ON DAVIS-JANUSZKIEWICZ HOMOTOPY TYPES I;
FORMALITY AND RATIONALISATION
DIETRICH NOTBOHM AND NIGEL RAY
Abstract.For an arbitrary simplicial complex K, Davis and Januszkiewicz *
*have de-
fined a family of homotopy equivalent CW-complexes whose integral cohomo*
*logy rings
are isomorphic to the Stanley-Reisner algebra of K. Subsequently, Buchst*
*aber and Panov
gave an alternative construction, which they showed to be homotopy equiv*
*alent to Davis
and Januszkiewicz's examples. It is therefore natural to investigate the*
* extent to which
the homotopy type of a space X is determined by having such a cohomology*
* ring. We
begin this study here, in the context of model category theory. In parti*
*cular, we extend
work of Franz by showing that the singular cochain algebra of X is forma*
*l as a differ-
ential graded noncommutative algebra. We then specialise to the rational*
*s, by proving
the corresponding property for Sullivan's commutative cochain algebra; t*
*his confirms that
the rationalisation of X is unique. In a sequel, we will consider the un*
*iqueness of X at
each prime separately, and apply Sullivan's arithmetic square to produce*
* global results in
special families of cases.
1.Introduction
Over the last decade, work of Davis and Januszkiewicz [7] has popularised hom*
*otopy
theoretical aspects of toric geometry amongst algebraic topologists. The resul*
*ts of [7]
have been surveyed by Buchstaber and Panov in [4], where several further applic*
*ations
were developed. Their constructions have led us to consider the uniqueness of *
*certain
associated homotopy types, and our aim is to begin that study here; we shall ad*
*dress
issues of formality and rationalisation, where the answers are most general. I*
*n a sequel
[19], we discuss the problem prime by prime, and deduce global results for fami*
*lies of
special cases by appealing to Sullivan's arithmetic square. The conclusions the*
*re are less
clear-cut, and suggest further problems that seem to be of considerable difficu*
*lty.
We work over an arbitrary commutative ring R with identity, and consider a un*
*iversal
set V of vertices v1, . . . , vm , ordered by their subscripts. The vertices *
*masquerade as
algebraically independent variables, which generate a graded polynomial algebra*
* SR(V )
over R. The grading is defined by assigning each of the generators a common dim*
*ension,
which we usually takePto be 2. A function M :V ! N is known as a multisetQon V *
*, with
cardinality |M| := jM(vj); it may be represented by the monomial vM := V vM*
*(v), or
by the n-tuple of constituent vertices (vj1, . .,.vjn), where j1 . . .jn and *
*n = |M|.
So SR(V ) is generated additively by the vM , and vM is squarefree precisely w*
*hen M is a
genuine subset.
___________
Key words and phrases. Formality, Davis-Januszkiewicz spaces, homotopy colimi*
*t, model category, ra-
tionalisation, Stanley-Reisner algebra.
1
2 DIETRICH NOTBOHM AND NIGEL RAY
A simplicial complex K on V consists of a finite set of faces oe V , closed*
* with respect
to the formation of subsetsQø oe. Alternatively, we may interpret K as the *
*set of
squarefree monomials vff:= ffv, which is closed under factorisation. Every s*
*implicial
complex generates a simplicial set Ko; for each n 0, the n-simplices Kn conta*
*in all M of
cardinality n + 1 whose support is a face of K. The face and degeneracy operato*
*rs delete
and repeat the appropriate vertices respectively.
The Stanley-Reisner algebra R[K] (otherwise known as the face ring of K) is a*
*n impor-
tant combinatorial invariant. It is defined as the quotient
(1.1) SR(V )=(vU : U =2K),
and is therefore generated additively by the simplices of Ko. The algebraic pro*
*perties of
R[K] encode a host of combinatorial features of K, and are discussed in detail *
*by Bruns
and Herzog [3] and Stanley [23], for example. If K is the simplex on V , then R*
*[K] is the
polynomial algebra SR(V ).
