COLIMITS, STANLEY-REISNER ALGEBRAS, AND LOOP SPACES TARAS PANOV, NIGEL RAY, AND RAINER VOGT Abstract.We study diagrams associated with a finite simplicial complex K,* * in various algebraic and topological categories. We relate their colimits to familia* *r structures in algebra, combinatorics, geometry and topology. These include: right-angled Artin a* *nd Coxeter groups (and their complex analogues, which we call circulation groups); Stanley-* *Reisner algebras and coalgebras; Davis and Januszkiewicz's spaces DJ(K) associated with toric * *manifolds and their generalisations; and coordinate subspace arrangements. When K is a * *flag complex, we extend well-known results on Artin and Coxeter groups by confirming th* *at the relevant circulation group is homotopy equivalent to the space of loops DJ(K). We* * define homotopy colimits for diagrams of topological monoids and topological groups, and * *show they commute with the formation of classifying spaces in a suitably generalised sense.* * We deduce that the homotopy colimit of the appropriate diagram of topological groups is a mo* *del for DJ(K) for an arbitrary complex K, and that the natural projection onto the orig* *inal colimit is a homotopy equivalence when K is flag. In this case, the two models are com* *patible. 1. Introduction In this work we study diagrams associated with a finite simplicial complex K,* * in various algebraic and topological categories. We are particularly interested in colimit* *s and homotopy colimits of such diagrams. We are motivated by Davis and Januszkiewicz's investigation [12] of toric man* *ifolds, in which K first arises as the boundary of the quotient polytope. In the course of* * their coho- mological computations, Davis and Januszkiewicz construct real and complex vers* *ions of a space whose cohomology ring is isomorphic to the Stanley-Reisner algebra of K, * *over Z=2 and Z respectively. We denote the homotopy type of these spaces by DJ (K), and foll* *ow Buch- staber and Panov [7] by describing them as colimits of diagrams of classifying * *spaces. In this context, an exterior version arises naturally as an alternative. Suggestively, * *the cohomology algebras and homology coalgebras of the DJ (K) may be expressed as the limits a* *nd colimits of analogous diagrams in the corresponding algebraic category. When colimits of similar diagrams are taken in a category of discrete groups,* * they yield right-angled Coxeter and Artin groups. These are more usually described by a co* *mplementary construction involving only the 1-skeleton K(1)of K. Whenever K is determined e* *ntirely by K(1)it is known as a flag complex, and results such as those of [12] and [22] m* *ay be interpreted as showing that the associated Coxeter and Artin groups are homotopy equivalent* * to the loop spaces DJ (K), in the real and exterior cases respectively. In other words, th* *e groups are discrete models for the loop spaces. These observations raise the possibility * *of modelling ___________ Key words and phrases. Colimit, flag complex, homotopy colimit, loop space, r* *ight-angled Artin group, right-angled Coxeter group, Stanley-Reisner ring, topological monoid. The first author was supported by a Royal Society/NATO Postdoctoral Fellowshi* *p at the University of Manchester, and also by the Russian Foundation for Basic Research, grant number* * 01-01-00546. 1 2 TARAS PANOV, NIGEL RAY, AND RAINER VOGT DJ (K) in the complex case, and for arbitrary K, by colimits of diagrams in a * *suitably defined category of topological monoids. Our primary aim is to carry out this p* *rogramme. Before we begin, we must therefore confirm that our categories are sufficient* *ly cocomplete for the proposed colimits to exist. We show that this is indeed the case (as p* *redicted by folklore), and explain how the complex version of DJ (K) is modelled by the co* *limit of a diagram of tori whenever K is flag. We refer to the colimit as a circulation gr* *oup, and consider it as the complex analogue of the corresponding right-angled Coxeter and Artin * *groups. Of course, it is also determined by K(1). On the other hand, there are simple exam* *ples of non- flag complexes for which the colimit groups cannot possibly model DJ (K) in an* *y of the real, exterior, or complex cases. More subtle constructions are required. Since we are engaged with homotopy theoretic properties of colimits, it is no* * great surprise that the appropriate model for arbitrary complexes K is a homotopy colimit. Con* *siderable care has to be taken in formulating the construction for topological monoids, b* *ut the outcome clarifies the status of the original colimits when K is flag; flag complexes ar* *e precisely those for which the colimit and the homotopy colimit coincide. Our main result is the* *refore that DJ (K) is modelled by the homotopy colimit of the relevant diagram of topologi* *cal groups, in all three cases and for arbitrary K. When K is flag, the natural projection * *onto the original colimit is a homotopy equivalence, and is compatible with the two model maps. O* *ur proof revolves around the fact that homotopy colimits commute with the classifying sp* *ace functor, in a context which is considerably more general than is needed here. For particular complexes K, our constructions have interesting implications f* *or traditional homotopy theoretic invariants such as Whitehead products, Samelson products, an* *d their higher analogues and iterates. We hope to deal with these issues in subsequent * *work [27]. We now summarise the contents of each section. It is particularly convenient to use the language of enriched category theory* *, so we devote Section 2 to establishing the notation, conventions and results that we need. T* *hese include a brief discussion of simplicial objects and their realisations, and verificati* *on of the cocom- pleteness of our category of topological monoids in the enriched setting. Read* *ers who are familiar with this material, or willing to refer back to Section 2 as necessary* *, may proceed directly to Section 3, where we introduce the relevant categories and diagrams * *associated with a simplicial complex K. They include algebraic and topological examples, amongs* *t which are the exponential diagrams GK ; here G denotes one of the cyclic groups C2 and C,* * or the circle group T , in the real, exterior, and complex cases respectively. We devote Section 4 to describing the limits and colimits of these diagrams. * * Some are identified with standard constructions such as the Stanley-Reisner algebra of K* * and the Davis-Januszkiewicz spaces DJ (K), whereas the GK yield right-angled Coxeter a* *nd Artin groups, or circulation groups respectively. In Section 5 we study aspects of t* *he diagrams involving associated fibrations and homotopy colimits. We note connections with* * coordinate subspace arrangements. We introduce the model map fK :colimtmgGK ! DJ (K) in Section 6, and determi* *ne the connectivity of its homotopy fibre in terms of combinatorial properties of * *K. The results confirm that fK is a homotopy equivalence whenever K is flag, and quantify its * *failure for general K. In our final Section 7 we consider suitably well-behaved diagrams D * *of topological monoids, and prove that the homotopy colimit of the induced diagram of classify* *ing spaces is homotopy equivalent to the classifying space of the homotopy colimit of D, t* *aken in the category of topological monoids. By application to the exponential diagrams GK * *, we deduce COLIMITS, STANLEY-REISNER ALGEBRAS, AND LOOP SPACES * * 3 that our generalised model map hK :hocolimtmgGK ! DJ (K) is a homotopy equival* *ence for all complexes K. We note that the two models are compatible, and homotopy e* *quivalent, when K is flag. We take the category top of k-spaces X and continuous functions f :X ! Y as* * our underlying topological framework, following [35]. Every function space Y X is e* *ndowed with the corresponding k-topology. Many of the spaces we consider have a distinguish* *ed basepoint *, and we write top + for the category of pairs (X, *) and basepoint preserving* * maps; the forgetful functor top+ ! top is faithful. For any object X of top, we may add a* * disjoint basepoint to obtain a based space X+ . The k-function space (Y, *)(X,*)has the * *trivial map X ! * as basepoint. In some circumstances we need (X, *) to be well-pointed, in* * the sense that the inclusion of the basepoint is a closed cofibration, and we emphasise t* *his requirement as it arises. Several other important categories are related to top+ . These include tmonh * *, consisting of associative topological monoids and homotopy homomorphisms [5] (essentially * *equivalent to Sugawara's strongly homotopy multiplicative maps [34]), and its subcategory * *tmon , in which the homorphisms are strict. Again, the forgetful functor tmon ! top+ i* *s faithful. Limiting the objects to topological groups defines a further subcategory tgrp, * *which is full in tmon . In all three cases the identity element e is the basepoint, and we may s* *ometimes have to insist that objects are well-pointed. The Moore loop space X is a typical o* *bject in tmonh for any pair (X, *), and the canonical inclusion M ! BM is a homotopy homomorp* *hism for any well-pointed topological monoid M. For each m 0 we consider the small categories id(m), which consist of m obj* *ects and their identity morphisms; in particular, we use the based versions id?