�(Q)j ! j�(P )j suchthat g(�) � f�1(Q�max�) for every chain � � Q. Then f � g(�) � Q�max� , and g � f(�) �f�1(Q�maxf(�)) for every chain � � P . Thus CP (�) = f�1(Q�maxf(�)) and CQ(�) = Q�max� arecontractible carriers for g�f , id�(P ) and f �g, id�(Q) respectively. Hence by the ContractibleCarrier Lemma there are homotopy equivalences HP : j�(P )j � [0; 1] ! j�(P )j and HQ :j�(Q)j � [0; 1]! j�(Q)j such that HP (x; 0) = g � f(x), HP (x; 1) = x and HQ(x; 0) = f � g(x),HQ(x; 1) = x.

In the second step we extend these homotopies to homotopies over the homotopy limits ofthe corresponding diagrams. First, we describe an alternative construction of the homotopylimit which is valid in the situation of the P -diagram f ]E . Namely, we introduce on the setUq2QEq � �(f�1(Q�q)) the equivalence relation � which is de�ned by identifying for q1 �

q2 2 Q the subspaces Eq1 ��(f�1(Q�q2)) � Eq1 ��(f�1(Q�q1)) and Eq2 � �(f�1(Q�q2)) �Eq2 ��(f�1(Q�q2)), via id� eq1q2 in the same way as in the usual description of the homotopylimit. The resulting space is homeomorphic to the homotopy limit holim

�!f ]E .

The map f induces a mapUf :Uq2QEq ��(f�1(Q�q)) !

Uq2QEq ��(Q�q) that sends

(a; b) 2 f ]Ep ��(P�p) to (a; f(b)) 2 Ep ��(Q�f(p)). The map g induces a map]g :]q2Q

Eq ��(Q�q) !]p2P

f ]Ep ��(f�1(Q�q))

that sends (a; b) 2 Eq ��(Q�q) to (a; g(b)) 2 Eq ��(f�1(Q�q)). We now check that the mapsU

f andUg are compatible with the identi�cations made in the construction of the homotopy

limits holim�!

f ]E and holim�!E .

(Uf) Assume p1 > p2 are elements of P . For (u; v) 2 f

]Ep1 ��(P�p2) we haveUf (u; v) =

(u; f(v)) 2 Ef(p1)��(Q�f(p2)) � Ef(p1)��(Q�f(p1)) andUf( f ]ep1p2(u); v ) = ( ef(p1)f(p2)(u);

f(v) ) 2 Ef(p2) � �(Q�f(p2)). From this it is easily seen thatUf is compatible with the

identi�cations made in the construction of holim�!

f ]E and holim�!E .

(Ug) Assume q1 > q2 are elements of Q. For (u; v) 2 Eq1 � �(Q�q2) we have

Ug(u; v) =

(u; g(v)) 2 Eq1��(f�1(Q�q2) � Eg(q1)��(f

�1(Q�q1)) andUg(eq1q2(u); v) = (eq1q2(u); g(v)) 2

Eq2 ��(f�1(Q�q2)).Thus both maps are compatible with the equivalence relation in the de�nition of the ho-

motopy limits (for holim�!

f ]E use the description given before in this proof). Hence they induce

maps holim�!

f : holim�!

f ]E ! holim�!E and holim

�!g : holim

�!E ! holim

�!f ]E . The map holim

�!f � g :

holim�!E ! holim

�!E is induced by (

Uf) � (

Ug) that sends (a; b) 2 Eq � Q�q to (a; f � g(b)) 2

Eq ��(Q�q). The map holim�!

g � f : holim�!

f ]E ! holim�!

f ]E is induced by (Ug) � (

Uf), which

sends (a; b) 2 Eq ��(f�1(Q�q)) to (a; g � f(b)) 2 Eq ��(f�1(Q�q)).

11

Analogous to holim�!

f � g and holim�!

g � f we construct maps

holim�!

HQ : holim�!E � [0; 1] ! holim

�!E ; holim

�!HP : holim

�!f ]E � [0; 1] ! holim

�!f ]E ;

where holim�!

HQ is induced by the map that sends ((a; b); t) 2 (Eq � �(Q�q)) � [0; 1] to the

point (a;HQ(b; t)) 2 Eq ��(Q�q) and holim�!

HP is induced by the map that sends ((a; b); t) 2

(Eq � �(f�1(Q�q))) � [0; 1] to (a;HP (b; t)) 2 Eq � �(f�1(Q�q)). By the ContractibleCarrier Lemma the maps are well de�ned. Again compatibility with the construction of thehomotopy limit is easy to check.

The condition \f�1(Q�q) is contractible" in the Inverse Image Lemmma cannot be re-placed by the order dual condition \f�1(Q�q) is contractible." The following diagrams providea counterexample to the conclusion of the lemma under this hypothesis.

a b f(a) = f(b)

P

Q

Da Db Ef(a)

c f(c)

Dc Ef(c)

f

f

�a; �b

�c

Let D be the P -diagram and let E be the Q-diagram in our Figure. Then Da = Db = Ef(a) isa point and Dc = Ef(c) is a circle. All maps are the constant maps to the point Da = Db = Ef(a).The homotopy limit of the P -diagram D is a 2-sphere and the homotopy limit of the Q-diagramE is a cone over a circle. Obviously, f�1(Q�q) is contractible for all q 2 Q, but holim

�!D 6' holim

�!E .

By a lemma of Babson [2] (see [35, Lemma 3.2]) the conditions of the Inverse Image Lemmaare satis�ed if (a) the inverse image f�1(q) is contractible for all q 2 Q, and (b) P�p \ f�1(q) iscontractible for all p 2 P and q 2 Q with f(p) < q.

Proposition 3.14 (Order Homotopy Theorem)Let D be a P -diagram, and let f : P ! P be a poset map which is decreasing (that is, f(p) � pfor all p 2 P ).

Then the inclusion i : f(P ) ,! P induces a homotopy equivalence

holim�!D[f(P )] ' holim

�!D:

Proof. The inclusion map i induces a diagram map I : D[f(P )] ,! D, which in turn inducesan embedding b� : holim

�!D[f(P )] ,! holim

�!D.

Now the restricted diagram D[f(P )] is the inverse image diagram of the inclusion map:D[f(P )] = i]D. Hence, to apply the Inverse Image Lemma it su�ces to see that i�1(P�p) isa contractible poset for all p 2 P .

12

To see this, we iterate the map f , setting f1(p) := f(p) and fk+1(p) := f(fk(p)) for allp 2 P and k � 1. Since fk is decreasing, p � f(p) � f2(p) � � � �, and P is �nite we concludethat fN = fN+1 = � � � holds for large enough N . Setting g := fN , we obtain that g is an orderpreserving, decreasing map with g � f = g.

Applying the order homotopy theorem (for posets) to i�1(P�p) and g, we compute

i�1(P�p) ' g(i�1(P�p)) = gff(p0) : p0 2 P; f(p0) � pg = fg(p0) : p0 2 P; f(p0) � pg:

Now the poset fg(p0) : p0 2 P; f(p0) � pg clearly contains g(p), and any other element g(p0)with f(p0) � p satis�es g(p0) = g(f(p0)) � g(p). Thus fg(p0) : p0 2 P; f(p0) � pg is a cone, andhence i�1(P�p) is contractible.

The Order Homotopy Theorem is de�nitely false if we use increasingmaps: see the caseof a single mapping cylinder.

Note that the Cone Lemma is a very special case of the Order Homotopy Theorem.We just remark that for any function satisfying f(f(p)) = f(p) � p for all p 2 P the homotopylimit holim

�!D[f(P )] is in fact a deformation retract of holim

�!D.

Another simple consequence of the Order Homotopy Theorem is that one can remove\join irreducibles" from a diagram without changing the homotopy type of the homotopy limit.That is, if Q is obtained from P by deleting an element that covers exactly one element, thenthe homotopy limits of a P -diagram D and of its deletion D[Q] are homotopy equivalent.

Lemma 3.15 (Lower Fiber Lemma)Let D be a P -diagram and E a Q-diagram. Let � : D ! E be a morphism of diagrams. If for allq 2 Q the induced map b� : holim

�!D[f�1(Q�q)]! holim

�!E [Q�q]

is a homotopy equivalence, then b� : holim�!D ! holim

�!E is a homotopy equivalence.

Proof. The basic idea of the proof is the same as that of the proof of the Upper Fiber Lemma.Therefore we will just outline the parts that deviate from the proof of the Upper Fiber Lemma.Again we proceed by induction on the cardinality jQj of Q. If there is a unique maximalelement q 2 Q, then the assertion follows trivially from the assumption holim

�!D[f�1(Q�q)] '

holim�!E [Q�q], since in this case Q�q = Q and f�1(Q�q) = P . Thus we may assume that Q

has more than one maximal element, in particular, jQj � 2. Let q be a maximal element of Q.Set Q1 = Q�q and Q2 = Qnfqg. The poset Q3 = Q1 \ Q2 equals Q<q . Set P1 = f�1(Q1)and P2 = f�1(Q2), P3 = P1 \ P2. Then by induction � induces a homotopy equivalenceholim�!D[Pi] ! holim

�!E [Qi] for i = 1; 2; 3. By the Gluing Lemma this induces a homotopy

equivalence from holim�!D to holim

�!E .

Again, as the special case when all the spaces Dp and Eq are one-point spaces, this containsthe Quillen Fiber Theorem (for lower ideals).

Note that in the setting of order complexes, the Upper and the Lower Quillen FiberTheorem are trivially equivalent: just apply one of them to the order dual map fop : P op ! Qop

to get the other one. However, there is no such simple equivalence between the Upper and theLower Fiber Lemma for diagrams.

