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Progress in Mathematics, Vol. 215: Algebraic Topology: Categorical Decomposition Techniques, 261­291 c 2003 Birkh¨ er Verlag Basel/Switzerland aus

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 top ological categories. We relate their colimits to familiar structures in algebra, combinatorics, geometry and top ology. These include: right-angled Artin and Coxeter groups (and their complex analogues, which we call circulation groups); Stanley-Reisner algebras and coalgebras; Davis and Januszkiewicz's spaces DJ(K ) associated with toric manifolds and their generalisations; and coordinate subspace arrangements. When K is a flag complex, we extend well-known results on Artin and Coxeter groups by confirming that the relevant circulation group is homotopy equivalent to the space of loops DJ(K ). We define homotopy colimits for diagrams of top ological monoids and top ological 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 top ological groups is a model for DJ(K ) for an arbitrary complex K , and that the natural pro jection onto the original colimit is a homotopy equivalence when K is flag. In this case, the two models are compatible.

1. Intro duction
In this work we study diagrams asso ciated with a finite simplicial complex K , in various algebraic and topological categories. We are particularly interested in colimits and homotopy colimits of such diagrams. We are motivated by Davis and Januszkiewicz's investigation [12] of toric manifolds, in which K arises from the boundary of the quotient polytope. In the course of their cohomological computations, Davis and Januszkiewicz construct real and complex versions of a space whose cohomology ring is isomorphic to the Stanley-Reisner algebra (otherwise known as the face ring [33]) of K , over Z/2 and Z respectively. We denote spaces of this homotopy type by DJ (K ), and follow Buchstaber and Panov [7] by describing them as colimits of diagrams of classifying
Key words and phrases. Colimit, flag complex, homotopy colimit, loop space, right-angled Artin group, right-angled Coxeter group, Stanley-Reisner algebra, topological monoid. The first author was supported by a Royal Society/NATO Postdoctoral Fellowship at the University of Manchester, and by the Russian Foundation for Basic Research, grant 01-01-00546.


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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 and 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 complementary construction involving only the 1-skeleton K (1) of K . Whenever K is determined entirely by K (1) it is known as a flag complex, and results such as those of [12] and [22] may be interpreted as showing that the asso ciated Coxeter and Artin groups are homotopy equivalent to the lo op spaces DJ (K ), in the real and exterior cases respectively. In other words, the groups are discrete mo dels for the loop spaces. These observations raise the possibility of mo delling 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 programme. Before we begin, we must therefore confirm that our categories are sufficiently cocomplete for the proposed colimits to exist. We show that this is indeed the case (as predicted by folklore), and explain how the complex version of DJ (K ) is mo delled by the colimit of a diagram of tori whenever K is flag. We refer to the colimit as a circulation group , 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 examples of non-flag complexes for which the colimit groups cannot possibly mo del DJ (K ) in any 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. Considerable care has to be taken in formulating the construction for topological monoids, but the outcome clarifies the status of the original colimits when K is flag; flag complexes are precisely those for which the colimit and the homotopy colimit coincide. Our main result is therefore that DJ (K ) is mo delled by the homotopy colimit of the relevant diagram of topological groups, in all three cases and for arbitrary K . When K is flag, the natural pro jection onto the original colimit is a homotopy equivalence, and is compatible with the two mo del maps. Our pro of revolves around the fact that homotopy colimits commute with the classifying space functor, in a context which is considerably more general than is needed here. For particular complexes K , our constructions have interesting implications for traditional homotopy theoretic invariants such as Whitehead pro ducts, Samelson pro ducts, and their higher analogues and iterates. We hope to deal with these issues in subsequent work [27]. We now summarise the contents of each section. It is particularly convenient to use the language of enriched category theory, so we devote Section 2 to establishing the notation, conventions and results that we need. These include a brief discussion of simplicial ob jects and their realisations,


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and verification of the co completeness of our category of topological monoids in the enriched setting. Readers who are familiar with this material, or willing to refer back to Section 2 as necessary, may pro ceed directly to Section 3, where we intro duce the relevant categories and diagrams asso ciated with a simplicial complex K . They include algebraic and topological examples, amongst which are the exponential diagrams GK ; here G denotes one of the cyclic groups C2 or 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 rightangled Coxeter and Artin groups, or circulation groups respectively. In Section 5 we study aspects of the diagrams involving asso ciated fibrations and homotopy colimits. We note connections with co ordinate subspace arrangements. We intro duce the mo del map fK : colimtmg GK DJ (K ) in Section 6, and determine 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 classifying spaces is homotopy equivalent to the classifying space of the homotopy colimit of D, taken in the category of topological monoids. By application to the exponential diagrams GK , we deduce that our generalised mo del map hK : ho colimtmg GK DJ (K ) is a homotopy equivalence for all complexes K . We note that the two mo dels are compatible, and homotopy equivalent, 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 endowed with the corresponding k -topology. Many of the spaces we consider have a distinguished basepoint , and we write top+ for the category of pairs (X, ) and basepoint preserving maps; the forgetful functor top+ top is faithful. For any ob ject 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 wel l pointed, in the sense that the inclusion of the basepoint is a closed cofibration, and we emphasise this requirement as it arises. Several other useful categories are related to top+ . These include tmonh, consisting of topological monoids and homotopy homomorphisms [5] (essentially equivalent to Sugawara's strongly homotopy multiplicative maps [34]), and its subcategory tmon, in which the homorphisms are strict. Again, the forgetful functor tmon top+ is faithful. Limiting the ob jects 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 sometimes have to insist that ob jects are well pointed. The Mo ore lo op space X is a typical ob ject in tmonh for any pair (X, ), and the canonical inclusion M BM is a homotopy homomorphism for any well-pointed topological monoid M .


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For each m 0 we consider the small categories id(m), which consist of m objects and their identity morphisms; in particular, we use the based versions id (m), which result from adjoining an initial ob ject . Given a topological monoid M , the asso ciated topological category c(M ) consists of one ob ject, and one morphism for each element of M . Segal's [32] classifying space B c(M ) then coincides with the standard classifying space BM . Given ob jects X0 and Xn of any category c,wedenotethe set of n-composable morphisms f1 f2 fn X0 - X1 - ··· - Xn by cn (X0 ,Xn ), for all n 0. Thus c1 (X, Y ) is the morphism set c(X, Y ) for all ob jects 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 topological group of unimo dular complex numbers, and S 1 for the circle. Similarly, we discriminate between the cyclic group C2 and the ring of residue classes Z/2, and between the infinite cyclic group C and the ring of integers Z. We reserve the symbol G exclusively for one of the groups C2 , C , or T , in contrast to an arbitrary topological group . 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 during June 2001. They are particularly grateful to the organisers for providing the opportunity to work in such magnificent surroundings.

2. Categorical prerequisites
We refer to the books of Kelly [21] and Borceux [3] for notation and terminology asso ciated with the theory of enriched categories, and to Barr and Wells [1] for background on the theory of monads (otherwise known as triples). For more specific results, we cite [14] and [18]. Unless otherwise stated, we assume that all our categories are enriched in one of the topological senses described below, and that functors are continuous. In many cases the morphism sets 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 ob jects of a category [a,r] , whose morphisms are natural transformations of functors. We may interpret any ob ject X of r as a constant diagram, which maps each ob ject of a to X and every morphism to the identity. Examples 2.1. Let be the category whose objects are the sets (n) = {0, 1,... ,n}, where n 0, and whose morphisms are the nondecreasing functions; then op - and -diagrams are simplicial and cosimplicial objects of r 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.