(m), whic* *h result from adjoining an initial object ?. Given a topological monoid M, the associated to* *pological category c(M) consists of one object, and one morphism for each element of M. S* *egal's [32] classifying space Bc(M) then coincides with the standard classifying space BM. Given objects X0 and Xn of any category c, we denote the set of n-composable * *morphisms X0 f1-!X1 f2-!. .f.n-!Xn by cn(X0, Xn), for all n 0. Thus c1(X, Y ) is the morphism set c(X, Y ) for a* *ll objects X and Y , and c0(X, X) consists solely of the identity morphism on X. In order to distinguish between them, we write T for the multiplicative topol* *ogical group of unimodular complex numbers, and S1 for the circle. Similarly, we discriminat* *e between the cyclic group C2 and the ring of residue classes Z=2, and between the infini* *te cyclic group C and the ring of integers Z. The first and second authors benefitted greatly from illuminating discussions* * with Bill Dwyer at the International Conference on Algebraic Topology held on the Island * *of Skye dur- ing June 2001. They are particularly grateful to the organisers for providing t* *he opportunity to work in such magnificent surroundings. 2. Categorical Prerequisites We refer to the books of Kelly [21] and Borceux [3] for notation and terminol* *ogy associated with the theory of enriched categories, and to Barr and Wells [1] for backgroun* *d on the theory of monads (otherwise known as triples). For more specific results, we cite [14]* * and [18]. Unless otherwise stated, we assume that all our categories are enriched in one of the * *topological senses 4 TARAS PANOV, NIGEL RAY, AND RAINER VOGT described below, and that functors are continuous. In many cases the morphism s* *ets are finite, and therefore invested with the discrete topology. Given an arbitrary category r, we refer to a covariant functor D :a! r as an * *a-diagram in r, for any small category a. Such diagrams are the objects of a category [a* *,r], whose morphisms are natural transformations of functors. We may interpret any object * *X of r as a constant diagram, which maps each object of a to X and every morphism to the * *identity. Examples 2.1. Let be the category whose objects are the ordinals (n) = {0, 1,* * . .,.n}, where n 0, and whose morphisms are the nondecreasing functions; then op- and -d* *iagrams are simplicial and cosimplicial objects of a respectively. In particular, : * * ! top is the cosimplicial space which assigns the standard n-simplex (n) to each object (n)* *. Its pointed analogue + is given by + (n) = (n)+ . If M is a topological monoid, then c(M)- and c(M)op-diagrams in top are left * *and right M-spaces respectively. We recall that (s, 2, ) is a symmetric monoidal category if the bifunctor 2:* * sx s! sis coherently associative and commutative, and is a coherent unit object. Such a* *n sis closed if there is a bifunctor sx sop! s, denoted by (Z, Y ) 7! [Y, Z], which satisfie* *s the adjunction s(X 2 Y, Z) ~=s(X, [Y, Z]) for all objects X, Y , and Z of s. A category r is s-enriched when its morphis* *m sets are identified with objects of s, and composition factors naturally through 2. A cl* *osed symmetric monoidal category is canonically self-enriched, by identifying s(X, Y ) with [X* *, Y ]. Henceforth, sdenotes such a category. Example 2.2. Any small s-enriched category a determines a diagram A: axa op! s,* * whose value at (a, b) is the morphism object a(b, a). An s-functor q ! r of s-enriched categories acts on morphism sets as a morphi* *sm of s. The category [q,r]of such functors has morphisms consisting of natural transfor* *mations, and is also s-enriched. The s-functors F :q ! r and U :r ! q are s-adjoint if there* * is a natural isomorphism r(F (X), Y ) ~=q(X, U(Y )) in s, for all objects X of q and Y of r. Examples 2.3. The categories top and top+ are symmetric monoidal under cartesia* *n prod- uct x and smash product ^ respectively, with unit objects the one-point space ** * and the zero- sphere *+ . Both are closed, and therefore self-enriched, by identifying [X, Y* * ] with Y X and (Y, *)(X,*)respectively. Since (Y, *)(X,*)inherits the subspace topology from Y X, the induced top-enr* *ichment of top+ is compatible with its self-enrichment. Both tmon and tgrp are top+ -enr* *iched by restriction. In certain situations it is helpful to reserve the notation t for either or b* *oth of the self- enriched categories top and top +. Similarly, we reserve tmg for either or b* *oth of the top+ -enriched categories tmon and tgrp . It is well known that top and top+ are complete and cocomplete, in the standa* *rd sense that every small diagram has a limit and colimit. Completeness is equivalent to* * the existence of products and equalizers, and cocompleteness to the existence of coproducts a* *nd coequal- izers. Both top and top+ actually admit indexed limits and indexed colimits [21* *], involving COLIMITS, STANLEY-REISNER ALGEBRAS, AND LOOP SPACES * * 5 topologically parametrized diagrams in the enriched setting; in other words, t * *is t-complete and t-cocomplete. A summary of the details for top can be found in [26]. Amongst indexed limits and colimits, the enriched analogues of products and c* *oproducts are particularly important. Definitions 2.4.An s-enriched category r is tensored and cotensored over s if t* *here exist bifunctors rx s! r and rx sop! r respectively, denoted by (X, Y ) 7-! X Y and (X, Y ) 7-! XY , together with natural isomorphisms (2.5) r(X Y, Z) ~=s(Y, r(X, Z)) ~=r(X, ZY ) * * __ in s, for all objects X, Z of r and Y of s. * * |__| For any such r, there are therefore natural isomorphisms (2.6) X ~=X ~=X and X (Y 2 W ) ~=(X Y ) W. Every sis tensored over itself by 2, and cotensored by [ , ]. Examples 2.7. The categories t are tensored and cotensored over themselves; so * *X Y and XY are given by X x Y and XY in top, and by X ^ Y and (Y, *)(X,*)in top+ . The r^ole of tensors and cotensors is clarified by the following results of K* *elly [21, (3.69)- (3.73)]. Here and henceforth, we take s to be complete and cocomplete in the s* *tandard sense. Theorem 2.8. An s-enriched category is s-complete if and only if it is complet* *e, and coten- sored over s; it is s-cocomplete if and only if it is cocomplete, and tensored * *over s. Theorem 2.8 asserts that standard limits and colimits may themselves be enric* *hed in the presence of tensors and cotensors, since they are special cases of indexed limi* *ts and colimits. Given an a-diagram D in r, where a is also s-enriched, we deduce that the natur* *al bijections (2.9) r(X, limD) ! [a,r](X, D) and r(colimD, Y ) ! [a,r](D, Y ) are isomorphisms in s, for any objects X and Y of r. It is convenient to formulate several properties of tmon and tgrp by observi* *ng that both categories are top +-complete and -cocomplete. We appeal to the monad associat* *ed with the forgetful functor U :tmg ! top +; in both cases it has a left top +-adjoin* *t, given by the free monoid or free group functor F . The composition U . F defines a top * *+-monad L: top+ ! top+ , whose category topL+of algebras is precisely tmg . We write V * *for the forgetful functor tgrp ! tmon , whose left top+ -adjoint is the universal group* * functor. Proposition 2.10.The categories tmon and tgrp are top +-complete and -cocompl* *ete; moreover, V preserves indexed colimits. Proof.We consider the forgetful functor topL+! top+ , noting that top+ is top+ * *-complete by Theorem 2.8. Part (i) of [14, VII, Proposition 2.10] asserts that the forgetful functor cr* *eates all indexed limits, confirming that tmg is top+-complete. Part (ii) asserts that topL+is to* *p+-cocomplete if L preserves reflexive coequalizers, which need only be verified for U becaus* *e F preserves colimits. The result follows for an arbitrary reflexive pair (f, g) in tmg by * *using the right 6 TARAS PANOV, NIGEL RAY, AND RAINER VOGT inverse to show that the coequalizer of (U(f), U(g)) in top+ is itself in the i* *mage of U, and lifts to the coequalizer of (f, g). Finally, V preserves indexed colimits because it is left adjoint to a top+ -f* *unctor tmon !