For each K, Davis and Januszkiewicz defined the notion of a toric space over *
*the cone on
the barycentric subdivision of K, and showed that the cohomology of such a spac*
*e is related
to R[K]. The relationship follows from their application of the Borel construct*
*ion, which
creates a family of spaces whose cohomology ring (with coefficients in R) is is*
*omorphic
to the Stanley-Reisner algebra. All members of the family are homotopy equivale*
*nt to a
certain universal example, and we refer to any space which shares their common *
*homotopy
type as a Davis-Januszkiewicz space. The isomorphisms equip R[K] with a natural*
* grad-
ing, which agrees with that induced from SR(V ). Subsequently, Buchstaber and P*
*anov [4]
defined a CW-complex whose cohomology ring is also isomorphic to R[K]. They con*
*firmed
that their complex is a Davis-Januszkiewicz space by giving an explicit homotop*
*y equiv-
alence with the universal example. In [21], their space is described as the poi*
*nted colimit
colim+(BK ) of a certain cat (K)-diagram BK , which assigns the cartesian produ*
*ct Bffto
each face oe of K. Here B denotes the classifying space of the circle, otherwis*
*e known as
complex projective space CP 1 .
We say that a space X realises the Stanley-Reisner algebra of K whenever ther*
*e is an
algebra isomorphism H*(X; R) ~=R[K]. We denote the rationalisation of X by X0, *
*and
write Autho(X) < Endho(X) for the homotopy classes of self-equivalences of X, c*
*onsidered
as a subgroup of the homotopy classes of self-maps with respect to composition.*
* We refer
to the group of unimodular complex numbers as T , in order to distinguish it fr*
*om the
underlying circle S1.
The contents of this article are as follows.
In Section 2 we describe our notation and prerequisites, including those aspe*
*cts of model
category theory which provide a useful context for exponential diagrams and the*
*ir coho-
mology. We also explain why it is more convenient to work with the unpointed c*
*olimit
c(K) :=colim(BK ). We introduce the Stanley-Reisner algebra in Section 3, and s*
*how that
the Bousfield-Kan spectral sequence for H*(c(K); R) collapses by analysing high*
*er limits
of certain cat(K)-diagrams. In Section 4 we apply similar techniques to prove t*
*he formal-
ity of the singular cochain algebra C*(c(K); R). Finally, we specialise to the *
*case R = Q
in Section 5, where we confirm that Sullivan's commutative cochain algebra APL *
*(c(K))
HOMOTOPY TYPES, FORMALITY AND RATIONALISATION 3
is formal in the commutative sense. We deduce that Q[K] determines the rational*
*isation
c(K)0 uniquely, and discuss Autho(c(K)0) in a few simple cases.
We would like to develop the results of Section 4 in the model category of E1*
* algebras,
which are emerging as the integral analogue of differential graded commutative *
*algebras
over Q. Recent work of Mandell [17] shows that c(K) is classified by the E1 a*
*lgebra
C*(c(K); Z), at least up to weak equivalence of nilpotent spaces. As we shall *
*explain
in the sequel, we do not believe that C*(c(K); R) is always formal in this stri*
*cter sense;
nevertheless, it may be true for our families of special cases.
Both authors are especially grateful to the organisers of the International C*
*onference on
Algebraic Topology, which was held on the Island of Skye in June 2001. The Conf*
*erence
provided the opportunity for valuable discussion with several colleagues, among*
*st whom
Octavian Cornea, Kathryn Hess, and Taras Panov deserve special mention. Without*
* that
remarkable and stimulating environment, our work could not have begun. We shou*
*ld
also thank the London Mathematical Society for supporting the Transpennine Topo*
*logy
Triangle, whose meetings have speeded up the completion of this article, and ar*
*e helping
us to prepare its sequel.
2. Background
We begin by establishing our notation and prerequisites, recalling various as*
*pects of
Davis-Januszkiewicz spaces. We refine results of [21] in the context of model *
*category
theory, referring readers to [9] and [14] for background details. Following [24*
*], we adopt
the model category top of k-spaces and continuous functions as our topological *
*workplace.