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We recall that (s, ,) is a symmetric monoidal category if the bifunctor : s â s s is coherently asso ciative and commutative, and is a coherent unit ob ject. Such an s is closed if there is a bifunctor s â sop s, denoted by (Z, Y ) [Y, Z ], which satisfies the adjunction s(X Y, Z ) s(X, [Y, Z ]) = for all ob jects X , Y , and Z of s. A category r is s-enriched when its morphism sets are identified with ob jects of s, and composition factors naturally through . A closed symmetric monoidal category is canonically self-enriched, by identifying s(X, Y ) with [X, Y ]. Henceforth, s denotes such a category. Example 2.2. A smal l s-enriched category a determines a diagram A : a â a whose value at (a, b) is the morphism object a(b, a). An s-functor q r of morphism of s. The category natural transformations, and U : r q are s-adjoint if ther
[ op

s,

s-enriched categories acts on morphism sets as a q,r] of such functors has morphisms consisting of is also s-enriched. The s-functors F : q r and e is a natural isomorphism r(F (X ),Y ) = q(X, U (Y ))

in s, for all ob jects X of q and Y of r. Examples 2.3. The categories top and top+ are symmetric monoidal under cartesian product â 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 topenrichment of top+ is compatible with its self-enrichment. Both tmon and tgrp are top+ -enriched by restriction. In certain situations it is helpful to reserve the notation t for either or both of the self-enriched categories top and top+ . Similarly, we reserve tmg for either or both of the top+ -enriched categories tmon and tgrp. It is well known that top and top+ are complete and co complete, in the standard sense that every small diagram has a limit and colimit. Completeness is equivalent to the existence of products and equalizers, and co completeness to the existence of copro ducts and co equalizers. Both top and top+ actually admit indexed limits and indexed colimits [21], involving 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 pro ducts and coproducts are particularly important. Definitions 2.4. An s-enriched category r is tensored and cotensored over s if there exist bifunctors r â s r and r â sop r respectively, denoted by (X, Y ) - X Y and (X, Y ) - X Y ,


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together with natural isomorphisms (2.5) r(X Y, Z ) s(Y, r(X, Z )) = r(X, Z Y ) = in s, for all ob jects 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 W ) (X Y ) W. = Every s is tensored over itself by , and cotensored by [ , ]. Examples 2.7. The categories t are tensored and cotensored over themselves; so X Y and X Y are given by X â Y and X Y in top, and by X Y and (Y, )(X,) in top+ . The role of tensors and cotensors is clarified by the following results of Kelly ^ [21, (3.69)­(3.73)]. Theorem 2.8. An s-enriched category is s-complete if and only if it is complete, and cotensored 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 enriched in the presence of tensors and cotensors, since they are special cases of indexed limits and colimits. Given an a-diagram D in r, where a is also s-enriched, we deduce that the natural bijections (2.9) r(X, lim D) [a,r] (X, D) and r(colim D, Y ) [a,r] (D, Y ) are isomorphisms in s, for any ob jects X and Y of r. Henceforth, we assume that s is complete and co complete in the standard sense. It is convenient to formulate several properties of tmon and tgrp by observing that both categories are top+ -complete and -co complete. We appeal to the monad asso ciated with the forgetful functor U : tmg top+ ; in both cases it has a left top+ -adjoint, 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. Proposition 2.10. The categories tmon and tgrp are top+ -complete and top+ cocomplete. 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 creates all indexed limits, confirming that tmg is top+ -complete. Part (ii) (whose origins lie in work of Hopkins) asserts that topL is top+ -co complete if L preserves reflex+ ive co equalizers, which need only be verified for U because F preserves colimits. The result follows for an arbitrary reflexive pair (f, g ) in tmg by using the right inverse to show that the co equalizer of (U (f ),U (g )) in top+ is itself in the image of U , and lifts to the coequalizer of (f, g ).


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Pioneering results on the completeness and co completeness of categories of topological monoids and topological groups may be found in [6]. Our main deduction from Proposition 2.10 is that tmon and tgrp are tensored over top+ . By studying the isomorphisms (2.5), we may construct the tensors explicitly; they are described as pushouts in [30, 2.2]. Construction 2.11. For any ob jects 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 )(m ,y ) = (mm ,y ) for all m, m M and y Y. For any ob ject of tgrp, V ( ) Y , where V denotes The cotensored monoid top+ (Y, M ) and top+ (Y, the tensored group Y is the forgetful functor tgrp M Y and cotensored group Y ) respectively, under pointwise the topological group tmon. are the function spaces multiplication.

Lemma 2.12. The forgetful functor V : tgrp tmon preserves indexed limits and colimits. Proof. Since V is right top+ -adjoint to the universal group functor, indexed limits. Construction 2.11 confirms that V preserves tensors, only show that it preserves coequalizers, by the results of [21]. But the of topological groups 1 - 2 in tmon is also a group; and inversion is - being induced by the continuous isomorphism -1 on 2 . it preserves so we need co equalizer continuous,

The constructions of Section 7 involve indexed colimits in tmg, and Lemma 2.12 ensures that these may be formed in tmon, even when working exclusively with topological groups. Given a category r which is tensored and cotensored over s, we may now describe several categorical constructions. They are straightforward variations on [18, 2.3], and initially involve three diagrams. The first is D : bop r, the second E : b s, and the third F : b r. Definitions 2.13. The tensor product D b E is the co equalizer of
g:b0 b1

- D(b1 ) E (b0 ) - - - - -



D(b) E (b)
b

in r, where g ranges over the morphisms of b, and |g = D(g ) 1 and |g = 1 E (g ). The homset Homb (E, F ) is the equalizer of F (b)E
b (b)

- - - - - -
g



F (b1 )E
g:b0 b1

(b0 )

in r, where =

g

·E (g ) and =

F (g )· .

We may interpret the elements of Homb (E, F ) as mappings from the diagram E to the diagram F , using the cotensor pairing.


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Examples 2.14. Consider the case r = s = top or top+ , with b = . Given simplicial spaces X· : op top and Y· : op top+ , the tensor products |X· | = X· â and |Y· | = Y· +

represent their topological realisation [24] in top and top+ respectively. If we choose r = tmg and s = top+ , a simplicial object M· : op tmg has internal and topological realisations |M· |tmg = M
·

+

and

|M· | = U (M )· +

in tmg and top+ respectively. Since | | preserves products, |M· | actual ly lies in tmg. If r = s, then D b is colim D, where is the trivial b-diagram. Also, Homb (E, F ) is the morphism set [b,r] (E, F ), consisting of the natural transformations E F . For Y· in Examples 2.14, its top- and top+ -realisations are homeomorphic because basepoints of the Yn represent degenerate simplices for n > 0. We identify |M· |tmg with |M· | in Section 7. We need certain generalisations of Definitions 2.13, in which analogies with homological algebra become apparent. We extend the first and second diagrams to D : a â bop r and E : b â cop s, and replace the third by F : c â dop s or G : a â cop r. Then D b E becomes an (a â cop )-diagram in r, and Homcop (E, G) becomes an (a â bop )-diagram in r. The extended diagrams reduce to the originals by judicious substitution, such as a = c = id in D and E . Example 2.15. Consider the case r = s = top+ , with a = c . Given E = + as before, and G a constant diagram Z : id Homcop (E, G) coincides with the total singular complex Sin (Z ) [op ,top+ ] . If r = tmg and N : id tmg is a constant diagram, an object of [op ,tmg] .