_ tgrp, which associates to each topological monoid its subgroup of invertible el* *ements. |__| In view of Proposition 2.10 we may form colimits of diagrams in tmg by applyi* *ng colimtmon, even when the diagram consists entirely of topological groups. Pioneering resu* *lts on the completeness and cocompleteness of categories of topological monoids and topolo* *gical groups may be found in [6]. Our main deduction from Proposition 2.10 is that tmon and tgrp are tensored* * over top+ . By studying the isomorphisms (2.5), we may construct the tensors explici* *tly; they are described as pushouts in [30, 2.2]. Construction 2.11. For any objects M of tmon and Y of top+, the tensored monoid* * M ~Y is the quotient of the free topological monoid on U(M) ^ Y by the relations (m, y)(m0, y) = (mm0, y) for all m, m02 M and y 2 Y. For any object G of tgrp , the tensored group G ~ Y is the topological group V * *(G) ~ Y . The cotensored monoid MY and cotensored group GY are the function spaces top+* * (Y, M) __ and top+ (Y, G) respectively, under pointwise multiplication. * * |__| Given a category r which is tensored and cotensored over s, we may now descri* *be several categorical constructions. They are straightforward variations on [18, 2.3], an* *d initially involve three diagrams. The first is D :bop! r, the second E :b! s, and the third F :b * *! r. Definitions 2.12.The tensor product D b E is the coequalizer of a ff a D(b1) E(b0) ---!---! D(b) E(b) g:b0!b1 fi b in r, where g ranges over the morphisms of b, and ff|g = D(g) 1 and fi|g = 1 * * E(g). The homset Hom b(E, F ) is the equalizer of Y ff Y F (b)E(b) ---!---! F (b1)E(b0) b fi g:b0!b1 Q Q * * __ in r, where ff = g.E(g) and fi = gF (g). . * * |__| We may interpret the elements of Hom b(E, F ) as mappings from the diagram E * *to the diagram F , using the cotensor pairing. Examples 2.13. Consider the case r = s = top or top +, with b = . Given simpl* *icial spaces Xo: op! top and Yo: op! top+ , the tensor products |Xo| = Xo x and |Yo| = Yo ^ + represent their topological realisation [24] in top and top+ respectively. If w* *e choose r= tmg and s= top+ , a simplicial object Mo: op! tmg has internal and topological rea* *lisations |Mo|tmg = Mo ~ + and |Mo| = U(M)o ^ + in tmg and top+ respectively. Since | | preserves products, |Mo| actually lies * *in tmg . If r = s, then D b is colimD, where is the trivial b-diagram. Also, Hom * *b(E, F ) is the morphism set [b,r](E, F ), consisting of the natural transformations E ! F . COLIMITS, STANLEY-REISNER ALGEBRAS, AND LOOP SPACES * * 7 For Yo in Examples 2.13, its top- and top+ -realisations are homeomorphic bec* *ause base- points of the Yn represent degenerate simplices for n > 0. We identify |Mo|tmg * *with |Mo| in Section 7. We need certain generalisations of Definitions 2.12, in which analogies with * *homological algebra become apparent. We extend the first and second diagrams to D :ax bop! * *r and E :bx cop! s, and replace the third by F :c x dop! sor G: ax cop! r. Then D b E becomes an (a x cop)-diagram in r, and Hom cop(E, G) becomes an (a x bop)-diagr* *am in r. The extended diagrams reduce to the originals by judicious substitution, such a* *s a = c= id in D and E. Example 2.14. Consider the case r = s= top+ , with a = c = idand b = . Given * *E = + as before, and G a constant diagram Z :id! top+ , then Hom cop(E, G) coincid* *es with the total singular complex Sin(Z) as an object of [ op,top+.]If r = tmg and N :* *id! tmg is a constant diagram, then Sin(N) is an object of [ op,tmg]. Important properties of tensor products are described by the natural equivale* *nces (2.15) D b B ~=D and (D b E) c F ~=D b (E c F ) of (a x bop)- and (a x dop)-diagrams respectively, in r. The first equivalence * *applies Example 2.2 with a = b, and the second uses the isomorphism of (2.6). The adjoint rela* *tionship between and Hom is expressed by the equivalences (2.16) [axcop,r](D b E, G) ~=[bxcop,s](E, [a,r](D, G)) ~=[axbop,r](D, Hom cop(* *E, G)), which extend the tensor-cotensor relations (2.5), and are a consequence of the * *constructions. Examples 2.17. Consider the data of Example 2.14, and suppose that D is a simpl* *icial pointed space Yo: op! top+ . Then the adjoint relation (2.16)provides a homeom* *orphism top+ (|Yo|, Z) ~=[ op,top+(]Yo, Sin(Z)). If r = tmg and s= top+ , and Mo is a simplicial object in tmg , we obtain a hom* *eomorphism tmg(|Mo|tmg, N) ~=[ op,tmg](Mo, Sin(N)) for any object N of tmg . If r = sand E = , the relations (2.16)reduce to the second isomorphism (2.9). The first two examples extend the classic adjoint relationship between | | an* *d Sin. We now assume r = s = top. We let D be an (a x bop)-diagram as above, and de* *fine Bo(*, a, D) to be a degenerate form of the 2-sided bar construction. It is a b* *op-diagram of simplicial spaces, given as a bopx op-diagram in top by G (2.18) (b, (n)) 7-! D(b, a0) x an(a0, an) a0,an for each object b of b; the face and degeneracy maps are described as in [18] b* *y composition (or evaluation) of morphisms and by the insertion of identities respectively. T* *he topological realisation B(*, a, D) is a bop-diagram in top. This definitions ensure the exi* *stence of natural equivalences (2.19) Bo(*, a, D) xb E ~=Bo(*, a, D xb E) and B(*, a, D) xb E ~=B(*, a, D x* *b E) of cop-diagrams in [ op,top]and top respectively. 8 TARAS PANOV, NIGEL RAY, AND RAINER VOGT Examples 2.20. If b = id, the homotopy colimit [4] of a diagram D :a! top is gi* *ven by hocolimD = B(*, a, D), as explained in [18]; using (2.15)and (2.19), it is homeomorphic to both of B(*, a, A) xa D ~=D xaopB(*, a, A). In particular, Bo(*, a, *) is the nerve [32] Boa of a, whose realisation is the* * classifying space Ba of a. The natural projection hocolimD ! colimD is given by the map D xaopB(*, a, A) -! D xaop*, induced by collapsing B(*, a, A) onto *. If a = c(M), where M is an arbitrary topological monoid, then D is a left M-s* *pace and B(*, c(M), C(M)) is a universal contractible right M-space EM [13]. So hocolimD = B(*, c(M), C(M)) xc(M)D is a model for the Borel construction EM xM D. 3. Basic Constructions We choose a universal set V of vertices v1, . .,.vm , and let K denote a simp* *licial complex with faces oe V . The integer |oe| - 1 is the dimension of oe, and the greate* *st such integer is the dimension of K. For each 1 j m, the faces of dimension less than or * *equal to j form a subcomplex K(j), known as the j-skeleton of K; in particular, the 1-skel* *eton K(1)is a graph. We abuse notation by writing V for the zero-skeleton of K, more properly* * described as {{vj} : 1 j m}. At the other extreme we have the (m - 1)-simplex, which* * is the complex containing all subsets of V ; it is denoted by 2V in the abstract setti* *ng and by (V ) when emphasising its geometrical realisation. Any simplicial complex K therefor* *e lies in a chain (3.1) V -! K -! 2V of subcomplexes. Every face oe may also be interpreted as a subcomplex of K, an* *d so mas- querades as a (|oe| - 1)-simplex. A subset W V is a missing face of K if every proper subset lies in K, yet * *W itself does not; its dimension is |W | - 1. We refer to K as a flag complex, or write* * that K is flag, when every missing face has two vertices. The boundary of a planar m-gon * *is therefore flag whenever m 4, as is the barycentric subdivision K0 of an arbitrary compl* *ex K. The flagification Fl(K) of K is the minimal flag complex containing K as a subcompl* *ex, and is obtained from K by adjoining every missing face containing three or more vertic* *es. Example 3.2. For any n > 2, the simplest non-flag complex on n vertices is the * *boundary of an (n - 1)-simplex, denoted by @(n); then Fl(@(n)) is (n - 1) itself. Given a subcomplex K L on vertices V , it is useful to define W V as a mi* *ssing face of the pair (L, K) whenever W fails to lie in K, yet every proper subset lies i* *n L. Every finite simplicial complex K gives rise to a finite category cat(K), who* *se objects are the faces oe and morphisms the inclusions oe ø. The empty face ? is an initia* *l object. For any subcomplex K L, the category cat(K) is a full subcategory of cat(L); in p* *articular, (3.1)determines a chain of subcategories (3.3) id?(m) -! cat(K) -! cat(2V ). COLIMITS, STANLEY-REISNER ALGEBRAS, AND LOOP SPACES * * 9 For each face oe, we define the undercategory oe # cat(K) by restricting attent* *ion to those objects ø for which oe ø; thus oe is an initial object. Insisting that the i* *nclusion oe ø be strict yields the subcategory oe + cat(K), obtained by deleting oe. The ove* *rcategories cat(K)#oe and cat(K)+oe are defined likewise. A complex K also determines a simplicial set S(K), whose nondegenerate simpli* *ces are ex- actly the faces of K [24]. So the nerve Bocat(K) coincides with the simplicial * *set S(Con (K0)), where Con(K0) denotes the cone on the barycentric subdivision of K, and the con* *e point cor- responds to ?. More generally, B(oe #cat (K)) is the cone on B(oe +cat (K)). Examples 3.4. If K = V , then Bid?