Weak equivalences induce isomorphisms in homotopy, fibrations are Serre fibrati*
*ons, and
cofibrations have the left lifting property with respect to acyclic fibrations.*
* Every function
space Y X is endowed with the corresponding k-topology. Many of the spaces we c*
*onsider
have a distinguished basepoint *, and we write top + for the model category of *
*pairs
(X, *) and basepoint preserving maps. We often require the inclusion of * to be*
* a closed
cofibration, in which case X is said to be well-pointed; this is automatic when*
* X is a
CW-complex and * its 0-skeleton.
Given a small category a, we refer to a covariant functor D :a ! r as an a-di*
*agram
in r. Such diagrams are themselves the objects of a category [a,r], whose morph*
*isms are
natural transformations of functors. We may interpret any object X of r as a c*
*onstant
diagram, which maps every object of a to X and every morphism to the identity. *
* We
describe a as finite whenever the total collection of morphisms is finite.
Examples 2.1. (1) For each integer n 0, the category ord (n) has objects 0, 1*
*, . . . , n,
equipped with a single morphism k ! m when k m. An ord (n)-diagram
f1 f2 fn
(2.2) X0 -! X1 -! . .-.! Xn
consists of n composable morphisms in r.
(2) The category has objects (n) := {0, 1, . .,.n} for n 0, and morphism*
*s the
nondecreasing functions; then op- and -diagrams are simplicial and cosimplici*
*al objects
4 DIETRICH NOTBOHM AND NIGEL RAY
of r respectively. In particular, : ! top is the cosimplicial space which as*
*signs the
standard n-simplex (n) to each object (n).
Given objects X0 and X1 of r, we write the set of morphisms X0 ! X1 as r(X0, *
*X1);
when r is small, the diagrams (2.2)also form a set rn(X0, Xn), which reduces to*
* r(X0, X1)
when n = 1. For any r it is often convenient to abbreviate [ op,r]to sr, and wr*
*ite a generic
simplicial object as Do. In particular, sset denotes the category of simplicial*
* sets Yo.
From this point on we work with an abstract simplicial complex K, whose faces*
* oe are
subsets of the vertices V . We assume that the empty face belongs to K, and wr*
*ite Kx
when it is expressly omitted. The integer |oe| - 1 is known as the dimension o*
*f oe, and
written dim oe; its maximum value is the dimension dim K of K. For any integer *
*d 1, the
d-skeleton K(d)of K is the subcomplex of faces whose dimension satisfies dim oe*
* d; it has
dimension d. When K contains every subset of V , we may call it the simplex (V*
* ) on V .
Each face of K therefore determines a subsimplex (oe), whose boundary (or (dim*
* oe - 1)-
skeleton) @(oe) is given by deleting the subset oe. We also require the link `*
*K (oe), whose
faces consist of those ø \ oe for which oe ø in K.
Definition 2.3. For any simplicial complex K, the small category cat (K) has ob*
*jects
the faces of K and morphisms the inclusions iff,fi:oe ø. The empty face ? is *
*an initial
object, and the maximal faces ~ admit only identity morphisms. The opposite ca*
*tegory
cat op(K) has morphisms pfi,ff:=iopff,fi:ø oe, and ? is final.
The nondegerate simplices of the nerve Nocat (K) form the cone on the barycen*
*tric
subdivision K0, and those of Nocat (Kx ) correspond to the subcomplex K0. So th*
*e classi-
fying space Bcat (K) (formed by realising the nerve) is a contractible CW-compl*
*ex, and
Bcat (Kx ) is a subcomplex homeomorphic to K. We shall study cat(K)- and catop(*
*K)-
diagrams D in various algebraic and topological categories r. Usually, r is po*
*inted by
an object *, which is both initial and final; unless stated otherwise, we then *
*assume that
D(?) = *.
For each face oe, the overcategories cat (K) # oe and cat (K) + oe are given *
*by restricting
attention to those objects ø for which ø oe and ø oe respectively. The unde*
*rcategories
oe #cat (K) and oe +cat (K) are defined likewise. It follows from the definitio*
*ns that
cat(K)#oe = cat( (oe)), cat(K)+oe = cat(@(oe)),
oe #cat (K) = cat(`K (oe)) and oe +cat (K) = cat(`K (oe)x).