= id and b = top+ , then as an object of then Sin (N ) is

Important properties of tensor pro ducts are described by the natural equivalences (2.16) D b B D = and (D b E ) c F D b (E c F ) =

of (a â bop )- and (a â 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 relationship between and Hom is expressed by the equivalences
[

(2.17)

aâcop ,r] (D b E, G) [bâcop ,s] (E, [a,r] (D, G)) = [aâbop ,r] (D, Homcop (E, G)), =

which extend the tensor-cotensor relations (2.5), and are a consequence of the constructions.


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Examples 2.18. Consider the data of Example 2.15, and suppose that D is a simplicial pointed space Y· : op top+ . Then the adjoint relation (2.17) provides a homeomorphism top+ (|Y· |,Z ) = [op ,top+ ] (Y· , Sin (Z )). If r = tmg and s = top+ , and M· is a simplicial object in tmg, we obtain a homeomorphism tmg(|M· |tmg ,N ) = [op ,tmg] (M· , Sin (N )) for any object N of tmg. If r = s and E = , the relations (2.17) reduce to the second isomorphism (2.9). The first two examples extend the classic adjoint relationship between | | and Sin . We now assume r = s = top. We let D be an (a â bop )-diagram as above, and define B· (, a,D) to be a degenerate form of the 2-sided bar construction. It is a bop -diagram of simplicial spaces, given as a bop â op -diagram in top by (2.19) (b, (n)) -
a0 ,an

D(a0 ,b) â an (a0 ,an )

for each ob ject b of b; the face and degeneracy maps are described as in [18] by composition (or evaluation) of morphisms and by the insertion of identities respectively. The topological realisation B (, a,D) is a bop -diagram in top. These definitions ensure the existence of natural equivalences B· (, a,D) âb E B· (, a,D âb E ) = (2.20) and B (, a,D) âb E B (, a,D âb E ) = of cop -diagrams in [op ,top] and top respectively. Examples 2.21. If b = id, the homotopy colimit [4] of a diagram D : a top is given by ho colim D = B (, a,D), as explained in [18]; using (2.16) and (2.20), it is homeomorphic to both of B (, a,A) âa D D âaop B (, a,A). = In particular, B· (, a, ) is the nerve [32] B· a of a, whose realisation is the classifying space B a of a. The natural projection ho colim D colim D is given by the map D âaop B (, a,A) - D âaop , induced by col lapsing B (, a,A) onto . If a = c(M ), where M is an arbitrary topological monoid, then D is a left M space and B (, c(M ),C (M )) is a universal contractible right M -space EM [13]. So ho colim D = B (, c(M ),C (M )) âc(M ) D is a model for the Borel construction EM âM D.


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3. Basic constructions
We cho ose a universal set V of vertices v1 ,. . . ,vm , and let K denote a simplicial complex with faces V . The integer | | - 1 is the dimension of , and the greatest 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-skeleton 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 al l subsets of V ; it is denoted by 2V in the abstract setting and by (V ) when emphasising its geometrical realisation. Any simplicial complex K therefore lies in a chain (3.1) V - K - 2
V

of subcomplexes. Every face may also be interpreted as a subcomplex of K , and so masquerades as a (| |- 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 K of an arbitrary complex K . The flagification Fl (K ) of K is the minimal flag complex containing K as a subcomplex, and is obtained from K by adjoining every missing face containing three or more vertices. 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 missing face of the pair (L, K ) whenever W fails to lie in K , yet every proper subset lies in L. Every finite simplicial complex K gives rise to a finite category cat(K ), whose ob jects are the faces and morphisms the inclusions . The empty face is an initial ob ject. For any subcomplex K L, the category cat(K ) is a full subcategory of cat(L); in particular, (3.1) determines a chain of subcategories (3.3) id (m) - cat(K ) - cat(2V ). For each face , we define the undercategory cat(K ) by restricting attention to those ob jects for which ; thus is an initial ob ject. Insisting that the inclusion be strict yields the subcategory cat(K ), obtained by deleting . The overcategories cat(K ) and cat(K ) are defined likewise, and may be rewritten as cat( ) and cat( ( )) respectively. A complex K also determines a simplicial set S (K ), whose nondegenerate simplices are exactly the faces of K [24]. So the nerve B· cat(K ) coincides with the simplicial set S (Con (K )), where Con (K ) denotes the cone on the barycentric subdivision of K , and the cone point corresponds to . More generally, B ( cat(K )) is the cone on B ( cat(K )).


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Examples 3.4. If K = V , then B id (m) is the cone on m disjoint points. If K = 2V , then B cat(2V ) is homeomorphic to the unit cube I V RV , and defines its canonical simplicial subdivision; the homeomorphism maps each vertex V to its characteristic function , and extends by linearity. If K is the subcomplex (m), then B cat( (m)) is obtained from the boundary I m by deleting al l faces which contain the maximal vertex (1,..., 1). The undercategories define a cat(K )op -diagram cat(K ) in the category of small categories. It takes the value cat(K ) on each face , and the inclusion functor cat(K ) cat(K ) on each reverse inclusion . The formation of classifying spaces yields a cat(K )op -diagram B ( cat(K )) in top+ , which consists of cones and their inclusions. It takes the value B ( cat(K )) on and B ( cat(K )) B ( cat(K ) on , and its colimit is the final space B cat(K ). Following [18], we note the isomorphism (3.5) B ( cat(K )) B (, cat(K ), CAT(K )) =

of cat(K )op -diagrams in top+ (using the notation of Example 2.2). We refer to the cones B ( cat(K )) as faces of B cat(K ), amongst which we distinguish the facets B (v cat(K )), defined by the vertices v . The facets determine the faces, according to the expression B ( cat(K )) =
v

B (v cat(K ))

for each K , and form a panel structure on B cat(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 . The polar P of P is a simple polytope of the same dimension, whose faces F are dual to those of P (it is convenient to consider F as P itself ). There is a homeomorphism B cat(KP ) P , which maps each vertex to the barycentre of F , and transforms each face B ( cat(K )) PL-homeomorphical ly onto F . Classifying the categories and functors of (3.3) yields the chain of subspaces (3.7) Con (V ) - B cat(K ) - I m .

So B cat(K ) contains the unit intervals along the co ordinate axes, and is a subcomplex of I m . It is therefore endowed with the induced cubical structure, as are all subspaces B ( cat(K )). In particular, 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 homomorphisms. 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 -co complete.