The following two corollaries control the change of the homotopy limit while passing to the\barycentric subdivision" of a poset. Here we use P (�P ) to denote the set of all nonempty chains

13

in P , ordered by inclusion. Clearly, the order complexes of P and of P (�P ) are homeomorphic(the second one is the barycentric subdivision of the �rst), and we have a surjective poset map

max : P (�P ) ! P;

which maps every chain to its largest element.We also consider the order dual to P (�P ), denoted P op(�P ), for which

min : P op(�P ) ! P;

is order preserving.

Corollary 3.16 (Upper Subdivision Lemma)Let D be a P -diagram. We de�ne a diagram �maxD on the poset P�(P ) by setting �maxD� :=Dmax� for every simplex � 2 �(P ). The maps in �maxD are induced from the maps in D. Thenholim�!

�maxD ' holim�!D.

Proof. The diagram �max is the inverse image max]D of the P -diagram D with respect to theposet map max. We check that, for all p 2 P , the preimages of ideals

max�1(P�p) = P (�(P�p))

are contractible: they are barycentric subdivisions of the cones �(P�p). Thus we can apply theInverse Image Lemma.

Corollary 3.17 (Lower Subdivision Lemma)Let D be a P -diagram. We de�ne a diagram �minD on the poset P op�(P ) by setting �minD� :=Dmin� for every simplex � 2 �(P ). The maps in �minD are induced from the maps in D. Thenholim�!

�minD ' holim�!D.

Proof. We proceed as in the previous proof. Thus, we have to check that min�1(P�p) iscontractible for all p 2 P . To see this, one considers the map

: min�1(P�p) ! min�1(P�p); � 7! � \ P�p:

which is easily checked to be well-de�ned, order preserving, and increasing. Thus the OrderHomotopy Theorem (in the poset version) implies that

min�1(P�p) ' (min�1(P�p)) �= P op(�(P�p));

where P op(�(P�p)) is the barycentric subdivision of a cone.

Corollary 3.18 (Crosscut Theorem)Let E be a Q-diagram, where Q is a join semilattice. De�ne P to be the poset of all nonemptysubsets of min(Q), ordered by inclusion. Let j : P ! Q be the poset map induced by \takingjoins," let j]E be the inverse image diagram over Q de�ned by the join map, with the corre-sponding diagram map J : j]E ! E.

Then J induces a homotopy equivalence bJ : holim�!

j]E ' holim�!E.

14

Proof. In this situation j�1(Q�q) is always a simplex, hence contractible. Therefore one canapply the Inverse Image Lemma.

The \usual" crosscut theorem for lattices is obtained from this in the case where Eq is aone point space for all q < 1, while E1 is taken to be empty. In this situation holim

�!j]E =

�(Pnj�1(1)) is the barycentric subdivision of the \usual" crosscut complex, while holim�!E =

�(Qn1) is the order complex of the proper part of Q.

We will call a P -diagram D a poset diagram if the spacesDp are order complexes, Dp = �(Qp)for posets Qp, and the maps dpp0 : Dp ! Dp0 are simplicial maps induced by poset morphismsfpp0 : Qp ! Qp0 .

Let D be a poset diagram. Then we de�ne the poset P lim�!D on the set

Up2P Qp � fpg =

f(q; p) : p 2 P; q 2 Qpg by setting (q; p)� (q0; p0) if and only if p � p0 and fpp0(q) � q0.

Proposition 3.19 (Simplicial Model Lemma)For every poset diagram D over a �nite poset P , there is a natural homotopy equivalence

: holim�!D ! �(P lim

�!D):

Proof. To obtain the map , one can use for each p 2 P the canonical homeomorphism

�(Qp)��(P�p) �= �(Qp � P�p)

(see Walker [41, Theorem 3.2]). Furthermore, we have a poset map Qp � P�p ! P lim�!D given

by (q; p0) 7! (fpp0(q); p0) for q 2 Qp. This poset map induces a map �(Qp� P�p)! �(P lim

�!D),

and thus we have a map

p : �(Qp)��(P�p) ! �(P lim�!D)

for each p 2 P . These maps p are easily checked to be compatible on the identi�cations thatcreate holim

�!D, which produces the required map .

To see that is a homotopy equivalence, we represent it by a sequence of two diagrammaps. Let E denote the diagram of one-point spaces over the poset limit P lim

�!D. We have

holim�!E = �(P lim

�!D) by Example 3.2(i). Now for p 2 P set

Rp := f(q0; p0) 2 P lim�!D : p0 � pg:

De�ne a P -diagram F by setting Fp := holim�!E [Rp] = �(Rp). The maps fpq are the natural

inclusion maps.Let f : P lim

�!D ! P be the poset map that sends (q; p) to p. Then F is the direct image

diagram f]E . Thus by the Direct Image Lemma we have a natural projection map � :holim�!F ' holim

�!E = �(P lim

�!D) that is a homotopy equivalence.

We also have (poset) inclusion maps ip : Qp ,! Rp, mapping Qp 3 q 7! (q; p). This producesa map of P -diagrams I : D ! F .

On the posets Rp we de�ne the maps hp : Rp ! Rp that send (q0; p0) to (fp0p(q

0); p). Obviouslyhp is monotone and hp(Rp) �= Qp. By the Order Homotopy Theorem (for posets) hp inducesa homotopy equivalence between �(Rp) and �(Qp). From hp � ip = idjQp we see that ip inducesa homotopy inverse to hp. Thus, by the Homotopy Lemma, the diagram map I induces ahomotopy equivalence bI : holim

�!D ' holim

�!F .

15

A simple computation shows that = � � bI . Thus we have shown that the natural map

: holim�!D

bI! holim

�!F

�! holim

�!E = �(P lim

�!D)

is a homotopy equivalence.

The Projection Lemma studies the e�ect of collapsing the order complexes �(P�p) in theconstruction of the homotopy limit of a P -diagram D Now we are going to study the e�ect ofcollapsing the spaces Dp, p 2 P . First we observe that this collapsing induces a map holim

�!D !

�(P ) for a P -diagram D. This map turns out to be a quasi�bration, a concept introduced byDold & Thom [9], under certain strong conditions. A map f : X ! Y between topological spacesis a quasi�bration if for any y 2 Y , x 2 f�1(y) the induced maps f� : �i(X; f

�1(y); x)! �i(Y; y)are isomorphism for all i � 0.

Proposition 3.20 (Quasifibration Lemma) [30],[12, (9.10)])If D is a P -diagram such that all the maps dpp0 are homotopy equivalences, then the naturalprojection map holim

�!D �! �(P ) is a quasi�bration. In particular, if �(P ) is contractible, then

for every p 2 P the inclusion of Dp into holim�!D induces a homotopy equivalence.

Proof. The �rst part of the Quasifibration Lemma is a result due to Quillen [30]. For thesecond part we have to use some standard facts about quasi�brations. Let f : holim

�!D ! �(P )

be the map induced by collapsing the spaces Dp in holim�!D. Then for p 2 P the �ber f�1(p) is

Dpi,! holim

�!D. By [9, 1.4] there is an exact sequence

� � � ! �l+1(�(P ))! �l(f�1(p))

i��! �l(holim�!D)

f��! �l(�(P ))! � � �

Since �(P ) is contractible the map i� is an isomorphism for all l. Results of Milnor [27] thenimply that the isomorphism i� in homotopy is induced by a homotopy equivalence i.

4 Bj�orner's generalized homotopy-complementation formula

Bj�orner's generalized homotopy-complementation formula [4] is an e�ective tool to computethe homotopy type of a simplicial complex � in the case when a large, contractible inducedsubcomplex �A is known, whose connections to the rest of the complex are not too complicated.

In the following, we provide a \homotopy limits" proof of Bj�orner's result, thereby demon-strating the applicability of some of our lemmas.

Let � � 2S be a �nite (abstract) simplicial complex with vertex set S, let A � S be a subsetof its vertex set, and denote by A the complement SnA of A.

Let �A := f� 2 � : � � Ag = � \ 2A be the induced subcomplex on A, and similarly for�A, the induced complex on A. In the following, the key assumption we will make is that �Ais contractible.

Theorem 4.1 (Generalized Homotopy-complementation Formula: Bj�orner [4])For any simplicial complex � � 2S and A � S, de�ne a new simplicial complex TA by takingthe union of all the simplicial complexes

(p] �) � (star�(�)\�A)

where p is an additional point p =2 S, and � denotes the join of two complexes.If �A is contractible, then � and TA are homotopy equivalent.

16

Proof. Let P be the poset of all nonempty faces of �, ordered by inclusion, and let

Q := f(�; �) 2 P � P : � � �g;

partially ordered by the condition

(�0; � 0) � (�; �) :() � � �0 � � 0 � �:

(Thus Q is isomorphic to the poset of all intervals in P , ordered by inclusion: Q �= Int(P ).)For � 2 P we de�ne

D� := star�(�)[�A:

Then we clearly have inclusion maps d�� : D� ,! D� whenever � � �. This de�nes a P -diagramD, with

� = lim�!D ' holim

�!D;

where the equality holds by de�nition of the limit (this is a subspace diagram!), and the homotopyequivalence is an application of the Projection Lemma.

Similarly, for (�; �) 2 Q we de�ne

E(�;�) := (p ] �) � (star�(�) \�A):

Then we get inclusion maps e(�;�)(�0;� 0) : E(�;�) ,! E(�0;� 0) whenever (�; �) � (�0; � 0). This de�nesa Q-diagram E , with

TA = lim�!E ' holim

�!E ;

by de�nition of the limit and the Projection Lemma.Thus, the claim of the theorem is reduced to proving that the homotopy limits of the P -

diagram D and the Q-diagram E are homotopy equivalent. This demands use of our newhomotopy lemmas, since the posets P and Q are quite di�erent.