The dimension function may be interpreted as a functor dim :cat(K) ! ord (m), w*
*hich
is a a linear extension in the sense of [14]; thus cat (K) is direct and cat op*
*(K) is inverse.
For any model category r, we may therefore follow Hovey and impose an associa*
*ted
model structure on the category of diagrams [cat(K),r]. Weak equivalences e: C *
*! D
are given objectwise, in the sense that e(oe): C(oe) ! D(oe) is a weak equivale*
*nce in r for
every face oe of K. Fibrations are also given objectwise. To describe the cofib*
*rations, we
consider the restrictions of C and D to the overcategories cat (@(oe)), and wri*
*te LffC and
LffD for their respective colimits; Lffis the latching functor of [14]. Then g *
*:C ! D is a
cofibration whenever the induced maps
(2.4) C(oe) qLoeCLffD -! D(oe)
HOMOTOPY TYPES, FORMALITY AND RATIONALISATION 5
are cofibrations in r for every face oe. Alternatively, the methods of Chachols*
*ki and Scherer
[6] lead to the same model structure on [cat(K),r].
There is a dual model category structure on [catop(K),r], where weak equivale*
*nces
and cofibrations are given objectwise. To describe fibrations f :C ! D, we cons*
*ider the
restrictions of C and D to the undercategories cat op(@(oe)), and write MffC an*
*d MffD for
their respective limits; Mffis the matching functor of [14]. Then f is a fibrat*
*ion whenever
the induced maps
(2.5) C(oe) -! D(oe) xMoeDMffC
are fibrations in r for every face oe.
Definition 2.6. For any CW-pair (X, *), the exponential pair of diagrams (XK , *
*XK )
consists of functors
(2.7) XK :cat (K) -! top + and XK :cat op(K) -! top +,
which assign the cartesian product Xffto each face oe of K; the value of XK on*
* iff,fiis the
cofibration Xff! Xfi, where the superfluous coordinates are set to *, and the v*
*alue of XK
on pfi,ffis the fibration Xfi! Xff, defined by projection. The pair are twins, *
*in the sense
that XK (i0) . XK (i) = XK (j0) . XK (j) for every pullback square
j0 0
oe \ oe0---!oe
? ?
j?y ?yi0
oe ---! ø
i
in cat (K).
The properties of twin diagrams are analogous to those of a Mackey functor. *
*They
include, for example, the fact that each XK (i) has left inverse XK (p), where *
*p = iop. Our
applications in Theorem 3.12 are reminiscent of [15], where the acyclicity of c*
*ertain Mackey
functors is established.
The colimit colimXK is a subcomplex of XV , whose inclusion rK is induced by *
*interpret-
ing the elements oe of K as faces of the (m - 1)-simplex (V ). Composing rK wi*
*th any of
the natural maps Xff! colimXK yields the standard inclusion Xff! XV . We note *
*that
colimXK is pointed by X? , otherwise known as the basepoint *, and is homeomor*
*phic to
the pointed colimit colim+XK of [21].
We wish to study homotopy theoretic properties of colimXK in favourable cases*
*. Yet the
colimit functor behaves particularly poorly in this context, because objectwise*
* equivalent
diagrams may well have homotopy inequivalent colimits. The standard procedure *
*for
dealing with this situation is to introduce the left derived functor, known as *
*the homotopy
colimit. Following [13], for example, hocolimXK may be described by the two-si*
*ded bar
construction B(*, cat(K), XK ) in top . We note that hocolimXK is also pointed*
*, and is
related to the pointed homotopy colimit hocolim+XK of [21] by the cofibration
f + K
Bcat (K) -! hocolimXK - ! hocolim X
6 DIETRICH NOTBOHM AND NIGEL RAY
of [2]. Since Bcat (K) is contractible, f is a weak equivalence. We may therefo*
*re concen-
trate on hocolimXK , and so avoid basepoint complications when working with fun*
*ction
spaces in [19].
Lemma 2.8. Every exponential diagram is cofibrant in [cat(K),top] .