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Given a commutative ring Q (usually the integers, or their reduction mo d 2), we consider the category Q mod of left Q-mo dules and Q-linear maps, which is symmetric monoidal with respect to the tensor pro duct Q and closed under (Z, Y ) Q mod(Y, Z ). We usually 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 cocommutative Q-coalgebras and their coalgebra maps. As an ob ject of Q mod, the polynomial algebra Q[V ] on V has a basis of monomials vW = W vj , for each multiset W on V . Henceforth, we assign a common dimension d(vj ) > 0 to the vertices vj for all 1 j m, and interpret Q[V ] as an ob ject of gQ calg; products are invested with appropriate signs if d(vj ) is o dd and 2Q = 0. Then the quotient map Q[V ] - Q[V ]/(v : V and K ) / is a morphism in gQ calg, whose target is known as the graded Stanley-Reisner Qalgebra of the simplicial complex K , and written SR Q (K ). This ring is a fascinating invariant 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 face is the discrete space + , obtained by adjoining + to the vertices, and the value on is the pro jection + + , which fixes the vertices of and maps the vertices of \ to +. Definition 3.8. Given ob jects (X, ) of top+ and M of tmg, the exponential diagrams X K and M K are the cotensor homsets Homid (DK ,X )and Homid (DK ,M ) respectively; they are cat(K )-diagrams in top+ and tmg. Alternatively, they are the respective compositions of the exponentiation functors X ( ) : topop top+ + op and M ( ) : topop tmg with DK . + So the value of X K on each face is the pro duct space X , whose elements are functions f : X , and the value of X K on is the inclusion X X obtained by extending f over by the constant map . The space X consists only of . In the case of M K , each M is invested with pointwise multiplication, so H K takes values in grp for a discrete group H . In gQ calg, we define a cat(K )op -diagram Q[K ] by analogy. Its value on is the graded polynomial algebra Q[ ], and on is the pro jection Q[ ] Q[ ]. We denote the dual cat(K )-diagram Homid (Q[K ],Q) by Q K , and note that it lies in gQ cocoa. Its value on is the free Q-mo dule Q S ( ) generated by simplices z in the simplicial set S ( ), and on is the corresponding inclusion of coalgebras. The copro duct is given by (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 therefore has dimension |U | in the graded algebra Z/2[ ], and every j -simplex in


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S ( ) has dimension j + 1 in Z/2 S ( ) . We refer to this as the real case. When Q = Z we consider two possibilities. First is the complex case, in which the vertices have dimension 2, so that the additive generators of Z[ ] and Z S ( ) 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 monomial vU then has dimension |U | in Z[ ], and anticommutativity ensures that every monomial containing a square is zero; every j -face of has dimension j +1 in Z S ( ) , and every degenerate j -simplex z represents zero. To distinguish between the complex and exterior cases, we write Q as Z and respectively. In the real and complex cases, Davis and Januszkiewicz [12] intro duce homotopy types DJ R (K ) and DJ C (K ). The cohomology rings H (DJ R (K ); Z/2) and H (DJ C (K ); Z) are isomorphic to the graded Stanley-Reisner algebras SR Z/2 (K ) and SRZ (K ) respectively. 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 homotopy types, and refer to them as DavisJanuszkiewicz spaces for K . They are represented by ob jects in top.

4. Colimits
In this section we intro duce the colimits which form our main topic of discussion, appealing to the completeness and co completeness of t and tmg as described in Section 2. We consider colimits of the diagrams X K , M K , GK , and Q K in the appropriate categories, and label them colim+ X K , colimtmg M K , colimtmg GK , and colim Q K respectively. Similarly, we write the limit of Q[K ] as lim Q[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 asso ciated to (3.3). Exponentiating with respect to (X, ) and taking colimits provides the chain of subspaces
m

(4.1)
j =1

Xj - colim+ X

K

- X m ,

thereby sandwiching colim+ X K between the axes and the cartesian power. On the other hand, using an ob ject M of tmg yields the chain of epimorphisms (4.2)
m j =1

Mj - colimtmg M K - M m ,

giving a presentation of colimtmg M K which lies between the m-fold free pro duct 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.


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Example 4.3. If K is the non-flag complex (m) of Example 3.2 (where m > 2), then colim+ X K is the fat wedge subspace {(x1 ,...,xm ) : some xj = }; on the other hand, colimtmg M K is isomorphic to M m itself.
K By construction, colimtmg C2 in grp enjoys the presentation

a1 ,... ,am : a2 = 1, (ai aj )2 = 1 for all {vi ,vj } in K j and is isomorphic to the right-angled 1-skeleton of K . Readers should not graph of the group, which is almost Similarly, colimtmg C K has the b1 ,...,b
m

Coxeter group Cox (K (1) ) determined by the confuse K (1) with the more familiar Coxeter its complement! presentation

: [bi ,bj ] = 1 for all {vi ,vj } in K

(where [bi ,bj ] denotes the commutator bi bj b-1 b-1 ), and so is isomorphic to the i j right-angled Artin group Art (K (1) ). Such groups are sometimes called graph groups, and are special examples of graph products [10]. As explained to us by Dave Benson, neither should be confused with the graphs of groups described in [31]. In the continuous case, we refer to colimtmg T K as the circulation group Cir (K (1) ) in tmg. Every element of Cir (K (1) ) may therefore be represented as a word (4.4) ti1 (1) ··· tik (k ), where tij (j ) lies in the ij th factor Tij for each 1 j k . Two elements tr Tr and ts Ts commute whenever {r, s} is an edge of K . Following (4.2), we abbreviate the generating subgroups Gvj < colim GK to Gj , where 1 j m, and call them the vertex groups. Since colimtmg GK is presented as a quotient of the free product m Gj , its elements g may be assigned j =1 a wordlength l(g ). In addition, the arguments of [8] apply to decompose every g from the right as
n

(4.5)

g=
j =1

sj (g )

for some n l(g ), where each subword sj (g ) contains the maximum possible number 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 colimtmg GK . Lemma 4.6. We have that (1) the subgroup colimtmg GKW colimtmg GK is abelian if and only if KW is 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 Gj for some face j of K .
(1)


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Other algebraic examples of our colimits relate to the Stanley-Reisner algebras and coalgebras of K . By construction, there are algebra isomorphisms (4.7) lim Z/2[K ] = SR Z/2 (K ), lim Z[K ] SR Z (K ), and lim [K ] SR (K ), = = where the limits are taken in gZ calg. Dually, there are coalgebra isomorphisms colim Z/2 K SR Z/2 (K ), colim Z K SR Z (K ), = = (4.8) and colim K SR (K ) = in gZ cocoa. The analogues of 4.1 display these limits and colimits as
m

Q[vj ] - lim Q[K ] - Q[V ] (4.9)
j =1 m

and
j =1

DP Q (vj ) - colim Q K - DP Q (V )