(m) is the cone on m disjoint points. If K = * *2V , then Bcat (2V ) is homeomorphic to the unit cube IV RV , and defines its canonical* * simplicial subdivision; the homeomorphism maps each vertex oe V to its characteristic fu* *nction Øff, and extends by linearity. If K is the subcomplex @(m), then Bcat (@(m)) is obta* *ined from the boundary @Im by deleting all faces which contain the maximal vertex (1, . * *.,.1). The undercategories define a cat(K)op-diagram # cat(K) in the category of sma* *ll cate- gories. It takes the value oe #cat (K) on each face oe, and the inclusion funct* *or ø #cat (K) oe # cat(K) on each reverse inclusion ø oe. The formation of classifying spa* *ces yields a cat(K)op-diagram B( # cat(K)) in top+ , which consists of cones and their inclu* *sions. It takes the value B(oe # cat(K)) on oe and B(ø # cat(K)) B(oe # cat(K) on ø o* *e, and its colimit is the final space Bcat (K). Following [18], we note the isomorphism (3.5) B( #cat (K)) ~=B(*, cat(K), CAT (K)) of cat(K)op-diagrams in top+ . We refer to the cones B(oe # cat(K)) as faces of Bcat (K), amongst which we d* *istinguish the facets B(v #cat (K)), defined by the vertices v. The facets determine the f* *aces, according to the expression " B(oe #cat (K)) = B(v #cat (K)) v2ff for each oe 2 K, and form a panel structure on Bcat (K) as described by Davis [* *11]. This terminology is motivated by our next example, which lies at the heart of recent* * developments in the theory of toric manifolds. Example 3.6. The boundary of a simplicial polytope P is a simplicial complex KP* * , with faces oe. The polar P *of P is a simple polytope of the same dimension, whose f* *aces Fffare dual to those of P (it is convenient to consider F? as P *itself). There is a h* *omeomorphism Bcat (KP ) ! P *, which maps each vertex oe to the barycentre of Fff, and trans* *forms each face B(oe #cat (K)) homeomorphically onto Fff. Classifying the categories and functors of (3.3)yields the chain of subspaces (3.7) Con (V ) -! Bcat (K) -! Im . So Bcat (K) contains the unit axes, and is a subcomplex of Im . It is therefore* * endowed with the induced cubical structure, as are all subspaces B(oe # cat(K)). In particul* *ar, the simple polytope P *of Example 3.6 admits a natural cubical decomposition. In our algebraic context, we utilise the category grp of discrete groups and * *homomor- phisms. Many constructions in grp may be obtained by restriction from those we * *describe in tmon , and we leave readers to provide the details. In particular, grp is a * *full subcategory of tmg , and is top+ -complete and -cocomplete. 10 TARAS PANOV, NIGEL RAY, AND RAINER VOGT Given a commutative ring Q (usually the integers, or their reduction mod 2), * *we consider the category Qmod of left Q-modules and Q-linear maps, which is symmetric mono* *idal with respect to the tensor product Q and closed under (Z, Y ) 7! Qmod (Y, Z). We us* *ually work in the related category gQ mod of connected graded modules of finite type, or more* * particularly in the categories gQ calg and gQ cocoa, which are dual; the former consists of * *augmented commutative Q-algebras and their homomorphisms, and the latter of supplemented * *cocom- mutative Q-coalgebras and their coalgebra maps. Q As an object of Qmod , the polynomial algebra Q[V ] on V has a basis of monom* *ials vW = W vj, for each multiset W on V . Henceforth, we assign a common dimension d(v* *j) > 0 to the vertices vj for all 1 j m, and interpret Q[V ] as an object of gQ calg;* * products are invested with appropriate signs if d(vj) is odd and 2Q 6= 0. Then the quotient * *map Q[V ] -! Q[V ]=(v~ : ~ =2K) is a morphism in gQ calg, whose target is known as the graded Stanley-Reisner Q* *-algebra of the simplicial complex K, and written SRQ (K). This ring is a fascinating in* *variant of K, and reflects many of its combinatorial and geometrical properties, as explained* * in [33]. Its Q-dual is a graded incidence coalgebra [20], which we denote by SRQ (K). We define a cat(K)op-diagram DK in top+ as follows. The value of DK on each f* *ace oe is the discrete space oe+ , obtained by adjoining + to the vertices, and the value* * on ø oe is the projection ø+ ! oe+ , which fixes the vertices of oe and maps the vertices of ø* * \ oe to +. Definition 3.8.Given objects (X, *) of top+ and M of tmg , the exponential diag* *rams XK and MK are the cotensor homsets Hom id(DK , X) and Hom id(DK , M) respectively;* * they are cat(K)-diagrams in top+ and tmg . Alternatively, they are the respective compos* *itions_of the exponentiation functors X( ):topop+! top+ and M( ):topop+! tmg with DopK. * * |__| So the value of XK on each face oe is the product space Xff, whose elements * *are functions f :oe ! X, and the value of XK on oe ø is the inclusion Xff Xfiobtained by e* *xtending f over ø by the constant map *. The space X? consists only of *. In the case of M* *K , each Mff is invested with pointwise multiplication, so HK takes values in grp for a dis* *crete group H. In gQ calg, we define a cat(K)op-diagram Q[K] by analogy. Its value on oe is * *the graded polynomial algebra Q[oe], and on ø oe is the projection Q[ø] ! Q[oe]. We deno* *te the dual cat(K)-diagram Hom id(Q[K], Q) by Q, and note that it lies in gQ cocoa . It* *s value on oe is the free Q-module Q generated by simplices z in S(oe), andPon o* *e ø is the corresponding inclusion of coalgebras. The coproduct is given by ffi(z) = z1 * * z2, where the sum ranges over all partitions of z into subsimplices z1 and z2. When Q = Z=2 we let the vertices have dimension 1. Every monomial vU therefor* *e has dimension |U| in the graded algebra Z=2[oe], and every j-simplex in S(oe) has d* *imension j+1 in Z=2. We refer to this as the real case. When Q = Z we consider two possi* *bilities. First is the complex case, in which the vertices have dimension 2, so that the additi* *ve generators of Z[oe] and Z have twice the dimension of their real counterparts. Second * *is the exterior case, in which the dimension of the vertices reverts to 1. Every squarefree mon* *omial vU then has dimension |U| in Z[oe], and anticommutativity ensures that every monomial c* *ontaining a square is zero; every j-face of oe has dimension j + 1 in Z, and every* * degenerate j-simplex z represents zero. To distinguish between the complex and exterior ca* *ses, we write Q as Z and ^ respectively. In the real and complex cases, Davis and Januszkiewicz [12] introduce homotop* *y types DJ R(K) and DJ C(K). The cohomology rings H*(DJ R(K); Z=2) and H*(DJ C(K); Z) * *are COLIMITS, STANLEY-REISNER ALGEBRAS, AND LOOP SPACES * * 11 isomorphic to the graded Stanley-Reisner algebras SR Z=2(K) and SR Z(K) respect* *ively. We shall deal with the exterior case below, and discuss alternative constructions * *for all three cases. We write DJ (K) as a generic symbol for Davis and Januszkiewicz's homoto* *py types, and refer to them as Davis-Januszkiewicz spaces for K. They are represented by * *objects in top. 4.Colimits In this section we introduce the colimits which form our main topic of discus* *sion, appealing to the completeness and cocompleteness of t and tmg as described in Section 2. * *We consider colimits of the diagrams XK , MK , GK , and Q in the appropriate categories,* * and label them colim+XK , colimtmgMK , colimtmgGK , and colimQ respectively. Similarl* *y, we write the limit of Q[K] as limQ[K]. As we shall see, these limits and colimits * *coincide with familiar constructions in several special cases. As an exercise in acclimatisation, we begin with the diagrams associated to (* *3.3). Expo- nentiating with respect to (X, *) and taking colimits provides the chain of sub* *spaces m` (4.1) Xj -! colim+XK - ! Xm , j=1 thereby sandwiching colim+XK between the axes and the cartesian power. On the* * other hand, using an object M of tmg yields the chain of epimorphisms m tmg K m (4.2) * Mj -! colim M - ! M , j=1 giving a presentation of colimtmgMK which lies between the m-fold free product* * of M and the cartesian power. The following example emphasises the influence of the underlying category on * *the formation of colimits, and is important later. Example 4.3. If K is the non-flag complex @(m) of Example 3.2 (where m > 2), th* *en colim+XK is the fat wedge subspace {(x1, . .,.xm ) : xj = * for some 1 j m* *}; on the other hand, colimtmgMK is isomorphic to Mm itself. By construction, colimtmgCK2in grp enjoys the presentation in K and is isomorphic to the right-angled Coxeter group Cox (K(1)) determined by th* *e 1-skeleton of K. Readers should not confuse K(1)with the more familiar Coxeter graph of th* *e group, which is almost its complement! Similarly, colimtmgCK has the presentation in K (where [bi, bj] denotes the commutator bibjb-1ib-1j), and so is isomorphic to t* *he right-angled Artin group Art(K(1)). Such groups are sometimes called graph groups, and are * *special examples of graph products [10]. As explained to us by Dave Benson, neither sh* *ould be confused with the graphs of groups described in [31]. 