Proof.The initial cat (K)-diagram in top is the constant diagram *, so XK is c*
*ofibrant
whenever the inclusion * ! XK is a cofibration. By (2.4), it suffices to show t*
*hat the map
X@(ff)! Xffis a cofibration for every face oe of K. But the map in question inc*
*ludes the
fat wedge in the cartesian product, and the result follows.
An immediate consequence of Lemma 2.8 is that the natural projection
(2.9) hocolim XK - ! colimXK
is a homotopy equivalence. This illustrates one of the fundamental results of m*
*odel category
theory, and is sometimes called the Projection Lemma [25].
3. Integral Cohomology and Limits
In this section we work in the category mod R of R-modules, and study the coh*
*omology
of exponential diagrams BK , where B is the classifying space of the circle T .*
* For this
case we abbreviate (2.9)to hc(K) ! c(K). We focus on the relationship between *
*the
Stanley-Reisner algebra R[K] and the Bousfield-Kan spectral sequence for H*(hc(*
*K); R).
We begin by investigating the cohomology of c(K). To simplify applications i*
*n later
sections, we consider an arbitrary pair of twin diagrams (DK , DK ),
(3.1) DK :cat op(K) -! mod R and DK :cat (K) -! mod R.
Thus DK (q0) . DK (q) = DK (p0) . DK (p) for every pullback square p . q = p0. *
*q0in catop(K);
in particular, DK (i) is left inverse to DK (p) for every morphism p = iop. Su*
*ch pairs
arise, for example, from any contravariant functor D :top ! mod R, by composing*
* with
the exponential twins (BK , BK ). Then (DK , DK ) = (D . BK , D . BK ), and fu*
*nctoriality
ensures the diagrams are twins. In this case we may apply D to the natural maps*
* Bff!
rK V
c(K) -! B , and obtain homomorphisms
D(rK) hK K
(3.2) D(BV ) --- ! D(c(K)) --! limD
in mod R.
By way of example, we consider the case D = H2j(-, R), for any j 0. For eve*
*ry face
oe of K, the space Bffis an Eilenberg-Mac Lane space H(Zff, 2), and may be expr*
*essed
as the realisation of a simplicial abelian group Ho(Zff, 2) whenever convenient*
* [18]. As a
CW-complex, the cells of Bffare concentrated in even dimensions, and correspond*
* to the
simplices vM of (oe)o. The cellular cohomology group H2j(Bff; R) is therefore*
* isomorphic
to the free R-module generated by those vM for which |M| = j and the support o*
*f M is a
subset of oe. The diagram DK of (3.1)becomes
(3.3) H2j(BK ; R): catop(K) -! mod R,
HOMOTOPY TYPES, FORMALITY AND RATIONALISATION 7
whose value on pfi,ffis the homomorphism which fixes vM whenever the support o*
*f M lies
in oe, and annihilates it otherwise; the left inverse is the inclusion induced *
*by DK . When
D = H2j+1(-, R), the diagram is zero.
In the case of cohomology, we may combine the diagrams (3.3)into a graded ver*
*sion
(3.4) H*(BK ; R): catop(K) -! gmod +R,
taking values in the category of augmented graded R-modules. The cup product on*
* each
of the constituent submodules H*(Bff; R) is given by the product of monomials x*
*LxM =
xL+M , as follows from the case of a single vertex. In other words, the cohomo*
*logy ring
H*(Bff; R) is isomorphic to the polynomial algebra SR(oe). So H*(BK ; R) actual*
*ly takes
values in the category gca +Rof augmented graded commutative R-algebras, and ma*
*ps the
morphism pfi,ffto the projection SR(ø) ! SR(oe); the left inverse is again incl*
*usion.
The homomorphisms (3.2)may similarly be combined as
r*K * hK * K
(3.5) SR(V ) --! H (c(K); R) --! limH (B ; R),
where the limit is taken in gmod +R. Since (3.4)is a diagram of algebras, the l*
*imit inherits
a multiplicative structure, and it is equally appropriate to interpret (3.5)in *
*gca +R. The
composition hK . r*Kis induced by the projections SR(V ) ! SR(oe). In this case*
*, we make
one further observation.