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 simple arguments with cellular chain complexes show that the cohomology rings H (colim+ (BC2 )K ; Z/2), H (colim+ (BT )K ; Z), and H (colim+ (BC )K ; Z)are isomorphic to the limits (4.7) respectively. Similarly, the homology coalgebras are isomorphic to the dual coalgebras (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-dimensional). In homology, they may be made in the context of incidence coalgebras, following [29]. In both cases, the maps of (4.1) induce the homomorphisms (4.9). Such calculations do not identify colim+ (BC2 )K and colim+ (BT )K with Davis and Januszkiewicz's constructions. Nevertheless, Buchstaber and Panov DJ R (K ) and colim+ (BT )K provide homotopy equivalences colim+ (BC2 )K DJ C (K ), which also follow from Corollary 5.4 below; the Corollary yields a corresponding equivalence in the exterior case. Of course, colim+ (BC )K is a subcomplex of the m-dimensional torus (S 1 )m , and is therefore finite. In due course, we shall use these remarks to interpret the following proposition in terms of Davis-Januszkiewicz spaces. The pro of 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 for any flag complex K . Since both cases Lane space; Charney BA, given any Artin as our next examples are discrete, B colimtmg GK is, of course, an Eilenberg-Mac and Davis [9] discuss the identification of go o d mo dels for group A. Proposition 4.10 fails for arbitrary complexes K , show. B colimtmg GK


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Examples 4.11. Proposition 4.10 applies when K = V , because the discrete complex is flag; then colimtmg GK is isomorphic to the free product of m copies of G, m whose classifying space is the m-fold wedge j =1 BGj (by [6], for example). On the other hand, when K is the non-flag complex (m), Example 4.3 confirms that B colimtmg GK 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 - colimtmg GK

are homotopy equivalences when K is flag. We therefore view the hK as mo delling the loop spaces; in the complex case, they express colim+ (BT )K in terms of the circulation groups colimtmg T K . In Section 7 we will use homotopy colimits to describe analogues of hK for all complexes K . Our interest in the lo op spaces colim+ (BG)K has been stimulated by several ongoing programmes in combinatorial algebra. For example, Herzog, Reiner, and Welker [17] discuss combinatorial issues asso ciated with calculating the k vector spaces TorSRk (K ) (k, k ) over an arbitrary ground field k , and refer to [16] for historical background. Such calculations have applications to diagonal subspace arrangements, as explained by Peeva, Reiner and Welker [28]. Since these Tor spaces also represent the E2 -term of the Eilenberg-Mo ore spectral sequence for H ( DJ (K ); k ), it seems well worth pursuing geometrical connections. 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 relevant fibrations and their geometrical interpretations. Some of the results appear in [7], but we believe that our approach offers an attractive and efficient alternative, and eases generalisation. We refer to [18] and [36] for the notation and fundamental properties of homotopy colimits. Several of the results we use are also summarised in [37], together with additional information on combinatorial applications. We begin with a general construction, based on a well-pointed topological group and a diagram H : a tmg of closed subgroups and their inclusions. We assume that the maps of the classifying diagram BH : a top+ are cofibrations, and that the Pro jection Lemma [37] applies to the natural pro jection ho colim+ BH colim+ BH , which is therefore 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 â c( ) diagram /H in top, and by Examples 2.21 the cofibration BH (a) B is equivalent to the


Colimits, Stanley-Reisner Algebras, and Lo op Spaces fibration B , c( ),C ( ) âc( ) /H (a) - B c( ) for each ob ject a of a. So fH is equivalent to ho colim+ B (, c( ),C ( ) â
c( )

277

/H ) - B

in the homotopy category of spaces over B , where the homotopy colimit is taken over a. Proposition 5.1. The homotopy fibre of f
H

is the homotopy colimit ho colim+ /H . /H ) - B.

Proof. We wish to identify the homotopy fibre of the pro jection B , a,B (, c( ),C ( ) â
c( )

But we may rewrite the total space as B (, a, /H ) âc( )op B (, c( ),C ( )), and therefore as B (, c( ),C ( )) âc( ) B (, a, /H ), using (2.20) and Examples 2.21. 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 cho ose = colimtmg GL and H = GK ; we also abbreviate the diagram /H to L/K . Then fH is the induced map (5.2) f
K,L

: colim+ (BG)K - B colimtmg GL ,

and the Pro jection Lemma applies to (BG)K because the maps colim+ (BG) () BG are closed cofibrations for each face . In particular, the canonical pro jection (5.3) ho colim+ (BG)K - colim+ (BG)K = DJ (K ) is a homotopy equivalence. We may also deduce the following corollary to Proposition 5.1. Corollary 5.4. The homotopy fibre of fK,L is the homotopy colimit ho colim L/K , and is homeomorphic to the identification space (5.5) B cat(K ) â colimtmg GL / , where (p, g h) (p, g ) whenever h G and p lies in the face B ( cat(K )). Proof. By (3.5), the homotopy colimit B (, cat(K ),L/K ) may be expressed as B ( cat(K )) â
cat(K )

L/K,

and the inclusions B ( cat(K )) B cat(K ) induce a homeomorphism with (5.5). We write the canonical action of colimtmg GL on B (, cat(K ),L/K ) as µ for future use. 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 ho colim 2V /K plays a significant role in [12], where it is described as the identification space of Corollary 5.4 and denoted by ZP (with P the dual of K , in the sense of Example 3.6). To emphasise this connection, we


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write ho colim L/K as ZG (K, L), which we abbreviate to ZG (K ) when K = L. It appears repeatedly below, by virtue of Proposition 5.1. Our examples assume that L = 2V , and continue the theme of Examples 4.11. Examples 5.6. If K = BGm , the inclusion of many years. If K is t equivalent to S m-1 for V then ZG (K, 2V ) is the axes; it has been he non-flag complex G = Z/2, and S 2m-1 the homotopy fibre of m BGj j =1 of interest to homotopy theorists for (m), then ZG (K, 2V ) is homotopy for G = T .

The second of these examples may be understo o d by noting that the inclusion of the fat wedge in BGm has the Thom complex of the external pro duct m of Hopf bundles as its cofibre. Davis and Januszkiewicz [12] prove that the mo d 2 cohomology ring of m m EC2 âC2 ZC2 (K, 2V ) and the integral cohomology ring of ET m âT m ZT (K, 2V ) are isomorphic to the Stanley Reisner algebras SR /2 (K )and SR (K ) respectively. Z Z In view of Corollary 5.4 (in the case L = 2V ), we regard the spaces colim+ (BG)K and the Davis-Januszkiewicz homotopy types as interchangeable from this point on. The canonical pro jection ZG (K, L) B cat(K ) is obtained by factoring out the action µ of colimtmg GL on ho colim L/K . The cubical structure (3.7) of the quotient lifts to an associated decomposition of ZG (K, L); when G = T and L = 2V , for example, we recover the description of [7] in terms of polydiscs and tori. The action µ has other important properties. Proposition 5.7. The isotropy subgroups of the action µ are given by the conjugates wG w-1 < colimtmg GL , as ranges over the faces of K . Proof. It suffices to note from Corollary 5.4 that each point [x, wG ] is fixed by wG w-1 < colimtmg GL , for any x B ( cat(K )). Corollary 5.8. The commutator subgroup of colimtmg GL acts freely on ZG (K, L) under µ. Proof. The isotropy subgroups are abelian, and so have trivial intersection with the commutator subgroup. When K = L and G = C2 , Proposition 5.7 strikes a familiar chord. The parabolic subgroups 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 therefore takes the form Cox (K (1) ), such subgroups are abelian. When L = 2V , each subgroup wG w-1 reduces to G . In this case, Proposition 5.7 implies that the isotropy subgroups form an exponential catop (K )-diagram in tgrp, which assigns G to the face and the quotient homomorphism G G to the reverse inclusion .


