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. There is a homotopy commutative square
_
hocolim+(BG)K --hK-!hocolimtmgGK
?? ?
y pK ?yqK
DJ (K) --hK-! colimtmgGK
__
of homotopy homomorphisms, where pK and hK are homotopy equivalences for any si*
*mplicial
complex K.
Proof.We apply Proposition 7.15 with D = GK , and loop the corresponding square*
*; the
projection pK :hocolim+(BG)K ! DJ (K) is a homotopy equivalence, as explained i*
*n Corol-
lary 5.3. The result follows by composing the horizontal maps with the canonica*
*l_homotopy
homomorphism BH ! H, where H = hocolimtmgGK and colimtmgGK respectively. |_*
*_|
It is an interesting challenge to_describe_good geometrical models for homoto*
*py homomor-
phisms which are inverse to hK and hK.
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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
~~