The canonical poset map to use is

f : Q! P; (�; �)! �:

This f induces a homotopy equivalence between the posets P and Q (Walker [41]). To see this,we show that the lower �bers f�1(P��) are canonically isomorphic to Int(P�� ). On the posetInt(P�� ) we have an increasing map g : Int(P�� )! Int(P�� ) given by (�0; � 0) 7! (�0; �). Thus,by the Order Homotopy Theorem the �ber Int(P�� ) is homotopy equivalent to the imageg(Int(P�� )) �= P�� , which is a cone.

Now we modify E and D a little. We de�ne a new Q-diagram E 0, whose spaces are

E0(�;�) := star�(�) [ (p ��A);

and whose maps are the obvious inclusions. Furthermore, there is a map 1 : E ! E 0, which isthe identity map on Q, and between the spaces uses the inclusion maps

(p ] �) � (star�(�)[�A) ,! (p ] �) � (star�(�) \�A) ,! star�(�) [ (p ��A)

which are clearly homotopy equivalences, since �A is contractible. Thus, by the HomotopyLemma, 1 induces a homotopy equivalence

b 1 : holim�!E ' holim

�!E 0:

17

Similarly, we de�ne a new P -diagram D0, whose spaces are

D0� := star�(�) [ (p ��A);

and whose maps are the obvious inclusions. Furthermore, there is a map 2 : D ! D0, which is

the identity map on P , and between the spaces uses the inclusion maps

star�(�) \ �A ,! star�(�) [ (p ��A)

which are clearly homotopy equivalences. Thus, by the Homotopy Lemma, 2 induces ahomotopy equivalence b 2 : holim

�!D ' holim

�!D0:

Finally, at this stage we see that E 0 is an inverse image diagram, E 0 = f ]D0, where f is aposet map whose lower �bers we have already checked to be contractible. Hence the InverseImage Lemma implies a homotopy equivalence

holim�!E 0 ' holim

�!D0

which completes the proof.Alternatively, one could derive holim

�!E 0 ' holim

�!D0 also from the Upper Fiber Lemma,

together with the Projection Lemma, since E 0 and D0 are subspace diagrams as well.

We note that there are alternative ways to describe the construction of TA. For example, onecan (as Bj�orner does in his manuscript [4]) start from a wedge (or a disjoint union) of the spacesE(�;�) = (p ] �) � (star�(�)\�A), and then check that all the identi�cations of the limit lim

�!E

are generated by identifying, for � � � , the identical subcomplexes (p ] �) � (star�(�)\�A) inE(�;�) and in E(�;�).

Also, there are countless variations possible, corresponding to di�erent coverings of the com-plex �. The beauty of Bj�orner's set-up is that his transformations of � end up with a subspacediagram, and thus with a limit instead of a homotopy limit, which leads to an e�ective modelfor �=�A.

5 Toric Varieties

In this section we give a representation of the topological space underlying a toric variety (seeDanilov [8], Fulton [16], and Ewald [14] for general background on toric varieties) as the ho-motopy limit of a diagram. For this we recall a description, due to MacPherson (see Yavin &Fischli [43], [15]), of a toric variety. A decomposition of Rn into a complex of closed, convex,polyhedral cones with apex 0 is called a complete fan. If all cones in � are generated by latticepoints in Zn, then � is called rational. Assume that � is a complete and rational fan in Rn.Then let P be a cell decomposition of the unit ball in Rn that is dual to the one induced by �.For � 2 � we denote by �� the cell in P that corresponds to �. Thus �� is a cell of dimensionn� dim(�).

We identify the n-torus T n with the image of the projection map � : Rn ! Rn=Zn. For allcones � 2 � the image of � under this projection is a subtorus �(�) = �(spanR(�)) = T� of T .Since � is rational, this is a closed subtorus of dimension dim(�). Thus the quotient T n=T� isa real torus of dimension n� dim(�).

The toric variety X� is obtained from P � T n by modding out (��)��T n by the action ofT� on T for each � 2 �. This leads to a nice (compact, Hausdor�) quotient space since we mod

18

out by larger tori on @��T n. In particular, we see that the toric variety X� has a well-de�nedmap � : X� ! P , for which the �ber over any interior point of �� is isomorphic to T =T�.

Let P� be the poset whose elements are in bijection with the cones in � and whose orderrelation is de�ned by the reversed inclusion of the cones in �. Thus P� is the poset of non-emptyfaces of P , ordered by inclusion. In particular, P has a largest element 1 corresponding to the0-dimensional cone f0g. We construct a diagram D� over the poset P� as follows. For � 2 �,set D� = T =T�. Topologically, D� is a n � dim(�) torus. The map d�;� for � � � is the mapinduced by the projection T =T� ! T =T�.

Proposition 5.1 Let � be a complete and rational fan in Rn. Then the toric variety X� ishomeomorphic to the homotopy limit of the diagram D� associated with �:

holim�!D��= X�:

Proof. Let � be the map that sends T =T� � P�� to its image in X�. By construction of D�

the map � is compatible with the equivalence relation � onU�2P T =T��P�� . Hence � induces

a map b� : holim�!D� ! X�. It is routine to check that b� is indeed a homeomorphism.

The resolution of singularities of a toric variety also �ts our homotopy limit framework.Namely, let �0, � be two complete, rational fans in Rn such that �0 is a re�nement of �(i.e., for every open cone � 0� in �0 there is an open cone �� in � such that � 0� � ��). Thusthere is an induced map f : P�0 ! P�. Also assume that � 0 is a cone in �0 whose interior iscontained in the interior of the cone � of �. Then the inclusion � 0 ,! � 0 induces a surjectivemap �� 0� : T =T� 0 ! T =T� . Is is easily seen that � induces a map of diagrams. Hence there isan induced map b� : holim

�!D�0�= X�0 ! holim

�!D��= X�. The map b� is surjective since f and all

�� 0� are surjective.

Proposition 5.2 Let �0 be complete rational fan which is the re�nement of the complete rationalfan �. Then there is a surjective map b� : X�0 ! X�.

It is well known that X� is non-singular if and only if � is simplicial (i.e., all cones aresimplicial) and unimodular (i.e., all full-dimensional cones are equivalent to fx 2 Rn : x � 0gunder unimodular transformations from GL(Rn;Z)). For an example that shows that eH�(X�) isnot a combinatorial invariant of � in general see [25]. It is also well known that for any completerational fan � there is a simplicial and unimodular complete rational fan �0 which re�nes �. Inthis case b� is a resolution of singularities.

One can also use our results to investigate the (co)homology of a toric variety. For this weset up a spectral sequence introduced by Segal [32], which uses the �ltration of holim

�!D by the

s-skeleta of the order complexes. For a simplicial complex � we denote by �s its s-skeleton.Assume D is a P -diagram for a poset P . Then we denote by holim

�!Ds the image of

Up2P Dp�

�(P�p)s in holim

�!D. The �ltration holim

�!D0 � holim

�!D1 � � � �holim

�!D de�nes a spectral sequence

with termination eH�(holim�!D) in the E2-term and E1

st =eHs+t(holim

�!Ds; holim

�!Ds�1). Following

Segal's arguments on �nds that E1st is given by

M�0<���<�s2�(P )n

eHt(T =T�s). Now assume (�0 <

� � � < �s) � c is a (s + t)-cell in holim�!Ds. Then the di�erential of the cell complex holim

�!D is

given by

19

@(�0 < � � � < �s)� c =s�1Xi=0

(�1)i(�0 < � � � < b�i < � � � < �s)� c +

+ (�1)s�1(�0 < � � � < �s�1)� d�s�s�1(c) + (�1)n(�0 < � � �< �s)� @c:

Thus the di�erential d1st : E1st ! E1

s�1;t applied to the cell (�0 < � � � < �s) � c equalss�1Xi=0

(�1)i(�0 < � � � < b�i < � � � < �s) � c + (�1)s�1(�0 < � � � < �s�1) � d�s�s�1(c), where c is a

cycle in Ht(T =T�s).From this it is easily seen that our spectral sequence is isomorphic to the deRahm-Hodge

spectral sequence applied by Danilov [8, Chap 3, x12] to compute the cohomology of a toricvariety.

6 Subspace Arrangements

Arrangements of a�ne subspaces in Rn also allow an application of the homotopy limit method.Let A be a �nite set of a�ne subspaces inRn. Let us denote by bA the corresponding arrangementof spheres in the one-point compacti�cation Sn of Rn. Under our assumptions intersections ofspheres in bA are again spheres (or the compacti�cation point). The following result can bededuced from the Projection, Homotopy and the Wedge Lemma.