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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 the real, complex, and exterior cases, and will feature below; we introduce them here as homotopy colimits. Given a pointed space (Y, 0), we let Yâ 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 K } is the associated arrangement of K , whose complement UY (K ) is given by the equivalent formulae (5.9) Y
V

\

W K YW /

= {f : f

-1

(0) K }.

The cat(K )-diagram Y (K ) associates the function space Y ( ) = {f : f -1 (0) } to each face , and the inclusion Y ( ) Y ( ) to each morphism . It follows V \ that Y ( ) is homeomorphic to Y â (Yâ ), and that UY (K ) is colim Y (K ). asso ciates Yâ to ; when Y is The exponential cat(K )-diagram Yâ contractible, we may therefore follow Proposition 5.1 by combining the Pro jection Lemma and Homotopy Lemma of [37] to obtain a homotopy equivalence (5.10) ho colim Yâ
V \K V \K V \

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 well-developed theory whose history is rich and colourful (see [2], for example). In the exterior case, we replace F by the union of a countably infinite collection of 1-dimensional cones 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 pro ducts. 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 Fâ is a cofibration, and Fâ retracts onto its image. So (5.10) applies, and may be replaced by the corresponding equivalence (5.11) ho colim GV
\K

UF (K ).

Proposition 5.12. The space ZG (K, 2V ) is homotopy equivalent to UF (K ), for any complex K . Proof. Substitute L = 2
V

in Corollary 5.4 and apply (5.11). FW \ 0 a more version Exam-

By specialising results of [37] and [38], we may also describe W K / as a homotopy colimit. This space is dual to UF (K ), and appears to have manageable homotopy type in many relevant cases. For G = C2 and T , a of Proposition 5.12 features prominently in [7]. The following examples illustrate Proposition 5.12, in the light of ples 5.6.


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Examples 5.13. 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 respectively; the corresponding complements are {z : zj = 0 zk = 0 for al l k = j } and Cm \ 0 .
2m-1

and

{0}

The former is homotopy equivalent to a wedge of spheres, and the latter to S

.

6. Flag complexes and connectivity
In this section, we examine the homotopy fibre ZG (K, L) more closely. The results form the basis of our mo del for DJ (K ) when K is flag, and enable us to measure the extent of its failure for general K . We consider a flag complex K , and substitute K = L into Corollary 5.4 to deduce that ZG (K ) is the homotopy fibre of the cofibration fK : DJ (K ) B colimtmg GK . It is helpful to abbreviate B ( cat(K )) to B ( ) throughout the following argument. Proposition 6.1. The cofibration f
K

is a homotopy equivalence whenever K is flag.

Proof. We prove that ZG (K ) is contractible. For any face K , the space (colimtmg GK )/G inherits an increasing filtration by subspaces (colimtmg GK )i /G , consisting of those cosets wG for which a representing element satisfies l(w) i. We may therefore define a cat(K )-diagram Ki /K , which assigns (colimtmg GK )i /G to each face and the corresponding inclusion to each inclusion . By construction, ZG (K ) is filtered by the subspaces hocolim Ki /K and each inclusion ho colim Ki-1 /K ho colim Ki /K is a cofibration. We proceed by induction on i. For the base case i = 0, we observe that (colimtmg GK )0 /G is the single point eG for all values of . Thus hocolim K0 /K is homeomorphic to B (), and is contractible. To make the inductive step, we assume that ho colim Ki /K is contractible for all i < n, and write the quotient space (ho colim Kn /K )/(ho colim Kn-1 /K ) as Qn . It then suffices to prove that Qn is contractible. Every point of Qn has the form (x, wG ), for some x B ( ) and some w of length n. If the final letter of w lies in G , then (x, wG ) is the basepoint of Qn . Otherwise, we rewrite w as w s by (4.5), where s contains the maximum possible number of mutually commuting letters. These determine a subset V , and Lemma 4.6 confirms that K (1) contains the complete graph on vertices . Since K is flag, we deduce that 2 K , and therefore that (x, w G ) is the basepoint of Qn . To describe a contraction of Qn , we may find a canonical path p in cat (K ), starting at x and finishing at some x in B (); of course p must vary continuously with (x, wG ), and lift to a corresponding path in Qn . If x is a vertex of B ( ), we choose p to run at constant speed along the edge from x to the cone point , and again from to the vertex B (). If x is an interior point of B ( ), we


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extend the construction by linearity. Then p lifts to the path through (p(t),w) for all 0 < t < 1, as required. We expect Proposition 6.1 to hold for more general topological groups . The Proposition also leads to the study of fK,L : DJ (K ) B colimtmg GL for any subcomplex 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 infinite. Finally, we define c(K, L) = c(K ) if L Fl (K ) 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 f in dimensions d(K, L). Proof. We may factorise f
K,L K,L

is an equivalence

as

DJ (K ) - DJ (Fl(K )) - DJ (Fl (L)) - B colimtmg GFl(L) . The first map is induced by flagification, and is a d(K )-equivalence by construction. The second is the identity if L Fl (K ); otherwise, it is 0-connected when G = C2 or C , and 2-connected when G = T . The third map is fFl(L) , and an equivalence by Proposition 6.1. Theorem 6.2 suggests our first mo del for DJ (K ). Proposition 6.3. For any complex K , there is a homotopy homomorphism and (d(K ) - 1)-equivalence hK : DJ (K ) colimtmg GK ; in particular, it is an equivalence when K is flag. Proof. We deduce that fK : DJ (K ) B colimtmg GK is a (d(K ) - 1)-equivalence by applying Theorem 6.2 with K = L. 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 . In this case only, we write their minimal dimension as c (K ). It is instructive to consider the homotopy commutative diagram --- ZG (K, L) - p (6.4) ZG (K, 2V ) - --- B [G, L]
id

ZG (K, L) DJ (K ) f

- ---



K,L

- ---- BGm
id Ba

fK,2V

- B colimtmg GL - BGm --- ---


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of fibrations, where a is the abelianisation homomorphism and [G, L] denotes the commutator subgroup of colimtmg GL . By Theorem 6.2, ZG (K, L) and ZG (K, 2V ) are (d(K, L) - 1)- and (d (K ) - 1)-connected respectively, where d(K, L) d (K ) by definition. In fact ZG (K, 2V ) is d (K )-connected, by considering the homotopy exact sequence of fK,2V . Corollary 5.8 confirms that (6.5) [G, L] - ZG (K, L) - ZG (K, 2V )
p

is a principal [G, L]-bundle, classified by . This bundle enco des a wealth of 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 co ordinate subspace arrangement AF (K ) by Corollary 5.12. Moreover, Theorem 6.2 implies that is also a d(K, L)-equivalence, and so sheds some light on the homotopy type of UF (K ). Lo oping (6.4) gives a homotopy commutative diagram of fibrations ZG (K, L) - ZG (K, L) --- p (6.6) UF (K ) [G, L] - ---
i id

- ---

1

DJ (K ) f

K,L

- ----- Gm - Gm ---
a

fK,2V

id

- colimtmg GL ---

in tmonh, which offers an alternative perspective on DJ (K ). Lemma 6.7. The loop space DJ (K ) splits as Gm â UF (K ) for any simplicial complex K ; the splitting is not multiplicative. Proof. The vertex groups Gj embed in DJ (K ) via homotopy homomorphisms, whose pro duct j : Gm DJ (K ) is left inverse to fK,2V (but not a homotopy homomorphism). The pro duct of the maps i and j is the required homeomorphism. The following examples continue the theme of Examples 5.6 and 5.13. They refer 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 lo op space S n with the free monoid F + (S n-1 ) for any n > 1. Examples 6.8. If equivalent to the non-flag complex G = Z/2, and F generating sphere K is the discrete flag complex V , then UF (K ) is homotopy commutator subgroup of the free product m Gj . If K is the j =1 (m), then UF (K ) is homotopy equivalent to F + (S m-2 ) for + (S 2m-2 ) for G = T ; the map i identifies the inclusion of the with the higher Samelson product (of order m) in ( DJ (K )).