12 TARAS PANOV, NIGEL RAY, AND RAINER VOGT In the continuous case, we define the circulation group Cir(K(1)) as colimtmg* *T K in tmg . Every element of Cir(K(1)) may therefore be represented as a word (4.4) ti1(1) . .t.ik(k), where tij(j) lies in the ijth factor Tijfor each 1 j k. Two elements tr 2 T* *r and ts 2 Ts commute whenever {r, s} is an edge of K. We shall use G as a generic symbol for any one of the groups C2, C, or T . Following (4.2), we abbreviate the generating subgroups Gvj < colimGK to Gj,* * where 1 j m, and call them the vertex groups. Since colimtmgGK is presented as a * *quotient of the free product *mj=1Gj, its elements g may be assigned a wordlength l(g). In * *addition, the arguments of [8] apply to decompose every g from the right as Yn (4.5) g = sj(g) j=1 for some n l(g), where each subword sj(g) contains the maximum possible numbe* *r of mutually commuting letters, and is unique. Given any subset W V of vertices, we write KW for the complex obtained by * *restricting K to W . The following Lemma is a simple restatement of the basic properties of* * colimtmgGK . Lemma 4.6. We have that 1.the subgroup colimtmgGKW colimtmgGK is abelian if and only if K(1)Wis a * *complete graph, in which case it is isomorphic to GW ; 2.when K is flag, each subword sj(g) of (4.5)lies in a subgroup Gffjfor some * *face oej of K. Other algebraic examples of our colimits relate to the Stanley-Reisner algebr* *as and coal- gebras of K. By construction, there are algebra isomorphisms (4.7) limZ=2[K] ~=SRZ=2(K), limZ[K] ~=SRZ(K), and lim^[K] ~=SR^ (K), where the limits are taken in gZcalg . Dually, there are coalgebra isomorphisms (4.8)colimZ=2 ~=SRZ=2(K), colimZ ~=SRZ(K), and colim^ ~=SR^ (K) in gZcocoa . The analogues of 4.1 display these limits and colimits as mM mM (4.9) Q[vj] - limQ[K] - Q[V ] and DP Q(vj) -! colimQ -! DP Q(V ) j=1 j=1 respectively; here DP Q(W ) denotes the divided power Q-coalgebra of multisets * *on W V , graded by dimension. If we let (X, *) be one of the pairs (BC2, *), (BT, *), or (BC, *), then simp* *le argu- ments with cellular chain complexes show that the cohomology rings H*(colim+(BC* *2)K ; Z=2), H*(colim+(BT )K ; Z), and H*(colim+(BC)K ; Z) are isomorphic to the limits (4.7* *)respec- tively. Similarly, the homology coalgebras are isomorphic to the dual coalgebr* *as (4.8). In cohomology, these observations are due to Buchstaber and Panov [7] in the real * *and complex cases, and to Kim and Roush [22] in the exterior case (at least when K is 1-dim* *ensional). In homology, they may be made in the context of incidence coalgebras, following [2* *9]. In both cases, the maps of (4.1)induce the homomorphisms (4.9). COLIMITS, STANLEY-REISNER ALGEBRAS, AND LOOP SPACES * * 13 Such calculations do not themselves identify colim+(BC2)K and colim+(BT )K wi* *th Davis and Januszkiewicz's constructions. Nevertheless, Buchstaber and Panov provide * *homotopy equivalences colim+(BC2)K ' DJ R(K) and colim+(BT )K ' DJ C(K), which also foll* *ow from Corollary 5.3 below; the Lemma yields a corresponding equivalence in the exteri* *or case. Of course, colim+(BC)K is a subcomplex of the m-dimensional torus (S1)m , and is * *therefore finite. In due course, we shall use these remarks to interpret the following proposit* *ion in terms of Davis-Januszkiewicz spaces. The proof for G = C2 is implicit in [12], and for G* * = C is due to Kim and Roush [22]. Proposition 4.10.When G = C2 or C, there is a homotopy equivalence colim+(BG)K ' B colimtmgGK for any flag complex K. Since both cases are discrete, B colimtmgGK is, of course, an Eilenberg-Mac L* *ane space; Charney and Davis [9] have since identified good models for BA, given any Artin* * group A. Proposition 4.10 fails for arbitrary complexes K, as our next examples show. Examples 4.11. Proposition 4.10 applies when K = V , because the discrete compl* *ex is flag; then colimtmgGK isWisomorphic to the free product of m copies of G, whose class* *ifying space is the m-fold wedge mj=1BGj (by [6], for example). On the other hand, when K is t* *he non-flag complex @(m), Example 4.3 confirms that B colimtmgGK is BGm , whereas colim+(BG* *)K is the fat wedge subspace. These examples apply unchanged to the case G = T , and serve to motivate our * *extension of Proposition 4.10 to the complex case in Proposition 6.1 below. So far as C2* * and C are concerned, the Proposition asserts that certain homotopy homomorphisms (4.12) hK : colim+(BG)K -! colimtmgGK are homotopy equivalences when K is flag. We therefore view the hK as modelling* * the loop spaces; in the complex case, they express colim+(BT )K in terms of the circul* *ation groups colimtmgT K. In Section 7 we will use homotopy colimits to describe analogues o* *f hK for all complexes K. Our interest in the loop spaces colim+(BG)K has been stimulated by several* * ongoing programmes in combinatorial algebra. For example, Herzog, Reiner, and Welker [1* *7] discuss combinatorial issues associated with calculating the k-vector spaces TorSRk(K)(* *k, k) over an arbitrary ground field k, and refer to [16] for historical background. Such cal* *culations have applications to diagonal subspace arrangements, as explained by Peeva, Reiner a* *nd Welker [28]. Since these Tor spaces also represent the E2-term of the Eilenberg-Moore* * spectral sequence for H*( DJ (K); k), it seems well worth pursuing geometrical connectio* *ns. We consider the algebraic implications elsewhere [27]. 5.Fibrations and homotopy colimits In this section we apply the theory of homotopy colimits to study various rel* *evant fibrations and their geometrical interpretations. Some of the results appear in [7], but w* *e believe that our approach offers an attractive and efficient alternative, and eases generali* *sation. We refer to [18] and [36] for the notation and fundamental properties of homotopy colimi* *ts. Several 14 TARAS PANOV, NIGEL RAY, AND RAINER VOGT of the results we use are also summarised in [37], together with additional inf* *ormation on combinatorial applications. We begin with a general construction, based on a well-pointed topological gro* *up and a diagram H :a ! tmg of closed subgroups and their inclusions. We assume that* * the maps of the classifying diagram BH :a ! top + are cofibrations, and that the Pr* *ojection Lemma [37] applies to the natural projection hocolim+BH ! colim+BH, which is th* *erefore a homotopy equivalence. The cofibrations BH(a) ! B correspond to the canonical* * map fH :colim+ BH ! B under the homeomorphism (2.9). By Examples 2.1 the coset spaces =H(a) define an a x c( ) diagram =H in to* *p, and by Examples 2.20 the cofibration BH(a) ! B is equivalent to the fibration B *, c( ), C( ) xc( ) =H(a) -! Bc( ) for each object a of a. So fH is equivalent to hocolim+ B(*, c( ), C( ) xc( ) =H) -! B in the homotopy category of spaces over B , where the homotopy colimit is take* *n over a. Proposition 5.1.The homotopy fibre of fH is the homotopy colimit hocolim+ =H. Proof.We wish to identify the homotopy fibre of the projection B *, a, B(*, c( ), C( ) xc( ) =H) -! B . But we may rewrite the total space as B(*, a, =H) xc( )opB(*, c( ), C( )), a* *nd therefore as B(*, c( ), C( )) xc( )B(*, a, =H), using (2.19)and Examples 2.20. So the * *homotopy_ fibre is B(*, a, =H), as required. * * |__| Given a pair of simplicial complexes (L, K) on vertices V , we let a = cat(K)* *, and choose = colimtmgGL and H = GK ; we also abbreviate the diagram =H to L=K. Then fH * *is the induced map (5.2) fK,L: colim+(BG)K -! B colimtmgGL, and the Projection Lemma applies to (BG)K because the maps colim+(BG)K+ff! BGff* *are closed cofibrations for each face oe. So we have the following corollary to Pro* *position 5.1. Corollary 5.3.The homotopy fibre of fK,L is the homotopy colimit hocolimL=K, an* *d is homeomorphic to the identification space tmgL (5.4) Bcat (K) x colim G = ~, where (p, gh) ~ (p, g) whenever h 2 Gffand p lies in the face B(oe #cat (K)). Proof.By (3.5), the homotopy colimit B(*, cat(K), L=K) may be expressed as B( #cat (K)) xcat(K)L=K, * * __ and the inclusions B(oe #cat (K)) Bcat (K) induce a homeomorphism with (5.4).* * |__| For future use, we write ~ for the canonical action of colimtmgGL on B(*, cat(K* *), L=K). We note that fK,L coincides with the right-hand map of (4.1)when L = 2V and X* * = BG; the cases in which K = L (abbreviated to fK ) and L = Fl(K) also feature below.