Theorem 6.1 (Ziegler & �Zivaljevi�c [44])Let A be a �nite set of a�ne subspaces in Rn. Let dUA be the one-point compacti�cation of theset-theoretic union of the subspaces in A and let P be the intersection poset of A. Then

dUA '_p2P

Sdim(p) ��(P<p):

An equivalent result can be found in Vassiliev [38, III. x6, Thm. 1]. In Vassiliev's formulationthe spaces �(P<p) are replaced by quotients of simplices by crosscut complexes, the spaces K(p)in his notation. More precisely, for an arbitrary subspace V corresponding to some point p = pVin the intersection lattice P of A, let V1; : : : ; Vt be the subspaces in A such that Vi containsV as a subspace. Let �(p) be the simplex which is spanned, in the abstract sense, by thevertices V1; : : : ; Vt. Vassiliev calls a face � of �(p) marginal if V is not the intersection of thesubspaces corresponding to the vertices of � . Thus the marginal faces are the simplices in thecrosscut complex �(P�p) of P�p. By the Crosscut Theorem the complex of marginal faces ishomotopy equivalent to �(P<p). In Vassiliev's formula the spaces S

dim(p) ��(P<p) are replacedby Sdim(p)�1 � �(p)=�(P�p). Let us analyze �(p)=�(P�p). If �(P�p) is the full simplex �(p),then �(p)=�(P�p) and by the Crosscut Theorem also �(P<p) are contractible. In particular,Sdim(p) � �(P<p) and S

dim(p)�1 � �(p)=�(P�p) are contractible. If �(P�p) is some non-emptypart of the boundary of �(p) then �(p)=�(P�p) is the suspension of �(P�p). Thus again theCrosscut Theorem shows that Sdim(p)��(P<p) and S

dim(p)�1��(p)=�(P�p) are homotopic. If�(P�p) is empty, then we have to \interpret" �(p)=�(P�p) as the suspension of the empty space,which is in our de�nition of the join with a two point space. Then the homotopy equivalencealso follows in this case.

By Alexander duality on Sn we infer from Theorem 6.1 the following formula of Goresky &MacPherson [18].

20

Corollary 6.2 (Goresky & MacPherson [18])Let A be a �nite set of a�ne subspaces in Rn. Let MA be the complement Sn �dUA and let Pbe the intersection poset of A. Then

eH i(MA;Z)�=Mp2P

eHcodim(p)�i�2(�(P<p);Z);

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

Analogous results for arrangements of spheres and projective spaces can be found in Goresky& MacPherson [18] and in [44]. In the following, we describe a simpler, more general, andmore powerful approach that provides combinatorial formulas for quite general \Grassmannianarrangements."

Let A be a central arrangement in Rn (or C n , Hn ) with intersection poset P and a dimensionfunction d : P ! N0. Let D = D(A) = fApgp2P be the corresponding P -diagram of linearspaces. In case each of the linear subspaces Ap, p 2 P , is invariant under the action of a�nite (or just closed) subgroup G � O(n;R) (resp. U(n), Sp(n)) of the orthogonal (unitary,symplectic) group, a natural step to make is to de�ne the associated orbit diagram. Moregenerally, if T is an operation (a functor) associating a space T (V ) to a linear subspace V � Kn ,where K = R; C or H , then T (A) denotes the diagram T (A) = fT (Ap)gp2P associated to thecorresponding arrangement of subspaces in T (Kn ). There are several examples that come upvery naturally in the mathematical practice. For example, if V 7! S(V ) is the operation ofassociating the unit sphere to the linear subspace V � Rn, then S(A) = fS(Ap)gp2P is theassociated spherical diagram. Similarly, functors V 7! RP (V ); V 7! CP (V ) or simply P (V ) inboth cases (and in the case of quaternionic projective spaces) lead to the corresponding projectivearrangements RP (A), CP (A) or HP (A), denoted simply by P (A). Projective diagrams arespecial cases of the associated Grassmann diagrams obtained with the aid of the functor V 7!Gk(V ) := fL � V : dim(V ) = kg. Lens space arrangements Lm(A) are de�ned similarly, whereLm(V ) = Lm(S(V )) := S(V )=Zm is the lens space associated to the unit sphere in a complexlinear space V .

We illustrate how the Homotopy Lemma can be applied to produce a combinatorial de-scription of the link of the related arrangement in all the special cases above. We obtain inparticular simpler, stronger, and more natural proofs of some results from [44, Thms. 2.11 and2.14] about projective arrangements. The advantage of the proof below is that it uses the Ho-motopy Lemma in its simplest form and allows a uniform treatment of all the special casesabove. It is clear that this method may be useful for other applications, say for other groupactions, since the argument no longer requires the arrangement to be shifted to a more specialposition by a dilatation ei 7! �iei, i = 1; : : : ; n, for some � > 0.

Our objective is to show that the homotopy type of the link has a purely combinatorialdescription in terms of the poset P and the associated rank function d : P ! N0. The link of a(spherical, a�ne, projective, lens, Grassmann etc.) arrangement is the union of all spaces in thearrangement. It follows from the Projection Lemma that the link has the same homotopytype as the homotopy limit of the corresponding diagram.

We uniformly construct a combinatorially de�ned diagram which serves for comparison withthe original one. Choose a ag F = fFig

ni=0, f0g = F0 � F1 � � � � � Fn = K

n . Let V 7! T (V )be one of the functors described above. Then T [F ] = T [F ](A), the ag diagram associated withT (A), is de�ned by T [F ]p := T (Fd(p)) where the morphisms T [F ]p ! T [F ]q are the obviousinclusion maps. Every two ag diagrams T [F ] and T [F 0] are naturally isomorphic, thus theisomorphism type of T [F ] depends only on P and d.

21

We want to compare our diagram T (A) with the combinatorially de�ned diagram T [F ](A).There does not seem to exist a natural map of diagrams between T (A) and T [F ](A) because e.g.the projection map is not natural. This di�culty was overcome in [44], in the case of projectivediagrams, by shifting the diagram T [F ](A) to a more special position and by applying a moregeneral version of the Homotopy Lemma by Vogt [39] that allows noncommutative diagramsif they commute up to coherent homotopies. The proof of Theorem 6.4 shows that in practicalproblems a very natural idea to use is the comparison with the third, so called \ample space"diagram that contains both T (A) and T [F ](A) as subdiagrams. We need a lemma which explainswhat is meant by \ample" in all the interesting cases above.

Lemma 6.3 Let T : Vect ! Top, V 7! T (V ), be one of the functors de�ned above whichto every vector space V � Kn associates the corresponding projective space P (V ), Grassmannmanifold Gk(V ), the lens space Lm(V ) (for V � C n), or the unit sphere S(V ). Then in eachof these cases there exists a functor RT : Vect(Kn)! Top which associates with each subspaceV � Kn an \ample" subspace RT (V ) � T (Kn ) and to each inclusion V ! W an inclusion ofspaces RT (V )! RT (W ) so that the following condition is satis�ed.

Let V � Kn , dim(V ) = k. Then for any W of dimension k with dim(W \ V ?) = 0, there isan inclusion T (W ) � RT (V ) 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 de�ned in a very similar way for all the examples above. For examplein the case of the functor V 7! S(V ), we put RT (V ) := S(Kn )nS(V ?). The construction inthe case of a projective functor V 7! P (V ) is analogous, one removes the projective spaceP (V ?) from P (Kn ). The inclusion map iV : P (V )! P (Kn)nP (V ?) is a homotopy equivalencesince P (Kn )nP (V ?) is the total space of a vector space bundle (a tubular neighborhood) overP (V ). Obviously P (W ) � P (Kn)nP (V ?) for any W with the property W \V ? = f0g. Finally,iW : P (W ) ! RT (V ) is a homotopy equivalence since there exists a linear map OV;W : Kn !Kn which maps V to W and leaves V ? invariant. Something similar is done in the case ofGrassmannians. Here, RGk(V ) is de�ned by RGk(V ) := fL 2 Gk(K

n ) : dim(L \ V ?) = 0g.In this case RGk(V ) is also the total space of a vector bundle over Gk(V ) which can be seenas follows. If Mk(V ) is the Stiefel manifold of all 1 � 1 linear maps (matrices) D : Kk !V then Gk(V ) = Mk(V )=G(k) where G(k) = GL(k;K) is the appropriate group of linearautomorphisms. Let p : Mk(K

n) ! Gk(Kn ) be the projection map. Note that p�1(RGk(V )) =

Mk(V )� L(Kk ; V ?); where L(Kk ; V ?) is the space of all linear maps, and that the group G(k)acts diagonally on the product Mk(V )�L(K

k ; V?). Now it is enough to recall that if X and Yare two G-spaces and if the action on Y is free, then the orbit space (X � Y )=G of the diagonalaction is represented as a �ber bundle

X ! (X � Y )=G! Y=G

which in our case means that RGk(V ) is �bered over Gk(V ) with the �ber L(Kk ; V ?). Hence,iV is a homotopy equivalence. It is shown that iW is also a homotopy equivalence analogouslyto the case of projective diagrams.

Finally, in the case of the \lens space" functor, let RLm(V ) := Lm(C n)nLm(V ?). The proofthat RLm(V ) is \ample" in the sense above is analogous and can rely on the fact that the sphereS(Cn) is a join of spheres S(V ) and S(V ?); and that the action of the cyclic group Zm respectsthis decomposition.

22

Theorem 6.4 (Homotopy types of Arrangements)Let A = fApgp2P be a linear subspace arrangement of Kn , where K is one of the (skew) �eldsR, C or H . Let P be the intersection poset of A, with the dimension function d : P ! N0,d(p) = dim(Ap), de�ned above, and set P[k] := fp 2 P : d(p) � kg for k � 0.

Let T : Vect ! Top, V 7! T (V ), be the projective, sphere, Grassmann or the lens spacefunctor de�ned above and let T (A) be the corresponding arrangement of subspaces of T (Kn).Then there is a homotopy equivalence

[p2P

T (Ap) 'n[

k=0

T (Kk )��(P[k]):

Proof. We start by choosing the ag F , Fi = spanfekgik=1, in su�ciently general position

with respect to the arrangement A. This requirement means that for any Fi and Ap 2 A, ifdim(Fi) = dim(Ap) then Ap \ F

?i = f0g. Let us de�ne the associated \ample" space diagram

R = R(A) by R = fRpgp2P ; Rp := RT (Ap) = RT (Fd(p), where V 7! RT (V ) is the \ample"space functor associated with T described in Lemma 6.3. Hence, there exist two naturallyde�ned maps of diagrams � and �,

� : T (A) ! R(A) � T [F ](A) : �

induced by the inclusion maps �p : T (Ap) ! Rp and �p : T [F ]p ! Rp. By Lemma 6.3 theseinclusion maps are homotopy equivalences. From here and the Homotopy Lemma it followsthat holim

�!T (A) ' holim

�!R and holim

�!T [F ](A) ' holim

�!R, which implies

holim�!