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283

Of course, both examples split topologically according to Lemma 6.7. The appearance of higher pro ducts in DJ ( (m)) shows that commutators alone cannot mo del DJ (K ) when K is not flag. More subtle structures are required, based on higher homotopy commutativity; they are related to Samelson and Whitehead products, as we explain elsewhere [27].

7. Homotopy colimits of topological monoids
We now turn to the lo op space DJ (K ) for a general simplicial complex K , appealing to the theory of homotopy colimits. Although the resulting mo dels are necessarily more complicated, they are homotopy equivalent to colimtmg GK when K is flag. The constructions depend fundamentally on the categorical ideas of Section 2, and apply to more general 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 forming the homotopy colimit ho colimtmg D in tmg, rather than top+ . This is made possible by the observation of Section 2 that the categories tmg are t-co complete, and therefore have sufficient structure for the creation of internal homotopy colimits. We confirm that ho colimtmg D is a mo del for the lo op space ho colim+ BD by proving that B commutes with homotopy colimits in the relevant sense. As usual, we work in tmg, but find it convenient to describe certain 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 (2.19). We write B· (, a,D) for the diagram bop â op top+ given by (b, (n)) -
a0 ,an

D(a0 ,b) an (a0 ,an )+ ,

where D is a diagram aâbop top+ . Following Examples 2.21, we define the homotopy top+ -colimit as ho colim+ D = B + (, a,D), and note the equivalent expressions B + (, a,A+ ) a D D aop B + (, a,A+ ). = For tmg, we proceed by categorical analogy. We replace the top-copro duct in (2.19) by its counterpart in tmg, and the internal cartesian pro duct in top by the tensored structure of tmg over top+ . For any diagram D : a tmg, the tmg simplicial topological monoid B· (, a,D) is therefore given by (7.1) (n) -
a0 ,an

D(a0 )

an (a0 ,an )+ ,

where denotes the free pro duct of topological monoids. The face and degeneracy operators are defined as before, but are now homomorphisms. When a is of the


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form cat(K ), the n-simplices (7.1) may be rewritten as the finite free pro duct
tmg Bn (, cat(K ),D) = n ···0



D(0 ),

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
tmg ho colimtmg D = |B· (, a,D)|tmg

in tmg, for any diagram D : a tmg. So ho colimtmg D is an ob ject of tmg. Following Construction (2.11), it may be described in terms of generators and relations as a quotient monoid of the form
n0

Bn (, a,D)

n +

i i di (b),s = b, n (s) , si (b),t = b, n (t) n n

,

i i for all b Bn (, a,D), and all s (n - 1) and t (n +1). Here n and n are 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 ho colimtmg D is its tmg mapping cylinder. It may be identified with the tmg-pushout of the diagram M where j (m) = (m, 0) in M (1)+ - M - N,
j

(1)+ for al l m M .

tmg An alternative expression for the simplicial topological monoid B· (, a,D) arises by analogy with the equivalences (2.20). + Proposition 7.4. There is an isomorphism D aop B· (, a,A+ ) B· (, a,D) = tmg of simplicial topological monoids, for any diagram D : a tmg.

Proof. By (2.17), the functor D aop : [aop âop ,top+ ] [op ,tmg] is left top+ adjoint to tmg(D, ), and therefore preserves copro ducts. So we may write + D aop B· (, a,A+ ) D aop (a( ,a)+ a· (a, b)+ ) =
a,b a,b

=

D(a)
aop

a· (a, b)+

as required, using the isomorphism D

a( ,a) D(a) of (2.16). =

tmg We must decide when the simplicial topological monoids B· (, a,D) are proper simplicial spaces (in the sense of [25]) because we are interested in the homotopy type of their realisations. This is achieved in Proposition 7.8, and leads on to the analogue of the Homotopy Lemma for tmg. These are two of the more memorable of the following sequence of six preliminaries, which precede the pro of of our main result. On several o ccasions we insist that ob jects of tmg are well pointed, and even that they have the homotopy type of a 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 .


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Lemma 7.5. Let fs : Ms Ns be a family of homomorphisms of wel l-pointed topological monoids, which are homotopy equivalences; then the coproduct homomorphism fs : Ms - Ns
s s s

is also a homotopy equivalence. Proof. Let f : M N denote the homomorphism in question, and write Fk M for the subspace of M of elements representable by words of length k . Hence F0 = {e}, and Fk+1 M is the pushout
K

(7.6)

WK (M ) - --- Fk M - ---
jk



K

PK (M )
+1

Fk

M

in top+ , where K runs through all (k + 1)-tuples (s1 ,...,sk+1 ) S k+1 such that si+1 = si , and WK (M ) PK (M ) is the fat wedge subspace of Ms1 â ··· â Msk+1 . Each Ms is well pointed, so WK (M ) PK (M ) is a closed cofibration, and therefore so is jk . Since M = colimk Fk M in top+ , it remains to confirm that the restriction fk : Fk M Fk N is a homotopy equivalence for all k . We pro ceed 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 construction. So the inductive hypothesis combines with Brown's Gluing Lemma [37, 2.4] to complete the pro of. Lemma 7.7. For any subset R S , the inclusion r Mr s Ms is a closed cofibration; in particular, s Ms is wel l pointed. Proof. Let B M be the inclusion in question, with Fk M as in the pro of of Lemma 7.5 and Fk M = B Fk M .Then Fk+1 M is obtained from Fk M by attaching spaces PK (M ), where K runs through all (s1 ,... ,sk+1 ) in S k+1 \ Rk+1 such that si+1 = si . Thus B Fk M is a cofibration for all k , implying the result. Proposition 7.8. Given any smal l category a, and any diagram D : a tmg of tmg wel l-pointed topological monoids, the simplicial space B· (, a,D) is proper, and its realisation is wel l pointed.
tmg tmg Proof. By Lemma 7.7, each degeneracy map Bn (, a,D) Bn+1 (, a,D) is a closed cofibration. The first result then follows from Lillig's Union Theorem [23] tmg tmg for cofibrations. So B0 (, a,D) |B· (, a,D)| is a closed cofibration and tmg B0 (, a,D) is well pointed, yielding the second result.

As described in Examples 2.14, every simplicial ob ject M· in tmg has two possible realisations. We now confirm that they agree, and identify their classifying space.