Proposition 3.6. The homomorphism r*Kis epic, and its kernel is the ideal (vU :*
* U =2K).
Proof.In each dimension 2j, the cells xM of BV correspond to the multisets on *
*V with
|M| = j. The cells of c(K) form a subset, given by those M for which xffdivide*
*s xM
for some face oe of K. Hence r*K is epic, and its kernel is generated by the r*
*emaining
cells. These are characterised by having no such factor, and therefore coincid*
*e with the
2j-dimensional additive generators of the ideal (vU : U =2K).
So the Stanley-Reisner algebra (1.1)admits an isomorphism R[K] ~=H*(c(K); R),*
* which
plays a central r^ole in [4].
Returning to our study of the twins (DK , DK ), the following definition iden*
*tifies an
important property.
Definition 3.7. A diagram F K:cat op(K) ! mod R of R-modules (graded or otherwi*
*se)
is fat if the natural map F K(æ) ! limF @(j)is an epimorphism for every face æ *
*of K.
The terminology acknowledges the relationship between @(æ) and the fat wedge *
*described
in Lemma 2.8.
Lemma 3.8. The twin DK is fat.
Proof.We consider an arbitrary face æ of K, whose vertices we label wk for 0 *
*k d;
thus d = dim æ. We write ~k := æ \ wk for the maximal faces of @(æ), and abbrev*
*iate the
morphism pj,~kto pk for 0 k d.
The definition of limensures that L := limD@(j)appears in an exact sequence
Y ffiY
0 -! L -! DK (oe) -! DK (oe),
j ff fi ff
8 DIETRICH NOTBOHM AND NIGEL RAY
Q
where ffi(u)(ø oe) = u(oe) - DK (pfi,ff)u(ø) for any u 2 ff jDK (oe). Henc*
*e u 2 L is
Q
determined by the values u(~k). The natural projection DK (æ) ! j ffDK (oe) t*
*herefore
factors through L, and it remains to find u(æ) 2 DK (æ) such that DK (pk)(u(æ))*
* = u(~k)
for every 0 k Pd.
We set u(æ) := j ff(-1)|j\ff|+1DK (pj,ff)u(oe). The fact that DK and DK *
*are twins
implies that
(
DK (p~k,ff\wk)u oe\wkif wk 2 oe
DK (pk)DK (pj,ff)u(oe) =
DK (p~k,ff)u(oe) otherwise
for every 0 k d; thus
X X
DK (pk)u(æ) = (-1)|j\ff|+1DK (p~k,ff)u(oe) + (-1)|j\fi|+1DK (p~k,fi\w*
*k)u(æ\wk).
ff63wk fi3wk
But we may write u(oe) as u (oe [ wk)\wk for any oe 63 wk other than ~k. So th*
*e summands
cancel in pairs, leaving u(~k) as required.
For cohomology, Lemma 3.8 contributes to our analysis of c(K). The homotopy e*
*quiv-
alence (2.9)provides a cohomology decomposition [8], in the sense that the coho*
*mology
algebra H*(c(K); R) may be computed by the Bousfield-Kan spectral sequence [2]
Ei,j2=) Hi+j(hc(K); R),
where Ei,j2is isomorphic to the ith derived functor limiHj(BK ; R) for every i,*
* j 0. The
vertical edge homomorphism coincides with the map hK of (3.5). Lemma 3.8 is req*
*uired
for our computation of these limits, and Corollary 3.14 will confirm that the c*
*ohomology
decomposition is sharp in Dwyer's language. Our proof uses the calculus of func*
*tors and
their limits; the appropriate prerequisites may be deduced from Gabriel and Zis*
*man [12,
Appendix II x3], by dualising their results for colimits.