T (A) ' holim�!

T [F ](A):

Finally, we notice that the embeddings T (Fk) ,! T (Kn ) induce a diagram map

� : T [F ](A) ,! F ;

where F denotes the \constant" P -diagram which has Fp = T (Kn) for all p 2 P , and identitymaps fpp0 for p � p0. From the Embedding Lemma together with Example 3.2(vi) we get

that b� is an embedding

b� : holim�!

T [F ](A) ,! holim�!F �= T (Kn)��(P );

and we easily identify the image of b� with the spaceSnk=0 T (K

k )��(P[k]).

Note that for the homotopy formula for Grassmann arrangements in Theorem 6.4 only thetruncated poset P[k] = fp 2 P : d(p) � kg is relevant: This corresponds to the fact that spaceswith d(p) < k do not have k-dimensional subspaces, so A and A[k] = fAp 2 A : d(p) � kg havethe same associated k-Grassmann arrangement.

The following proposition shows that there is a general decomposition formula for the ho-mology of ag diagrams and, a posteriori, of the T -links of K-arrangements for K = R, Cor H .

23

Proposition 6.5 (Homology of Flag Diagrams)Let A be an arrangement of linear subspaces in Kn and let T be one of the functors describedabove. Let T [F ](A) be the combinatorially de�ned ag diagram associated to the arrangementA and the functor T . We set s = minfd(Ap)gp2P and t = maxfd(Ap)gp2P . Assume that for thecoe�cient ring R and for all s � k � m � t

� the exact sequence of the pair (T (Km ); T (Kk )) splits in homology and

� the homology groups H�(T (Km );R) are free R-modules.

Then, H�( holim�!

T [F ](A);R) �=

�H�(�(P[s]);R) H�(T (K

s);R)��

tMk=s+1

�H�(�(P[k]);R) H�(T (K

k ); T (Kk�1);R)�:

Proof. For an arbitrary poset Q and a natural number k we introduce \constant" diagramsMk;Q over Q, de�ned by Mk;Q : Q ! Top;Mk;Q(q) := T (Kk ). If it is clear from the contextwhich poset Q is used, we write Mk for the diagram Mk;Q. Note that our diagram T [F ](A)can be squeezed in between two constant diagrams Ms;P and Mt;P . By Example 3.2 (vi)we have holim

�!Mk;Q

�= �(Q) � T (Kk ). The K�unneth theorem and the freeness of R-modules

H�(T (Kk );R) imply

H�( holim�!Ms;Q;R) �= H�(�(Q);R)H�(T (K

s );R):

From this observation and from the fact that the map H�(T (Kk );R) ! H�(T (K

m);R) is amonomorphism for k � m, we conclude that the map holim

�!Mk ! holim

�!Mm, for k � m;

induces a monomorphism of homology groups. Then the composition of homomorphisms

H�( holim�!Ms;R)! H�( holim

�!T [F ](A);R)! H�( holim

�!Mt;R)

is a monomorphism too. Thus the �rst of them is also a monomorphism. It follows that thelong exact sequence of the pair ( holim

�!T [F ](A) ; holim

�!Ms ) splits and

H�( holim�!

T [F ](A);R)�= H�( holim�!Ms;R)�H�( holim

�!T [F ](A) ; holim

�!Ms ;R):

Informally speaking, we peel from the homotopy limit of the diagram T [F ](A) the part of itshomology coming from the constant subdiagramMs. Let E[i] be the restriction of the diagramT [F ](A) on the poset P[s+i]; i � 0: By excision, recalling the de�nition of the homotopy limitholim�!

T [F ](A), we have H�( holim�!

T [F ](A) ; holim�!Ms;R) �= H�(holim

�!E[1] ; holim�!

Ms;P[s+1];R):

By induction on i we may assume that for some i � 0

H�( holim�!

T [F ](A);R) �=�H�(�(P[s]);R) H�(T (K

s );R)��

s+iMk=s+1

�H�(�(P[k]);R) H�(T (K

k ); T (Kk�1);R)��H�(holim

�!E[i+1]; holim�!

Ms+i;P[s+i+1];R)

The long exact sequence of the triple

( holim�!E[i+1] ; holim�!

Ms+i+1;P[s+i+1]; holim�!Ms+i;P[s+i+1]

)

24

splits by the same argument as above. This means that we can peel from the homologyH�( holim

�!E[i+1] ; holim�!

Ms+i;P[s+i+1];R) the part isomorphic to

H�( holim�!Ms+i+1;P[s+i+1]

; holim�!Ms+i;P[s+i+1]

;R) �= H�(�(P[s+i+1]))H�(T (Ks+i+1); T (Ks+i)):

The part that remains is isomorphic to the group H�( holim�!E[i+1] ;Ms+i+1;P[s+i+1]

;R). The last

group is by excision isomorphic to H�( holim�!E[i+2] ; holim�!

Ms+i+1;P[s+i+2];R). So the K�unneth

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 and t = maxfd(Ap)gp2P = n), then both K�1

and T (K�1) are interpreted as empty spaces and the formula given in Proposition 6.5 can berewritten as follows.

Corollary 6.6 Let A be an essential arrangement. Then

H�( holim�!

T [F ](A);R)�=nM

k=0

H�(�(P[k]);R)H�(T (Kk ); T (Kk�1);R):

For example if T = P is the functor which associates the complex projective space P (V ) toevery complex linear space V , then the formula given in Corollary 6.6 has the following form

Hr( holim�!

T [F ](A);R) �=nM

k=0

Hr�2k(�(P[k]);R)

where r � 0 and H�k(X) := 0 for all k > 0: This formula together with its counterpart for thereal projective arrangements was formulated and proved in [44]. Unfortunately, as it was kindlypointed to us by Anders Bj�orner and Karanbir Sarkaria, the formulation there su�ers from somemisprints. The following example, which we also owe to A. Bj�orner and K. Sarkaria, show howa formula of the type above arises in connection with the Stanley-Reisner ring of a simplicialcomplex �.

Corollary 6.7 Let � be a simplicial complex over an n-element ground set f1; : : : ; ng. LetA� be the arrangement of complex linear subspaces in C n de�ned by A� = fA�g�2�, whereA� := spanC fejgj2�, ei the ith unit coordinate vector in C n . Let P (A�) = fP (A�)g�2�; bethe associated projective arrangement. Then the homology of the union

S�2� P (A�) of the

arrangement P (A) is given by

Hr([�2�

P (A�) ;Z)�=nM

k=0

Hr�2k(�(�[k]);Z)

where �[k] := f� 2 � : dimC (�) � kg is a subposet of (�;�) and �(�[k]) its order complex.

Note that the union of the arrangement A� is a projective variety whose homogeneouscoordinate ring is the Stanley-Reisner ring of �.

Note that Proposition 6.5 deals with the case when H�(T (Ki)) ! H�(T (Ki+1)) is injectivein contrast to the Goresky-MacPherson formula (Proposition 6.2). We already mentioned thatthe Goresky-MacPherson formula for the cohomology of the complement of an arrangement Aof (a�ne) subspaces can be proved by Alexander duality from the homology of an associatedarrangement S(A) of spheres. In the case when T = S the map eH�(T (Ki)) ! eH�(T (Ki+1)) istrivial. Although it does not completely �t in the setting of this section one may now regardtoric varieties | seen from the point of view of Section 5 | as an interesting third case, whenthe map in homology induced to the diagram maps are surjective.

25

7 Subgroup Complexes

The order complex of the poset Sp(G) = fP � Q : jP j = pi 6= 1g of non-trivial p-subgroups of a�nite group G has received considerable interest over that past few years (see for example [1]).It was already observed by Quillen [31] that Sp(G) is homotopy equivalent to the poset Ap(G)of non-trivial elementary abelian p-subgroups of G. In [28] the authors consider the coveringof �(Ap(G)) by the subcomplexes �(Ap(NA)) for a �xed solvable normal p0-subgroup N andmaximal elementary abelian subgroup A of G. Then they use the following facts :

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

(b) For a solvable normal p0-group N and an elementary abelian p-subgroup A the complex�(Ap(NA)) are homotopic to a wedge of spheres of dimension rank(D)� 1.

Observation (a) follows by basic group theoretical argumentation. Assertion (b) is much lessobvious. It was established by Quillen [31, Theorem 11.2], but also follows by applications ofthe homotopy limit methods (see [28, Theorem (A)]). Using facts (a) and (b), the Projection,Homotopy and the Wedge Lemma the following wedge decomposition of �(Ap(G)) for �nitesolvable groups G with non-trivial normal p0-group is proved in [28].

Theorem 7.1 (Pulkus & Welker [28, Theorem (B)])Let G be a �nite group and let p be a prime. Let N be a solvable normal p0-subgroup. LetCN=N be the intersection of all maximal elementary abelian p-subgroups of G=N . For AN=N 2Ap(G=N) let cAN=N be an arbitrary but �xed point in �(Ap(G=N)>AN=N). Then �(Sp(G)) ishomotopy equivalent to _

AN=N2Ap(G=N)>CN=N[fCN=Ng

�(Ap(NA)) ��(Ap(G=N)>AN=N):

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

the point cAN=N 2 �(Ap(G=N)>AN=N) �Ap(NA)

with the point AN=N 2 �(Ap(G=N)) ��(Ap(N1)):

In particular, if A is a maximal elementary abelian group of rank r in G, then �(Ap(NA)) ��(Ap(G=N)>AN=N) is homotopic to a wedge of (r� 1)-spheres.