* * The space hocolim2V =K plays a significant r^ole in [12], where it is described as the id* *entification space of Corollary 5.3 and denoted by ZP (with P the dual of K, in the sense of Examp* *le 3.6). To emphasise this connection, we write hocolimL=K as ZG (K, L), which we abbrev* *iate to COLIMITS, STANLEY-REISNER ALGEBRAS, AND LOOP SPACES * * 15 ZG (K) when K = L. It appears repeatedly below, by virtue of Proposition 5.1. O* *ur examples assume that L = 2V , and continue the theme of Examples 4.11. W m Examples 5.5. If K = V then ZG (K, 2V ) is the homotopy fibre of j=1BGj ! BGm* * , the inclusion of the axes; it has been of interest to homotopy theorists for many y* *ears. If K is the non-flag complex @(m), then ZG (K, 2V ) is homotopy equivalent to Sm-1 for G = * *Z=2, and S2m-1 for G = T . The second of these examples may be understood by noting that the inclusion o* *f the fat wedge in BGm has the Thom complex of the external product im of Hopf bundles * *as its cofibre. Davis and Januszkiewicz [12] prove that the mod 2 cohomology ring of ECm2xCm2* *ZC2(K, 2V ) and the integral cohomology ring of ET mxTm ZT(K, 2V ) are isomorphic to the St* *anley Reisner algebras SR *Z=2(K) and SR *Z(K) respectively. In view of Corollary 5.3 (in the* * case L = 2V ), we regard the spaces colim+(BG)K and the Davis-Januszkiewicz homotopy types as* * inter- changeable from this point on. The canonical projection ZG (K, L) ! Bcat (K) is obtained by factoring out th* *e action ~ of colimtmgGL on hocolimL=K. The cubical structure (3.7)of the quotient lift* *s to an associated decomposition of ZG (K, L); when G = T and L = 2V , for example, we * *recover the description of [7] and [12] in terms of polydiscs and tori. The action ~ has other important properties. Proposition 5.6.The isotropy subgroups of ~ are the conjugates wGffw-1 < colimt* *mgGL, where oe ranges over the faces of K. Proof.It suffices to note from Corollary 5.3 that each point [x, wGff] is fixed* * by wGffw-1_< colimtmgGL, for any x 2 B(oe #cat (K)). * * |__| Corollary 5.7.The commutator subgroup of colimtmgGL acts freely on ZG (K, L) un* *der ~. Proof.The isotropy subgroups are abelian, and so have trivial intersection with* * the_commu- tator subgroup. * *|__| When K = L and G = C2, Proposition 5.6 strikes a familiar chord. The parabol* *ic sub- groups of a Coxeter group H are the conjugates w w-1 of certain subgroups , * *generated by subsets of the defining Coxeter system; when H is right-angled, and therefor* *e takes the form Cox(K(1)), such subgroups are abelian. When L = 2V , each subgroup wGffw-1* * reduces to Gff. In this case, Proposition 5.6 implies that the isotropy subgroups form * *an exponential catop(K)-diagram in tgrp, which assigns Gffto the face oe and the quotient homo* *morphism Gfi! Gffto the reverse inclusion ø oe. As detailed in [7], the homotopy fibre ZG (K, 2V ) is closely related to the * *theory of subspace arrangements and their auxiliary spaces. These spaces are defined in each of th* *e real, complex, and exterior cases, and will feature below; we introduce them here as homotopy * *colimits. Given a pointed space (Y, 0), we let Yx denote Y \ 0. For any subset W V , * *we write YW Y V for the coordinate subspace of functions f :V ! Y for which f(W ) = 0* *. The set of subspaces AY (K) = {YW : W =2K} 16 TARAS PANOV, NIGEL RAY, AND RAINER VOGT is the associated arrangement of K, whose complement UY (K) is given by the equ* *ivalent formulae S -1 (5.8) Y V\ W2=KYW = {f : f (0) 2 K}. The cat(K)-diagram Y (K) associates the function space Y (oe) = {f : f-1(0) o* *e} to each face oe, and the inclusion Y (oe) Y (ø) to each morphism oe ø. It follows * *that Y (oe) is homeomorphic to Y ffx (YxV \ff), and that UY (K) is colimY (K). The exponential cat(K)-diagram YxV \Kassociates YxV \ffto oe; when Y is contr* *actible, we may therefore follow Proposition 5.1 by combining the Projection Lemma and Homo* *topy Lemma of [37] to obtain a homotopy equivalence (5.9) hocolimYxV \K' UY (K). Now let us write F for one of the fields R or C. The study of the coordinate* * subspace arrangements AF(K), together with their complements, is a special case of a wel* *l-developed theory whose history is rich and colourful (see [2], for example). In the exte* *rior case, we replace F by the union of a countably infinite collection of 1-dimensional cone* *s in R2, which we call a 1-star and write as E. So EV is an m-star; it is homeomorphic to the* * union of countably many m-dimensional cones in (R2)V , obtained by taking products. As G ranges over C2, T and C, we let F denote R, C and E respectively. In all* * three cases, the natural inclusion of G into Fx is a cofibration, and Fx retracts onto its i* *mage. So (5.9) applies, and may be replaced by the corresponding equivalence (5.10) hocolimGV \K ' UF(K). Proposition 5.11.The space ZG (K, 2V ) is homotopy equivalent to UF(K), for any* * complex K. * * __ Proof.Substitute L = 2V in Corollary 5.3 and apply (5.10). * * |__| S By specialising certain results of [37] and [38], we may also describe W2=* *KFW \ 0 as a homotopy colimit. This space is dual to UF(K), and appears to have a more man* *ageable homotopy type in many relevant cases. For G = C2 and T , a version of Proposit* *ion 5.11 features prominently in [7]. The following examples illustrate Proposition 5.11, in the light of Examples * *5.5. Examples 5.12. For m > 2 and G = T , the subspace arrangements of the discrete * *complex V and the non-flag complex @(m) are given by {z : zj = zk = 0} : 1 j < k m and {0} respectively; the corresponding complements are {z : zj = 0 ) zk 6= 0} and Cm \ 0. The former is homotopy equivalent to a wedge of spheres, and the latter to S2m-* *1. 6. Flag complexes and connectivity In this section, we examine the homotopy fibre ZG (K, L) more closely. The re* *sults form the basis of our model for DJ (K) when K is flag, and enable us to measure the* * extent of its failure for general K. COLIMITS, STANLEY-REISNER ALGEBRAS, AND LOOP SPACES * * 17 We consider a flag complex K, and substitute K = L into Corollary 5.3 to dedu* *ce that ZG (K) is the homotopy fibre of the cofibration fK :DJ (K) ! B colimtmgGK . It * *is helpful to abbreviate B(oe #cat (K)) to B(oe) thoughout the following argument. Proposition 6.1.The cofibration fK is a homotopy equivalence whenever K is flag. Proof.We prove that ZG (K) is contractible. For any face oe 2 K, the space (colimtmgGK )=Gffinherits an increasing filtra* *tion by subspaces (colimtmgGK )i=Gff, consisting of those cosets wGfffor which a repres* *enting el- ement satisfies l(w) i. We may therefore define a cat (K)-diagram Ki=K, whic* *h as- signs (colimtmgGK )i=Gffto each face oe and the corresponding inclusion to each* * inclusion oe ø. By construction, ZG (K) is filtered by the subspaces hocolimKi=K and ea* *ch inclusion hocolimKi-1=K hocolimKi=K is a cofibration. We proceed by induction on i. For the base case i = 0, we observe that (colimtmgGK )0=Gffis the single poin* *t eGfffor all values of oe. Thus hocolimK0=K is homeomorphic to B(?), and is indeed contracti* *ble. To make the inductive step, we assume that hocolimKi=K is contractible for all i <* * n, and write Qn for the quotient space (hocolimKn=K)=(hocolimKn-1=K). It then suffices to pr* *ove that Qn is contractible. Every point of Qn has the form (x, wGff), for some x 2 B(oe) and some w of le* *ngth n. If the final letter of w lies in Gff, then (x, wGff) is the basepoint of Qn. Other* *wise, we rewrite w as w0s by (4.5), where s contains the maximum possible number of mutually com* *muting letters. These determine a subset Ø V , and Lemma 4.6 confirms that K(1)conta* *ins the complete graph on vertices Ø. Since K is flag, we deduce that 2ffl2 K, and ther* *efore that (x, w0Gffl) is the basepoint of Qn. To describe a contraction of Qn, we may fin* *d a canonical path p in cat? (K), starting at x and finishing at some x0 in B(Ø); of course p* * must vary continuously with (x, wGff), and lift to a corresponding path in Qn. If x is a * *vertex of B(oe), we choose p to run at constant speed along the edge from x to the cone point ?,* * and again from ? to the vertex Ø 2 B(Ø). If x is an interior point of B(oe), we extend th* *e construction __ by linearity. Then p lifts to the path through (p(t), w) for all 0 < t < 1, as * *required. |__| Proposition 6.1 leads to the study of fK,L:DJ (K) ! B colimtmgGL for any subc* *omplex K L. We consider the missing faces of K with three or more vertices and write* * c(K) 2 for their minimal dimension. We let d(K) denote c(K) - 1 when G = C2 or C, and * *2c(K) when G = T ; thus K is flag if and only if c(K) (and therefore d(K)) is infinit* *e. Finally, we define ( c(K) if L Fl(K) c(K, L) = 1 otherwise, and let d(K, L) be given by c(K, L) - 1 or 2c(K, L) as before. Theorem 6.2. For any subcomplex K L, the cofibration fK,L is a d(K, L)-equiv* *alence. Proof.We may factorise fK,L as DJ(K) -! DJ (F l(K)) -! DJ (Fl(L)) -! B colimtmgGFl(L). The first map is induced by flagification, and is a d(K)-equivalence by constru* *ction. The second is the identity if L Fl(K); otherwise, it is 0-connected when G = C2 o* *r C, and 2-_ connected when G = T . The third map is fFl(L), and an equivalence by Propositi* *on 6.1. |__| Theorem 6.2 suggests our first model for DJ (K). 