In particular, we follow [12] (as expounded in [20], for example) by calculat*
*ing limiDK
as the i th cohomology group of a certain cochain complex C*(DK ), ffi of R-m*
*odules. The
groups are defined by
Y
(3.9) Cn(DK ) := DK (oen) for n 0,
ff0 ...ffn
P n
and the differential ffi := k=0(-1)kffik is defined on u 2 Cn(DK ) by
(
u(oe0 . . .boek . . .oen+1)for k 6= *
*n + 1
(3.10) ffik(u)(oe0 . . .oen+1) :=
DK (pffn,ffn+1)u(oe0 . . .oen)for k =*
* n,+ 1
We may replace C*(DK ) by its quotient N*(DK ) of normalised cochains, in which*
* the faces
oe0, . . . , oen of (3.9)and (3.10)are required to be distinct.
Lemma 3.11. Given a diagram D :catop(K) ! mod R, and a maximal face ~ of K, then
(
D(~) for i = 0
limiD =
0 for i > 0
HOMOTOPY TYPES, FORMALITY AND RATIONALISATION 9
whenever D(oe) = 0 for all oe 6= ~.
Proof.Since ~ is maximal, the only morphism oe ~ is the identity. So the nor*
*malised
chain complex N*(D) is D(~) in dimension 0, and 0 in higher dimensions, as requ*
*ired.
Theorem 3.12. For any fat diagram F K:cat op(K) ! mod R, we have that limiF K =*
* 0
for all i > 0; in particular, limiDK = 0 for every twin DK .
Proof.We proceed by induction on the total number of faces f(K); the result obv*
*iously
holds for the initial example K = ?, where f(K) = 0. Our inductive hypothesis i*
*s that
limiF K vanishes whenever K satisfies f(K) f.
We therefore consider an arbitrary complex K with f(K) = f + 1, and write J *
* K for
the subcomplex obtained by deleting a single maximal face ~, The inclusion of J*
* defines
a functor G: catop(J) ! cat op(K), whose induced functor G*: [catop(K),modR] !
[catop(J),modR] acts by restriction, and admits a right adjoint G*, known as th*
*e right
Kan extension [16]. In particular, G*F Jis given on oe 2 K by limF @(~)when oe *
*= ~, and
F ffotherwise.
But F ~! limD@(~)is an epimorphism, by Lemma 3.8, so the natural transformati*
*on
F K ! G*F Jis epic on every face of K, and its kernel H is zero on every face e*
*xcept ~.
We acquire a short exact sequence of functors
0 -! H -! F K -! G*F J-! 0,
which induces a long exact sequence of higher limits. By Lemma 3.11, this colla*
*pses to a
sequence of isomorphisms
(3.13) limiF K ~= limiG*F J,
for i 1. We now apply the composition of functors spectral sequence [5], [12]
limnGi*F J=) limn+iF J.
Here Gi*denotes the ith derived functor of G*; it may be evaluated on any face *
*oe of K
as limiF @(ff), and therefore vanishes for i > 0, by inductive hypothesis. So *
*the spectral
sequence collapses onto the first row of the E2 page, from which we obtain isom*
*orphisms
limnG*F J~=limnF Jfor all n 0. Since the inductive hypothesis applies to J, w*
*e deduce
that limnG*F J= 0 for every n > 0. Combining this with (3.13)concludes the proo*
*f.
Corollary 3.14. The Bousfield-Kan spectral sequence for BK collapses at the E2*
* page; it
is concentrated along the vertical axis, and given by
(
limHj(BK ; Q) if i = 0
limiHj(BK ; Q) =
0 otherwise.
Proof.The result follows immediately from Theorem 3.12 by letting DK be Hj(-; R*
*) for
every j 0.
10 DIETRICH NOTBOHM AND NIGEL RAY
Corollary 3.14 confirms that the edge homomorphism hK is an isomorphism in gc*
*a +R.
When combined with (3.5)and Proposition 3.6 it implies that the natural map
R[K] ~= H*(c(K); R) -h! limH*(BK ),
which is induced by the projections R[K] ! SR(oe), is also an isomorphism. This*
* may be
proven directly, by refining the methods of Proposition 3.6.
4. Integral Formality
In this section we study the formality of c(K) over our arbitrary commutative*
* ring
R, and construct a zig-zag of quasi-isomorphisms between the singular cochain a*
*lgebra
C*(c(K); R) and its cohomology ring.