8 Appendix: Comparison Lemmas over Small Categories

We focused our attention on diagrams of spaces over �nite partially ordered sets P . A moregeneral approach would be to choose diagrams over small categories as basic objects. This wouldbe closer to the original approach as in Segal [32], in the book of Bous�eld & Kan [6], or inmore recent papers of Hollender & Vogt [19], Dwyer & Kan [13] and Dror Farjoun & Zabrodsky[11]. Namely, for any diagram of spaces D de�ned as a functor D : S ! Top from a smallcategory S to the category of spaces, there exists the corresponding space holim

�!D; which in the

special case of a diagram over a �nite poset, seen as a small category, reduces to the de�nitionfrom Section 3. There is also the concept of the \homotopy inverse limit" (a de�nition is givenbelow) which is dual to the construction above, see [6], [11], [19]. There exist other naturalgeneralizations. One can de�ne the homotopy limit for diagrams over topological categories, for

26

diagrams commuting up to coherent homotopies [19], or in di�erent direction a very interesting\homotopy mixing" bifunctor B(E ; S;D); de�ned for a pair of diagrams D and E , see [19].

As in the case of a diagram over a poset P we write D� to denote the space attached to� 2 Ob(S) by the S-diagram D, and df for the map df : D� ! D� assigned by D to the

morphism �f! � in S.

A map � : D ! E from an S-diagram D to a T -diagram E is a natural transformationof functors; that is, it is given by a functor � : S ! T and a collection of continuous maps

�� : D� ! E� so that for all �f! � 2 Mor(S) and �; � 2 Ob(S) the following diagram

commutes :

D����!E�(�)

df # # e�(f)

D����!E�(�)

The level of generality chosen in the body of our paper is motivated and justi�ed by intendedapplications in combinatorics and should make these ideas more accessible to non-specialists.The situation resembles the theory of transformation groups where the language and methodsof diagrams are also very useful, see [10], but usually the level of generality and the language iscarefully chosen to match intended applications. Recall that every G-space X can be seen as adiagram BG;X over a one point category SG; Ob(SG) = fptg and Mor(SG) = G, where BG;Xassigns to the point pt the G-space X and to each g 2 G the the homeomorphism g : X ! X

induced by g. Then the fundamental objects such as the classifying space BG, the associateduniversal G-space EG and the space EG�GX(= (EG�X)=G) are as spaces of the type above.In particular, we have EG �G X �= holim

�!BG;X . With a little abuse of the language we will

silently, wherever convenient, denote the category SG above simply by G.This appendix is intended to serve as a link with the general theory. We outline proofs of

some of the theorems proved earlier for the special case of P -diagrams. The cases of a �niteposet diagram and a G-space seen as a diagram serve as test examples for the general theory.

The general construction of the homotopy limit �ts in the well known scheme of constructingspaces out of simplicial data. More precisely, for every simplicial space X : Ord ! Top, onede�nes its geometric realization jjXjj, where a simplicial space X is a contravariant functor fromthe category Ord of all ordered simplices to topological spaces [17] [19] [26] [32].

If X : S ! Top is an S-diagram of spaces, then a natural simplicial space X arises as follows.For the standard n-simplex [n] := f0; 1; : : : ; ng 2 Ord, a typical n-simplex in Xn := X([n]) isviewed informally as a chain or \molecule" of length n starting at a point x 2 X�; � 2 Ob(S).More precisely, by a molecule of length n we mean a point x 2 X� and a chain of morphisms

�0f1!�1

f2!� � �fn�1! �n�1

fn!�n so that dom(f1) = �0 = �. The collection of spaces Xn, n 2 N, is

naturally turned into a simplicial space with the usual face and degeneracy maps de�ned by theusual equalities, see [17] or the equations given below. Then the homotopy limit of the diagramX : S ! Top is by de�nition the geometric realization jjXjj of this simplicial space, i.e.,

holim�!X = jjXjj:

The reader can view molecules or the cells de�ned by them as the basic building blocks ofthis space jjXjj.

The holim�!

-functor is a mono functor (i.e., a functor in one variable from the category of all

S-diagrams to spaces). A useful generalization of this construction can be found in [19], where

27

a de�nition of a \homotopy mixing" bifunctor B(E ; S;D) is given. Here D : S ! Top is a usualdiagram, or a left S-module in the terminology of [19], and E : Sop ! Top is a diagram overthe opposite category Sop, or a right S-module as it is called in [19]. B(E ; S;D) is de�ned asthe geometric realization of the simplicial space B(E ; S;D) which has for a typical n-simplex (n-

molecule) a chain of morphisms �nfn �n�1

fn�1 � � �

f3 �2

f2 �1

f1 �0 and a pair of points x 2 D�0

and y 2 E�n. The face and degeneracy maps are de�ned as follows.

di(y; fn; : : : ; f1; x) =

8><>:

(y; fn; : : : ; f2; f1x) for i = 0;(y; fn; : : : ; fi+1 � fi; : : : ; f1; x) for 0 < i < n;(yfn; fn�1; : : : ; f1; x) for i = n:

si(y; fn; : : : ; f1; x) = (y; fn; : : : ; id; fi; : : : ; f1; x) for 0 � i � n:

It is clear thatholim�!D �= jjB(PT ; S;D)jj = B(PT ; S;D);

where \PT " is the diagram which associates a one point space fptg to each object in S. Thesimplicial space (set) B(PT ; S;PT ) = fBngn2N is called the nerve of S and denoted by N(S):The space holim

�!PT = jjN(S)jj is called the classifying space BS of the category S. In the case

of a group G, seen as a small category, this space agrees [32] with the usual classifying spaceBG. If S is a �nite partially ordered set, then BS = �(S) (see Example 3.2 (i)).

Both holim�!D and B(E ; S;D) are geometric realizations of appropriate simplicial spaces, and

it is useful to have versatile tools for comparing geometric realizations of di�erent simplicialtopological spaces. Such a tool is provided by the following proposition, which follows from [33,Prop. A.1]. We need a concept of a \good" simplicial space introduced by G. Segal [33]. (Seealso [23] and [24], where a similar concept was introduced under the name \proper" simplicialspace. R. Vogt has remarked that a weaker property is su�cient for all our applications: thatthe inclusion DAn �! An of the space of degenerate simplices into the space of all simplices isa closed co�bration.)

De�nition A.1 ([33])A simplicial topological space A : Ord ! Top; An := A([n]); is called \good" if for everysurjective map � : [n]! [n� 1] the map A(�) : An�1 ! An is a (closed) co�bration.

Proposition A.2 ([33])If a (continuous) simplicial map F : A ! B; Fn : An ! Bn; between two \good" simplicialtopological spaces has the property that Fn : An ! Bn is a (weak) homotopy equivalence then

jjF jj : jjAjj ! jjBjj

is also a (weak) homotopy equivalence.

Let us note that the proof of this proposition is inductive and that the inductive step isbased on a \push out" lemma which is a form of the Gluing Lemma 2.1.

Proposition A.3

For any two diagrams E and D; the simplicial space B(E ; S;D) is \good."

28

Proof. Indeed, this is a consequence of the fact that in this presentation we restricted ourattention on diagrams over discrete categories S, (i.e., categories without a topology). It fol-lows that Bn(E ; S;D) is a disjoint topological sum of spaces of the form M�

�= E�n � D�0 ,

where � 2 Nn(S) is a molecule �nfn �n�1

fn�1 � � �

f3 �2

f2 �1

f1 �0 and M� � Bn(E ; S;D) con-

sists of all molecules having � as its underlying chain of morphisms. The degeneracy maps : Bn�1(E ; S;D)! Bn(E ; S;D); associated to a surjective map � : [n] ! [n � 1]; is obviouslya co�bration since Bn�1 is sent to a closed and open subspace of Bn: Note that in general, see[19], for this to be true in a topological category S one should assume, as part of the de�nitionof a topological category, that Ob(S) �Mor(S) is a (closed) co�bration.

The original de�nition of the homotopy limit of a diagram given in [6] is much closer inspirit to our de�nition given in Section 3. It utilizes the concept of the \undercategory" of acategory S. We emphasize the parallelism with the concept of the lower cone P�p of a poset P atan element p 2 P . The lower cone S#� or the \undercategory" of the category S at � 2 Ob(S)is de�ned as follows. An object of this category is any arrow � ! � and a morphism between

objects �f! � and �

g! is an arrow �

h! so that h � f = g. A more general and also very

useful concept is the \undercategory" ��1T#� associated to a functor � : S ! T from a categoryS to a category T and � 2 Ob(T ) which is roughly, at least in the case of posets, the �-\inverse

image" of T#�. In the general case the objects of ��1T#� are pairs (u; f) where u 2 S and �

f!�(u).

A morphism between pairs (u; f) and (v; g) is an arrow uh! v so that �(h) � f = g. The functor

� 7! S#� is an Sop-diagram of categories which, by passing to the corresponding classifyingspaces, leads to an Sop-diagram of spaces B(S#) : Sop ! Top; � 7! B(S#�). Similarly, theT op-diagram � 7! B(��1T#�) is denoted by B(��1T#). The de�nition of the homotopy limit of anS-diagram D : S ! Top given by Bous�eld & Kan [6] turns out to be an S-\tensor product" or\mixing" of diagrams B(S#) and D de�ned as the disjoint union of spaces B(S#�)�D� divided

by the equivalence relation � where (yf; x) � (y; dfx) for �f! �, x 2 B(S#�), y 2 D� (see [6]

and [19]). It is shown in [19] that these two de�nition of homotopy limits coincide. The readeris invited to check this fact her-himself.