18 TARAS PANOV, NIGEL RAY, AND RAINER VOGT Proposition 6.3.There is a homotopy homomorphism hK : DJ (K) ! colimtmgGK , whi* *ch is a (d(K) - 1)-equivalence for any complex K; in particular, it is an equivale* *nce if K is flag. Proof.Applying Theorem 6.2 with K = L implies that fK : DJ (K) ! B colimtmgGK is a (d(K) - 1)-equivalence. The result follows by composing with the canonical* * homotopy_ homomorphism BH ! H, which exists for any topological group H. * * |__| When L = 2V , the missing faces of (2V , K) are precisely the non-faces of K. I* *n this case only, we write their minimal dimension as c0(K). It is instructive to consider the homotopy commutative diagram ZG (K, L)- id--! ZG (K, L) - --! * ?? ? ? y p ?y ?y fK,2V (6.4) ZG (K, 2V )---! DJ (K) ----! BGm ?? ? ? y fl ?yfK,L ?yid B[G, L] - --! B colimtmgGL - Ba--!BGm of fibrations, where a is the abelianisation homomorphism and [G, L] denotes th* *e commutator subgroup of colimtmgGL. By Theorem 6.2, ZG (K, L) and ZG (K, 2V ) are (d(K, L) * *- 1)- and (d0(K) - 1)-connected respectively, where d(K, L) d0(K) by definition. In fac* *t ZG (K, 2V ) is d0(K)-connected, by considering the homotopy exact sequence of fK,2V. Corollary 5.7 confirms that (6.5) [G, L] -! ZG (K, L) -p!ZG (K, 2V ) is a principal [G, L]-bundle, classified by fl. This bundle encodes a wealth o* *f geometrical information on the pair (L, K). Its total space measures the failure of fK,L to* * be a homotopy equivalence, and its base space is the complement of the coordinate subspace ar* *rangement AF(K) by Corollary 5.11. Moreover, Theorem 6.2 implies that fl is also a d(K, L* *)-equivalence, and so sheds some light on the homotopy type of UF(K). Looping (6.4)gives a homotopy commutative diagram of fibrations ZG (K, L) --id-! ZG (K, L) ---! 1 ?? ? ? y p ?y ?y (6.6) fK,2V UF(K) --i-! DJ (K) -----! Gm ?? ? ? y fl ?y fK,L ?yid [G, L] ---! colimtmgGL --a-! Gm in tmonh , which offers an alternative perspective on DJ (K). Lemma 6.7. The loop space DJ (K) splits as Gm x UF(K) for any simplicial comp* *lex K; the splitting is not multiplicative. Proof.The vertex groups Gj embed in DJ (K) via homotopy homomorphisms, whose p* *rod- uct j :Gm ! DJ (K) is left inverse to fK,2V (but not a homotopy homomorphism)* *. The__ product of the maps i and j is the required homeomorphism. * * |__| COLIMITS, STANLEY-REISNER ALGEBRAS, AND LOOP SPACES * * 19 The following examples continue the theme of Examples 5.5 and 5.12. They refe* *r to the second horizontal fibration of the diagram (6.6), which is homotopy equivalent * *to the third whenever K = L is flag, by Proposition 6.1. The second examples also appeal to* * James's Theorem [19], which identifies the loop space Sn with the free monoid F +(Sn-1* *) for any n > 1. Examples 6.8. If K is the discrete flag complex V , then UF(K) is homotopy equ* *ivalent to the commutator subgroup of the free product *mj=1Gj. If K is the non-flag compl* *ex @(m), then UF(K) is homotopy equivalent to F +(Sm-2 ) for G = Z=2, and F +(S2m-2) for G =* * T ; the map i identifies the generators of each free monoid with higher Samelson produc* *ts (of order m) in DJ (K). Of course, both examples split topologically according to Lemma 6.7. The appe* *arance of higher products in DJ (@(m)) shows that commutators alone cannot model DJ (K)* * when K is not flag. More subtle structures are required, based on higher homotopy co* *mmutativity; they are related to Samelson and Whitehead products, as we explain elsewhere [2* *7]. 7. Homotopy colimits of topological monoids We now turn to the loop space DJ (K) for a general simplicial complex K, app* *ealing to the theory of homotopy colimits. Although the resulting models are necessar* *ily more complicated, they are homotopy equivalent to colimtmgGK when K is flag. The con* *structions depend fundamentally on the categorical ideas of Section 2, and apply to more g* *eneral spaces than DJ (K). We therefore work with an arbitrary diagram D :a ! tmg for most * *of the section, and write BD :a ! top+ for its classifying diagram. Our applications* * follow by substituting GK for D. We implement proposals of earlier authors (as in [36], for example) by formin* *g the homotopy colimit hocolimtmgD in tmg , rather than top+ . This is made possible by the ob* *servation of Section 2 that the categories tmg are t-cocomplete, and therefore have suffici* *ent structure for the creation of internal homotopy colimits. We confirm that hocolimtmgD is * *a model for the loop space hocolim+BD by proving that B commutes with homotopy colimits i* *n the relevant sense. As usual, we work in tmg , but find it convenient to describe c* *ertain details in terms of topological monoids; whenever these monoids are topological groups,* * so is the output. We recall the standard extension of the 2-sided bar construction to the based* * setting, with reference to (2.18). We write B+o(*, a, D) for the diagram bopx op! top+ given* * by ` (b, (n)) 7-! D(b, a0) ^ an(a0, an)+ , a0,an where D is a diagram axbop ! top+ . Following Examples 2.20, we define the hom* *otopy top+ -colimit as hocolim+D = B+ (*, a, D), and note the equivalent expressions B+ (*, a, A+ ) ^a D ~=D ^aopB+ (*, a, A+ ). For tmg , we proceed by categorical analogy. We replace the top-coproduct in * *(2.18)by its counterpart in tmg , and the internal cartesian product in top by the tenso* *red struc- ture of tmg over top+ . For any diagram D :a ! tmg , the simplicial topologica* *l monoid 20 TARAS PANOV, NIGEL RAY, AND RAINER VOGT Btmgo(*, a, D) is therefore given by (7.1) (n) 7-! a* D(a0) ~ an(a0, an)+ , 0,an where * denotes the free product of topological monoids. The face and degenerac* *y operators are defined as before, but are now homomorphisms. When a is of the form cat(K)* *, the n-simplices (7.1)may be rewritten as the finite free product Btmgn(*, cat(K), D) = * D(oe0), ffn ... ff0 where there is one factor for each n-chain of simplices in K. Definition 7.2.The homotopy tmg -colimit of D is given by hocolimtmgD = |Btmgo(*, a, D)|tmg * * __ in tmg , for any diagram D :a! tmg . * * |__| So hocolimtmgD is an object of tmg . Following Construction (2.11), it may be* * described in terms of generators and relations as a quotient monoid of the form i j . D * * E * Bn(*, a, D) ~ n+ din(b), s = b, ffiin(s) , sin(b), t = b, * *oein(t) , n 0 for all b 2 Bn(*, a, D), and all s 2 (n-1) and t 2 (n+1). Here ffiinand oeina* *re the standard face and degeneracy maps of geometric simplices. Example 7.3. Suppose that a is the category . ! . , with a single non-identity.* * Then an a- diagram is a homomorphism M ! N in tmg , and hocolimtmgD is its tmg mapping cyl* *inder. It may be identified with the tmg -pushout of the diagram M ~ (1)+ -j M -! N, where j(m) = (m, 0) in M ~ (1)+ for all m 2 M. An alternative expression for the simplicial topological monoid Btmgo(*, a, D* *) arises by analogy with the equivalences (2.19). Proposition 7.4.There is an isomorphism D ~aopB+o(*, a, A+ ) ~=Btmgo(*, a, D) o* *f simpli- cial topological monoids, for any diagram D :a! tmg . Proof.By (2.16), the functor D~aop :[aopx op,top+!][ op,tmg]is left top+ -adjo* *int to tmg (D, ), and therefore preserves coproducts. So we may write D ~aopB+o(*, a, A+~)=*D ~aop(a( , a)+ ^ ao(a, b)+ ) a,b ~=* D(a) ~ ao(a, b)+ a,b * * __ as required, using the isomorphism D ~aopa( , a) ~=D(a) of (2.15). * * |__| It is important to establish when the simplicial topological monoids Btmgo(*,* * a, D) are proper simplicial spaces, in the sense of [25], because we are interested in th* *e homotopy type of their realisations. This is achieved in Proposition 7.8, and leads on to the* * analogue of the Homotopy Lemma for tmg . These are two of the more memorable of the following s* *equence of six preliminaries, which precede the proof of our main result. On several o* *ccasions we insist that objects of tmg are well-pointed, and even that they have the homoto* *py type of a COLIMITS, STANLEY-REISNER ALGEBRAS, AND LOOP SPACES * * 21 CW-complex. Such conditions certainly hold for our exponential diagrams, and do* * not affect the applications. We consider families of monoids indexed by the elements s of an arbitrary set* * S. Lemma 7.5. Let fs:Ms ! Ns be a family of homomorphisms of well-pointed monoids* *, which are homotopy equivalences; then the coproduct homomorphism *sfs: *sMs -! *sNs is also a homotopy equivalence. Proof.Let f :M ! N denote the homomorphism in question, and write FkM for the s* *ubspace of M of elements representable by words of length k. Hence F0 = {e}, and Fk+1* *M is the pushout W W K WK?