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Lemma 7.9. The realisations |M· |tmg and |M· | are natural ly isomorphic objects of tmg, whose classifying space is natural ly homeomorphic to |B (M· )|. Proof. We apply the techniques of [14, VII §3] and [26, §4] to the functors | |tmg 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 |M· | |B (M· )| = arises by considering the bisimplicial ob ject (k, n) (Mn )k in top+ , and forming its realisation in either order. We may now establish our promised Homotopy Lemma. Proposition 7.10. Given diagrams D1 , D2 : a tmg of wel l-pointed topological monoids, and a map f : D1 D2 such that f (a): D1 (a) D2 (a) is a homotopy equivalence of underlying spaces for each object a of a, the induced map ho colimtmg D1 - ho colimtmg D 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 X· : a â op top+ of simplicial spaces, and D· : a â op tmg of simplicial topological monoids. Proposition 7.11. With X· and D· as above, there are natural isomorphisms ho colim+ |X· | | ho colim+ X· | and ho colimtmg |D· | | ho colimtmg D· | = = in top+ and tmg respectively. Proof. The isomorphisms arise from realising the bisimplicial ob jects
+ (k, n) - Bk (, a,Xn ) 2

and

tmg (k, n) - Bk (, a,Dn )

in either order. In the case of D· , we must also apply the first statement of Lemma 7.9. Parts of the pro ofs above may be rephrased using variants of the equivalences (2.16). They lead to our first general result, which states that the formation of classifying spaces commutes with homotopy colimits in an appropriate sense. Theorem 7.12. For any diagram D : a tmg of wel l-pointed topological monoids with the homotopy types of CW-complexes, there is a natural map gD : hocolim+ BD - B ho colimtmg D which is a homotopy equivalence. Proof. For each ob ject a of a, The natural map |D· (a)| and a homotopy equivalence, BD(a) under the formation let D· (a) be the singular simplicial monoid of D(a). D(a) is a homomorphism of well-pointed monoids so it passes to a homotopy equivalence B |D· (a)| of classifying spaces. By Proposition 7.10 and the


Colimits, Stanley-Reisner Algebras, and Lo op Spaces corresponding Homotopy for diagrams of realisatio Let D· : a â op and Proposition 7.11, we

287

Lemma for top+ , it therefore suffices to prove our result ns of simplicial monoids. tmg be a diagram of simplicial monoids. By Lemma 7.9 must exhibit a natural homotopy equivalence

| ho colim+ BD· | - |B ho colimtmg D· |. Since both simplicial spaces are proper, it suffices to find a natural map ho colim+ BD· - B ho colimtmg D
·

which is a homotopy equivalence in each dimension n. But ho colim+ BDn is the + realisation of the proper simplicial space B· (, a,B Dn ), and Lemma 7.9 confirms tmg that B ho colim Dn is naturally homeomorphic to the realisation of the proper tmg simplicial space B (B· (, a,Dn )); so gD may be specified by a sequence of maps + tmg Bk (, a,B Dn ) B (Bk (, a,Dn ), where k 0. They are most easily described as maps (7.13)
a0 ···ak

BDn (a0 ) - B

a0 ···ak



Dn (a0 ) ,

and are induced by including each of the Dn (a0 ) into the free pro duct. Since (7.13) is a homotopy equivalence by a theorem of Fiedorowicz [15, 4.1], the pro of is complete. Various steps in the pro of of Theorem 7.12 may be adapted to verify the following, which answers a natural question about tensored monoids. Proposition 7.14. For any wel l-pointed topological monoid M and based space Y , the natural map BM Y - B (M Y ) is a homotopy equivalence if M and Y have the homotopy type of CW-complexes. Proof. As |Y· | of the realisation Y is discre in Theorem 7.12, we need only work with the realisations |M· | and total singular complexes. Since B |M· | |Y· | B (|M· | |Y· |) is the of the natural map BMn Yn B (Mn Yn ), it suffices to assume that te; in this case, BM Y - B

M
y

y

is a homotopy equivalence by the same result of Fiedorowicz [15]. We apply Theorem 7.12 to construct our general mo del for DJ (K ), but require a commutative diagram to clarify its relationship with the special case hK of Proposition 6.3. We deal with aop â op -diagrams X· in top+ , and certain of their morphisms. These include : X· top+ (BD, B (D aop X· )), defined + for any X· by (x) = B (d d x), and the pro jection : B· (+ )· , where


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+ + B· and (+ )· denote B· (, a,A) and the trivial diagram respectively. Under the homeomorphism [op

,top+ ] BD a

op

X· ,B (D aop X· ) = [aop âop ,top+ ] X· , top+ (BD, B (D
op

aop

X· ))

of (2.17), corresponds to a map : BD a spaces.
gD

X· B (D

aop

X· ) of simplicial

Proposition 7.15. For any diagram D : a tmg, there is a commutative square ho colim+ BD - B ho colimtmg D --- Bptmg , p+
+

where p

and p

tmg

colim+ BD - B colimtmg D --- are the natural projections.
fD + --- B· - top+ (BD, B (D B (1 + B· ))

Proof. By construction, the diagram
aop )· a
op

(+ )· - top+ (BD, B (D --- op op is commutative in [a â ,top+ ] , and has adjoint
+ BD aop B· 1



(+ )· ))
+ B· )

- B (D ---



(7.16)

B
a
op

aop

(1 )

BD a
op

op

(+ )· - B (D ---



(+ )· )

+ in [ ,top+ ] . By Proposition 7.4, the upper is the map B· (, a,B D) tmg B (B· (, a,D)) obtained by applying the relevant map (7.13) in each dimension. By Examples 2.14, the lower is given by the canonical map fD : colim+ BD B colimtmg D in each dimension. Since realisation commutes with B , the topological realisation of (7.16) is the diagram we seek; for Lemma 7.9 identifies the upper right-hand space with B ho colimtmg D, and Examples 2.21 confirms that the vertical maps are the natural pro jections.

Theorem 7.17. There is a homotopy commutative square ho colim+ (BG)K - K ho colimtmg GK --- pK qK
h

DJ (K )

- K ---
K

h

colimtmg GK
K

of homotopy homomorphisms, where p any simplicial complex K .

and h

are homotopy equivalences for


Colimits, Stanley-Reisner Algebras, and Lo op Spaces

289

Proof. We apply Proposition 7.15 with D = GK , and lo op the corresponding square; the pro jection pK : hocolim+ (BG)K DJ (K ) is a homotopy equivalence, as explained in (5.3). The result follows by composing the horizontal maps with the canonical homotopy homomorphism BH H , where H = ho colimtmg GK and colimtmg GK respectively. It is an interesting challenge to describe go o d geometrical mo dels for homotopy homomorphisms which are inverse to hK and hK .

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Taras Panov Department of Mathematics and Mechanics Moscow State University 119899 Moscow, Russia and Institute for Theoretical and Experimental Physics 117259 Moscow, Russia e-mail: tpanov@mech.math.msu.su Nigel Ray Department of Mathematics University of Manchester Manchester M13 9PL, England e-mail: nige@ma.man.ac.uk Rainer Vogt Fachbereich Mathematik/Informatik Universit¨t Osnabruck a ¨ D-49069 Osnabruck, Germany ¨ e-mail: rainer@mathematik.uni-osnabrueck.de