Although, guided by the intended applications, we in this paper put more emphasis on theholim�!

functor, here is an appropriate place to say a few words about the dual concept of the

inverse homotopy limit functor holim �

: One starts with the \overcategory" S"� which is just the

undercategory of the opposite category Sop: In the case of a poset this reduces to the constructionof the upper cone P�p: The functor � 7! S"� is an S-diagram of categories which, by passingto the corresponding classifying spaces, leads to an S-diagram of spaces B(S") : S ! Top; � 7!B(S"�). For a functor � : S ! T and � 2 Ob(T ) one de�nes in a completely analogousway the \overcategory" ��1T"� and the diagram B(��1T"). The homotopy inverse limit holim

�D �=

HomS(B(S");D) is then de�ned as the functional space of all S-\equivariant maps" or naturaltransformations f := ff�g�2S from B(S") to D: More precisely, f� : B(S"�) 7! D�; � 2 Ob(S),

is a collection of continuous maps so that for every morphism �g! � the following diagram is

commutative :

B(S"�)f��!D�

B(S")(g) # # dg

B(S"�)f��!D�

29

The notation above, especially the use of the word \equivariant," turns out to have the usualmeaning if the category S is chosen to be a group G: Occasionally it is useful to emphasize thisconnection and motivate some general constructions by the corresponding constructions whicharise naturally in the case of group diagrams. For example, as in Theorem A.6 below, we candenote B(S") by ES since in the case S = G the space EG, seen as a diagram over a one-elementcategory G, has the usual meaning, while holim

�D �= HomS(ES;D) �= HomG(EG;X) is just the

space of all equivariant maps from the space EG to a G-space X . Below we will see that EGitself can be seen as a homotopy limit of a suitable diagram EG.

The two concepts of the direct and inverse homotopy limits are dual to each other. Thisduality is made precise by the following proposition, see [6], which shows that in principle, ifconvenient, one can prove a result by proving �rst the dual statement.

Proposition A.4For every S-diagram D of topological spaces and a topological space Z there is a natural home-omorphism

Hom(holim�!D; Z) �! holim

�Hom(D; Z):

[For this it is silently assumed that all spaces belong to a category of spaces satisfying the equiv-alence Hom(X � Y; Z) �= Hom(X;Hom(Y;Z)).]

Proof. We observe �rst that the holim�!

construction is a \di�erence cokernel" (coequalizer, direct

limit) of two obvious \parallel" maps. Indeed, holim�!D is the di�erence cokernel of the following

mappings :

]�!�

B(S#�)�D� �

]�2S

B(S#�)�D�

where the two parallel maps are induced by the obvious diagram maps S#� ! S#� and D� ! D�

determined by the S-morphism �! �:By applying the functor Hom( � ; Z) we see that the space Hom(holim

�!D; Z) is an inverse

limit (di�erence kernel, equalizer), as shown by the diagram

Hom(holim�!D; Z) 9 9 K

Y�2S

Hom(B(S#�)�D�; Z)�Y�!�

Hom(B(S#�)�D�; Z):

By the de�nition of the inverse homotopy limit given above, taking into account that the diagramHom(D; Z) is an Sop-diagram of spaces, the space holim

�Hom(D; Z) is a di�erence kernel, as

shown by the diagram

holim �

Hom(D; Z) 9 9 KY�2S

Hom(B(Sop"�); Hom(D�; Z))�Y�!�

Hom(B(Sop"�); Hom(D�; Z)):

By comparing these two diagrams we obtain the desired result.

Here and elsewhere some care is needed, since we rely on the equivalence

Hom(X � Y; Z) �= Hom(X;Hom(Y;Z))

30

which is valid only under some restrictions, say if we work with compactly generated spaces.But as usual these restrictions are not very important if we focus our attention on the weakhomotopy type.

The de�nitions and concepts introduced above have a clear special meaning for a diagramover a (�nite) poset. It is interesting to recall their usual meaning in the case of a G-space Xseen as a G-diagram. For example the \undercategory"(\overcategory") of G turns out to be a

small category which has G itself for the set of objects and a single morphism gg�1h�! h between

each pair (g; h) of objects g; h 2 G. Let us denote by EG the corresponding diagram (i.e., EGis the functor that assigns to each g 2 G the one point space fgg and to each pair (g; h) ofelements of G the obvious map egh : fgg ! fhg). It is easily seen, see [32], that the functorwhich maps this category into a one element category with a single (identity) morphism is anequivalence. As a consequence we obtain that the classifying space holim

�!EG of this category

is contractible and identi�ed as the space EG (i.e., a contractible space where G acts freely).We de�ne the diagram BG;pt which maps the point \pt" to a singleton and maps each elementg 2 G to a distinguished morphism bg from \pt" to \pt." Then holim

�!BG;pt = BG is the usual

classifying space BG and the obvious natural transformation of diagrams EG ! BG;pt inducesby passing to homotopy limits the usual universal bundle map EG ! BG. Finally, if X is aG-space, then its homotopy limit is identi�ed as the space EG�GX which plays an exceptionallyimportant role in the theory of transformation groups, while the homotopy inverse limit is thespace HomG(EG;X) of homotopy �xed points in X:

Now we are ready to give an outline of extensions of some of the results which have beenestablished earlier in the case of a diagram over a �nite poset.

We start with the \homotopy lemma" which directly extends our Corollary 3.9. It playscertainly a very important role in the theory and serves as a tool for proving of other relatedresults. Because of this we will outline two di�erent approaches to this result, one based onPropositions A.2 and A.3 and the other on Proposition A.4.

Theorem A.5 (Homotopy Lemma [39] [6] [11] [19])Let � : D ! E be a map of two S-diagrams D and E so that � : S ! S is the identity andthe map �� : D� ! E� is a (weak) homotopy equivalence for every � 2 S. Then the inducedmap b� : holim

�!D ! holim

�!E is also a (weak) homotopy equivalence. More generally, the bifunctor

B(E ; S;D) de�ned above, which generalizes the holim�!

-functor, has similar homotopy properties

with respect to both arguments D and E [19].

Proof. The proof easily follows from Propositions A.2 and A.3. The description of the simplicialspacesB(PT ; S; E) andB(PT ; S;D) given in the proof of Proposition A.3 shows that the map �induces a homotopy equivalence �n :Bn(PT ; S; E)! Bn(PT ; S;D) for all n.

The Homotopy Lemma is a manifestation of the principle, see the Gluing Lemma 2.1,that a \local" homotopy equivalence between spaces is a global (weak) homotopy equivalence.A very interesting approach to the dual statement is described in [11]. We believe that itdeserves our attention here since it illuminates the \homotopy lemma" from the point of viewof transformation groups and serves as a link to other results in [11].

31

Theorem A.6Let � : D ! E be a map of two S-diagrams D and E so that � : S ! S is the identity and themap �� : D� ! E� is a (weak) homotopy equivalence for every � 2 S. Then there is an inducedhomotopy equivalence b�1 : holim

�D ! holim

�E :

Proof. Recall that holim �D can be thought of as the space HomS(ES;D) of all S-equivariant

maps between the two S-diagrams. Let us make a few useful observations about this space. Thediagram ES is a CW-complex which in the case S = G is a free CW-complex, i.e., each cell inEG is viewed inside a free orbit G � e: In short, EG is a free G-CW-complex which is �ltered byits n-skeletons EGn. Actually it is a direct limit of the sequence of subcomplexes

: : :! EGn�1 ! EGn ! EGn+1 ! : : :

As a consequence the space HomG(EG;D) is the inverse limit of the tower of �brations

: : : HomS(ESn�1;D) HomS(ESn;D) HomS(ESn+1;D) : : :

The proof of the desired homotopy equivalence, at least in the case S = G; then can go byinduction and then by \passing to the limit." The result which is needed for the inductive stepis the Cogluing Lemma (see [7] for details) which is dual to our Gluing Lemma 2.1 and alimit result which is needed for the limit step is a result dual to Theorem 1 in [37]. One observes�rst that EGn is obtained as a pull-back

:�n� n �! �n � n

# #

EGn�1 �! EGn

where the set n denotes the G-set of all (free) G-orbits. As a space n has the discretetopology. Then the space HomG(EGn;D) of all equivariant maps from EGn to the G-space Dis the push-out of the following diagram.

HomG(EGn;D) �! HomG(EGn�1;D)

# #

HomG(�n � n;D) �! HomG(

:�n� n;D)

We observe that the lower horizontal map is a �bration as a dual map to a co�bration from theprevious diagram. Let us note that exactly here we need the assumption of the theorem aboutthe \local" homotopy equivalence. Namely, there is a homeomorphism

HomG(�n � n;D) �= Hom(�n; HomG(n;D))

and the space HomG(n;D) is naturally homeomorphic to the space D:From here and the assumption of the theorem we conclude that the conditions of the

Cogluing Lemma [7] are ful�lled. So the inductive step of the homotopy equivalence betweenHomG(EGn;D) and HomG(EGn; E) is established.

32

Both the limit result needed to establish the homotopy equivalence between HomG(EG;D)and HomG(EG; E) and the Cogluing Lemma follow from the dual of a basic lemma which isdual to Lemma 2 in [37]. This lemma is essentially a most basic form of the Theorem A.2 inthe case of the two element category S = f� ! �g.