(M) ---! KPK?(M) (7.6) ?y ?y FkM --jk-! Fk+1M in top+ , where K runs through all (k + 1)-tuples (s1, . .,.sk+1) 2 Sk+1 such t* *hat si+16= si, and WK (M) PK (M) is the fat wedge subspace of Ms1x. .x.Msk+1. Each Msis well* *-pointed, so WK (M) PK (M) is a closed cofibration, and therefore so is jk. Since M = c* *olimkFkM in top+ , it remains to confirm that the restriction fk: FkM ! FkN is a homotopy e* *quivalence for all k. We proceed by induction, based on the trivial case k = 0. The map f induces a homotopy equivalence WK (M) ! WK (N) because Ms and Ns are well-pointed, and a further homotopy equivalence PK (M) ! PK (N) by constructio* *n. So the inductive hypothesis combines with Brown's Gluing Lemma [37, 2.4] to comple* *te_the proof. |* *__| Lemma 7.7. For any subset R S, the inclusion *rMr ! *sMs is a closed cofibra* *tion; in particular, *sMs is well-pointed. Proof.Let B ! M be the inclusion in question, with FkM as in the proof of Lemma* * 7.5 and Fk0M = B [ FkM. Then Fk0+1M is obtained from Fk0M by attaching spaces PK (M), w* *here K runs through all (s1, . .,.sk+1) in Sk+1 \ Rk+1 such that si+16= si. Thus B !* * Fk0M is_a_ cofibration for all k, implying the result. * * |__| Proposition 7.8.Given any small category a, and any diagram D :a! tmg of well-p* *ointed monoids, the simplicial space Btmgo(*, a, D) is proper, and its realisation Btm* *g(*, a, D) is well- pointed. Proof.By Lemma 7.7, each degeneracy map Btmgn(*, a, D) ! Btmgn+1(*, a, D) is a * *closed cofi- bration. The first result then follows from Lillig's Union Theorem [23] for cof* *ibrations. So Btmg0(*, a, D) Btmg(*, a, D) is a closed cofibration and Btmg0(*, a, D) is we* *ll-pointed,_yield- ing the second result. * * |__| As described in Examples 2.13, every simplicial object Mo in tmg has two poss* *ible reali- sations. We now confirm that they agree, and identify their classifying space. Lemma 7.9. The realisations |Mo|tmg and |Mo| are naturally isomorphic objects * *of tmg , whose classifying space is naturally homeomorphic to |B(Mo)|. 22 TARAS PANOV, NIGEL RAY, AND RAINER VOGT Proof.We apply the techniques of [14, VII x3] and [26, x4] to the functors | |t* *mg and the restriction of | | to [ op,tmg]. Both are left top+ -adjoint to Sin:tmg ! [ op* *,tmg], and so are naturally equivalent. The homeomorphism B|Mo| ~=|B(Mo)| arises by consider* *ing the __ bisimplicial object (k, n) 7! (Mn)k in top+ , and forming its realisation in ei* *ther order. |__| We may now establish our promised Homotopy Lemma. Proposition 7.10.Given diagrams D1, D2: a! tmg of well-pointed topological mono* *ids, and a map f :D1 ! D2 such that f(a): D1(a) ! D2(a) is a homotopy equivalence of* * under- lying spaces for each object a of a, the induced map hocolimtmgD1 -! hocolimtmgD2 is a homotopy equivalence. * * __ Proof.This follows directly from Lemmas 7.5 and 7.9, and Proposition 7.8. * * |__| We need one more technical result concerning homotopy limits of simplicial ob* *jects. We work with diagrams Xo: ax op! top+ of simplicial spaces, and Do: ax op! tmg * *of simplicial topological monoids. Proposition 7.11.With Xo and Do as above, there are natural isomorphisms hocolim+ |Xo| ~=| hocolim+Xo| and hocolimtmg|Do| ~=| hocolimtmgDo| in top+ and tmg respectively. Proof.The isomorphisms arise from realising the bisimplicial objects (k, n) 7-! B+k(*, a, Xn) and (k, n) 7-! Btmgk(*, a, Dn) * * __ in either order. In the case Do, we must also apply the first statement of Lemm* *a 7.9. |__| Parts of the proofs above may be rephrased using variants of the equivalences* * (2.15). They lead to our first general result, which states that the formation of classifyin* *g spaces commutes with homotopy colimits in an appropriate sense. Theorem 7.12. For any diagram D :a! tmg of well-pointed topological monoids wi* *th the homotopy types of CW-complexes, the map gD :hocolim+ BD -! B hocolimtmgD is a homotopy equivalence. Proof.For each object a of a, let Do(a) be the singular simplicial monoid of D(* *a). The natural map |Do(a)| ! D(a) is a homomorphism of well-pointed monoids and a homo* *topy equivalence, so it passes to a homotopy equivalence B|Do(a)| ! BD(a) under the * *formation of classifying spaces. By Proposition 7.10 and the corresponding Homotopy Lemma* * for top+, it therefore suffices to prove our result for diagrams of realisations of simpl* *icial monoids. So let Do: ax op! tmg be a diagram of discrete simplicial monoids. By Lemma* * 7.9 and Proposition 7.11, we must show that the canonical map | hocolim+BDo| -! |B hocolimtmgDo| is a homotopy equivalence. Since the simplicial spaces hocolim+BDo and B hocoli* *mtmgDo are proper, this reduces to proving that hocolim+BDn -! B hocolimtmgDn COLIMITS, STANLEY-REISNER ALGEBRAS, AND LOOP SPACES * * 23 is a homotopy equivalence in each dimension. But hocolim+BDn is the realisatio* *n of the proper simplicial space B+o(*, a, BDn), and B hocolimtmgDn is naturally homeomo* *rphic to the proper simplicial space B(Btmgo(*, a, Dn)) by Lemma 7.9. Moreover, for each* * k 0 the natural map B+k(*, a, BDn) ! B(Btmgk(*, a, Dn) coincides with the map ` (7.13) BDn(a0) -! B * Dn(a0) a0!...!ak a0!...!ak induced by the inclusion of each Dn(a0) into the free product. Since (7.13)is a* * homotopy __ equivalence by a theorem of Fiedorowicz [15, 4.1], the proof is complete. * * |__| Various steps in the proof of Theorem 7.12 may be adapted to verify the follo* *wing, which answers a natural question about tensored monoids. Proposition 7.14.For any well-pointed topological monoid M and based space Y , * *the nat- ural map BM ^ Y -! B(M Y ) is a homotopy equivalence when M and Y have the homotopy type of CW-complexes. Proof.As in Theorem 7.12, we need only work with the realisations |Mo| and |Yo|* * of the total singular complexes. Since B|Mo| ^ |Yo| ! B(|Mo| |Yo|) is the realisation of t* *he natural map BMn ^ Yn ! B(Mn Yn), it suffices to assume that Y is discrete; in this case, BM ^ Y -! B *yMy * * __ is a homotopy equivalence by the same result of Fiedorowicz [15]. * * |__| We apply Theorem 7.12 to construct our general model for DJ (K), but require* * a com- mutative diagram to clarify its relationship with the special case hK of Propos* *ition 6.3. We deal with aopx op-diagrams Xo in top +, and certain of their morphisms. These* * include ` :Xo ! top+ (BD, B(D ~aopXo)), defined for any Xo by `(x) = B(d 7! d ~ x), and* * the projection ß :B+o! (*+ )o, where B+oand (*+ )o denote B+o(*, a, A) and the triv* *ial diagram respectively. Under the homeomorphism [ op,top+ ](BD ^aopXo, B(D ~aopXo))~=[aopx op,top+X]o, top+(BD, B(D ~aopXo)) of (2.16), ` corresponds to a map OE: BD ^aopXo ! B(D ~aopXo) of simplicial spa* *ces. Proposition 7.15.For any diagram D :a! tmg , there is a commutative square hocolim+BD - gD--!B hocolimtmgD ?? ? yp+ ?yBptmg, colim+BD - fD--! B colimtmgD where p+ and ptmg are the natural projections. Proof.By construction, the diagram B+o - -`-! top+ (BD, B(D ~aopB+o)) ?? ? y i ?yB(1~i). (*+ )o--`-! top+ (BD, B(D ~aop(*+ )o)) 24 TARAS PANOV, NIGEL RAY, AND RAINER VOGT is commutative in [aopx op,top+,]and has adjoint BD ^aopB+o --ffi-!B(D ~aopB+o) ?? ? (7.16) y1^i ?yB(1~i) BD ^aop(*+ )o --ffi-!B(D ~aop(*+ )o) in [ op,top+.]By Proposition 7.4, the upper OE is the map B+o(*, a, BD) ! B (Bt* *mgo(*, a, D)) obtained by applying the relevant map (7.13)in each dimension. By Examples 2.13* *, the lower OE is given by the canonical map fD :colim+BD ! B colimtmgD in each dimension. * *Since realisation commutes with B, the topological realisation of (7.16)is the diagra* *m we seek; for Lemma 7.9 identifies the upper right-hand space with B hocolimtmgD, and Example* *s 2.20 __ confirms that the vertical maps are the natural projections. * * |__| Theorem 7.17. 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[37]Volkmar Welker, Günter M Ziegler, and Rade T ~Zivaljevi'c. Homotopy colimit* *s - comparison lemmas for combinatorial applications. Journal für die reine und angewandte Mathematik,* * 509:117-149, 1999. [38]Günter M Ziegler and Rade T ~Zivaljevi'c. Homotopy types of subspace arrang* *ements via diagrams of spaces. Mathematische Annalen, 295:527-548, 1993. Department of Mathematics and Mechanics, Moscow State University, 119899 Mosc* *ow, Russia and Institute for Theoretical and Experimental Physics, 117259 Moscow, Russia E-mail address: tpanov@mech.math.msu.su Department of Mathematics, University of Manchester, Manchester M13 9PL, Engl* *and E-mail address: nige@ma.man.ac.uk Fachbereich Mathematik/Informatik, Universität Osnabrück, D-49069 Osnabrück, * *Germany E-mail address: rainer@mathematik.uni-osnabrueck.de