The key observation and possibly a surprise is the fact that the proof above can be carriedwithout essential changes in the case of any small category S. The papers [11] and [10] explainin detail how the usual concepts from the theory of G-sets like (free) orbits, �xed point sets, etc.can be meaningfully extended to the case of general diagrams. All we need here is an observationthat the diagram ES is a \free" S-CW-complex. The reader is encouraged to �ll in the detailsfollowing and extending the pattern of the proof of the special case S = G. We skip the detailsand make a brief comment on the use of the \local homotopy equivalence" assumption in thetheorem. One observes [10] that a typical cell e in the CW-decomposition of ES belongs tothe S-\orbit" of a unique (Sop)-molecule of the \overcategory" S"�: This molecule has the form

�nfn!�n�1

fn�1! � � �

f2!�1

f1!�0

i! d. This observation and Lemma 2.2 from [10] permit us to use

the Cogluing Lemma again as in the proof of the special case S = G.

Homotopy lemmas and their proofs should give a hint how to generalize some of the otherresults from this paper. Let us illustrate their power by the following elegant and short proof(cf. [19]), of the Inverse Image Lemma in the form that generalizes our Theorem 3.13.

Theorem A.8 (Inverse Image Lemma)Let � : S ! T be a functor from the small category S to the small category T . Let E : T ! Topbe a diagram over T . Then �](E) := � � E is a diagram over S. Assume that for all � 2 Ob(T )the \overcategory" ��1T#� is contractible (i.e., its classifying space B(��1T#�) is contractible). Then,

holim�!

�](E) ' holim�!E :

Proof. The Inverse Image Lemma is in [19] called the co�nality theorem. We essentiallyfollow the proof given in that paper.

Let us start with di�erent descriptions of spaces B(��1T#) and holim�!

�](E): It is useful to

introduce an auxiliary bidiagram �# : S � T op ! Top; �#(�; �) = MorT (�; �(�)); of discretetopological spaces (sets). Then by de�nition

holim� �

�#(�; �) �= B(��1T#�)�= B(PT ; S;�#(�; �)) :

Here, �#(�; �) is the S-diagram arising for �xed �. Also, by inspection of the de�nition of the\homotopy mixing" bifunctor one has B(�#(�; �); T;E)�= D�(�) (i.e., B(holim

� �

�#; T; E)�= E �� =

�](E) ). Since �# is a bidiagram or a bimodule in the terminology of [19], it can serve both asthe right and the left argument in the homotopy mixing bifunctor. If V is a bifunctor the readershould convince himself that the homotopy mixing operation has the following associativityproperty B(B(U ; S;V); T;W)�= B(U ; S; B(V ; T;W)): These preliminary facts together with theassumption that B(��1T#�) is contractible allow us to use Theorem A.5 which leads us to thefollowing sequence of equivalences

holim�!E = B(PT ; T; E)

' �B(B(��1T#); T; E)

�= B(B(PT ; S;�#); T; E)�=

�= B(PT ; S; B(�#; T; E)) = B(PT ; S; �](E)) = holim�!

�](E) :

33

Acknowledgements

Thanks to Eva-Maria Feichtner for a careful reading and for lots of helpful comments on themanuscript. We are grateful to Emmanuel Dror Farjoun and Rainer Vogt for some correctionsand very valuable remarks and references. We all thank the Konrad-Zuse-Zentrum in Berlin forits hospitality.

References

[1] M. Aschbacher and S. Smith. On Quillen's conjecture for the p-subgroups complex. Ann. of Math.(2), 137:473{529, 1993.

[2] E. Babson. A Combinatorial Flag Space. PhD thesis, MIT, 1993.

[3] A. Bj�orner. Topological methods. In R. Graham, M. Gr�otschel, and L. Lov�asz, editors, Handbookof Combinatorics. North-Holland, Amsterdam, 1995.

[4] A. Bj�orner. A generalized homotopy complementation formula. (Preprint 1995). Manuscript 1989.

[5] A. Bj�orner and J.W. Walker. A homotopy complementation formula for partially ordered sets.European J. Combin., 4:11{19, 1983.

[6] M. Bous�eld and D. M. Kan. Homotopy Limits, Completions and Localizations, volume 304 ofLecture Notes in Math. Springer, Berlin, Heidelberg, New York, 1972.

[7] R. Brown and P.R. Heath. Coglueing homotopy equivalences. Math. Z., 113:313{325, 1970.

[8] V.I. Danilov. The geometry of toric varieties. Russian Math. Surveys, 33(2):97{151, 1978.

[9] A. Dold and R. Thom. Quasifaserungen und unendliche symmetrische Produkte. Ann. of Math. (2),67(2), 1958.

[10] E. Dror Farjoun. Homotopy and homology of diagrams of spaces. In Proc. Workshop \AlgebraicTopology" (Seattle/Wash. 1985), volume 1286 of Lecture Notes in Math., pages 93{134, Berlin,Heidelberg, New York, 1987. Springer.

[11] E. Dror Farjoun and A. Zabrodsky. Homotopy equivalence between diagrams of spaces. J. PureAppl. Algebra, 41:169{182, 1986.

[12] W.G. Dwyer and D.M. Kan. Function complexes for diagrams of simplicial sets. Indag. Math.,45:139{147, 1983.

[13] W.G. Dwyer and D.M. Kan. A classi�cation theorem for diagrams of simplicial sets. Topology,23(2):139{155, 1984.

[14] G. Ewald. Combinatorial Convexity and Algebraic Geometry. Springer-Verlag, New York, to appear.

[15] S. Fischli and D. Yavin. Which 4-manifolds are toric varieties? Math. Z., 215:179{185, 1994.

[16] W. Fulton. Introduction to Toric Varieties, volume 131 of Annals of Mathematics Studies. PrincetonUniversity Press, 1993.

[17] P. Gabriel and M. Zisman. Calculus of Fractions and Homotopy Theory, volume 35 of Ergebnisseder Mathematik und ihrer Grenzgebiete. Springer, Berlin, Heidelberg, New York, 1967.

[18] M. Goresky and R. MacPherson. Strati�ed Morse Theory, volume 14 of Ergebnisse der Mathematikund ihrer Grenzgebiete. Springer, Berlin, Heidelberg, New York, 1988.

[19] J. Hollender and R. M. Vogt. Modules of topological spaces, applications to homotopy limits andE1 structures. Arch. Math. (Basel), 59:115{129, 1992.

[20] J.F. Kraus. Mayer-Vietoris-Prinzipien f�ur Cofaserungen. Math. Ann., 231:47{53, 1977.

[21] J. Lillig. A union theorem for co�brations. Arch. Math. (Basel), 24:410{415, 1973.

[22] A. T. Lundell and S. Weingram. The Topology of CW Complexes. Van Nostrand Reinhold, NewYork, 1969.

[23] J.P. May. The Geometry of Iterated Loop Spaces, volume 271 of Lecture Notes in Math. Springer,Berlin, Heidelberg, New York, 1972.

34

[24] J.P. May. E1-spaces, group completions and permutative categories, new developments in topology.volume 11 of London Math. Soc. Lecture Note Ser., pages 61{93. Cambridge Univ. Press, Cambridge,1974.

[25] M. McConnell. Rational homology of toric varieties. Proc. Amer. Math. Soc., 105(4):986{991, 1989.

[26] J. Milnor. The geometric realization of a semi-simplicial complex. Ann. of Math. (2), 65:357{362,1957.

[27] J. Milnor. On spaces having the homotopy type of a cw-complex. Trans. Amer. Math. Soc., 90:272{280, 1959.

[28] J. Pulkus and V. Welker. On the homotopy type of the p-subgroup complex for �nite solvable groups.Preprint 1994.

[29] D. Puppe. Homotopiemengen und induzierte Abbildungen, I. Math. Z., 69:299{344, 1958.

[30] D. Quillen. Higher Algebraic K-Theory: I, volume 341 of Lecture Notes in Math. Springer, Berlin,Heidelberg, New York, 1973.

[31] D. Quillen. Homotopy properties of the poset of nontrivial p-subgroups of a group. Adv. in Math.,28:101{128, 1978.

[32] G. B. Segal. Classifying spaces and spectral sequences. Publ. Math. I.H.E.S., 34:105{112, 1968.

[33] G. B. Segal. Categories and cohomology theories. Publ. Math. I.H.E.S., 13:293{312, 1974.

[34] R. P. Stanley. Enumerative Combinatorics, I. Wadsworth & Brooks/Cole, Monterey, CA, 1986.

[35] B. Sturmfels and G.M. Ziegler. Extension spaces of oriented matroids. Disc. Comp. Geom., 10:23{45,1993.

[36] S. Sundaram and V. Welker. Group actions on arrangements of linear subspaces and applicationsto con�guration spaces. Preprint, 1994.

[37] T. tom Dieck. Partitions of unity in homotopy theory. Compositio Math., 23:159{161, 1973.

[38] V.A. Vassiliev. Complements of Discriminants of Smooth Maps : Topology and Applications, vol-ume 98 of Transl. of Math. Monographs. Amer. Math. Soc., Providence, RI, 1994. Revised Edition.

[39] R. M. Vogt. Homotopy limits and colimits. Math. Z., 134:11{52, 1973.

[40] J. W. Walker. Homotopy type and Euler characteristic of partially ordered sets. European J.Combin., 2:373{384, 1981.

[41] J. W. Walker. Canonical homeomorphisms of posets. European J. Combin., 9:97{107, 1988.

[42] G. W. Whitehead. Elements of Homotopy Theory, volume 61 of Graduate Texts in Mathematics.Springer, Berlin, Heidelberg, New York, 1978.

[43] D. Yavin. The Intersection Homology with Twisted Coe�cients of Toric Varieties. PhD thesis, MIT,1990.

[44] G. M. Ziegler and R. T. �Zivaljevi�c. Homotopy types of subspace arrangements via diagrams ofspaces. Math. Ann., 295:527{548, 1993.

35