We work in the model category dga +Rof differential graded R-algebras with au*
*gmen-
tation. The model structure arises by interpreting dga +Ras the category of mo*
*noids in
the monoidal model category dgmod R of unbounded cochain complexes over R. The*
* lat-
ter is isomorphic to Hovey's category of unbounded chain complexes [14], and th*
*e model
structure is induced on dga +Rby checking that it satisfies the monoid axiom of*
* [22]. As
Schwede and Shipley confirm, weak equivalences are zig-zags of quasi-isomorphis*
*ms and
fibrations are epimorphisms. Cofibrations are defined by the appropriate liftin*
*g property,
and are necessarily degreewise split injections. We emphasise that the objects *
*of dga +Rare
not necessarily commutative.
A differential graded R-algebra C* is formal in dga +Rwhenever there is a zig*
*-zag of
quasi-isomorphisms
(4.1) H(C*) -~! . .-.~ C*,
where we follow the convention of assigning the zero differential to the cohomo*
*logy algebra
H(C*). Our aim is to show that the cochain algebra C*(c(K); R) is always formal*
* in dga +R.
This extends Franz's result [11], which only applies to complexes arising from *
*smooth fans.
We begin by choosing D to be the j-dimensional cochain functor Cj(-; R) in (3*
*.1),
thus creating twin diagrams Cj(BK ; R), Cj(BK ; R) for each j 0. As in (3.4*
*), we may
consider the graded version C*(-; R) in dgmod R. In fact its values are always*
* R-algebras,
with respect to the cup product of cochains. This product is no longer commutat*
*ive, but
the procedure for forming the limit of a dga +R-diagram remains the same; work *
*in dgmod +R,
and superimpose the induced multiplicative structure.
For the Eilenberg-Mac Lane space B = H(Z, 2), we let v denote a generator of *
*H2(B; R),
which is isomorphic to R. We choose a cocycle _v representing v in C2(B; R), an*
*d define a
homomorphism _ :H*(B; R) ! C*(B; R) by _(vk) = _kv, for all k 0. By construct*
*ion,
_ is multiplicative, and is a quasi-isomorphism in dga +R. We extend this proc*
*edure to
H*(BV ; R) via the zig-zag
~= * V _ * V ez * V
(4.2) H*(BV ; R) --! H (B; R) - -! C (B; R) - - C (B ; R)
of quasi-isomorphisms in dga +R. Here the leftmost map is the Künneth isomorphi*
*sm, and
ez is the Eilenberg-Zilber map.
HOMOTOPY TYPES, FORMALITY AND RATIONALISATION 11
By restriction, we may also interpret the _j as cocycles in C2(Bff; R) for ev*
*ery face
oe; they represent vj in H2(Bff; R) when oe contains vj, and 0 otherwise. Beca*
*use the
restrictions are compatible, we obtain homomorphisms
~= * K _ * K ez * K
H*(BK ; R) --! H (B; R) - -! C (B; R) - - C (B ; R)
of cat op(K)-diagrams, whose components are quasi-isomorphisms on each Bff. Ta*
*king
limits in dga +Ryields homomorphisms
~= * K _ * K ez * K
(4.3) lim H*(BK ; R) --! limH (B; R) - -! limC (B; R) - - limC (B ; R).
Lemma 4.4. The leftmost homomorphism in (4.3)is an isomorphism in dga +R, and t*
*he
remaining two are quasi-isomorphisms.
Proof.The limit of the Künneth isomorphisms is automatically an isomorphism.
A diagram DK :cat op(K) ! dga +Ris fibrant whenever the projection onto the c*
*onstant
diagram 0 is a fibration. By (2.5)this occurs precisely when DK is fat, and the*
*refore holds
for H*(BK ; R) and C*(BK ; R) by Lemma 3.8; it follows for H*(B; R) K by the K*
*ünneth
isomorphism. So far as C*(B; R) K is concerned, we note that singular cochains *
*determine
a pair of diagram twins (C K , C K ) in dga +R. Both functors assign C*(B; R) f*
*fto the face
oe. The value of C K on pfi,ffis the projection C*(B; R) fi! C*(B; R) ff, and *
*the value
of CK on iff,fiis the inclusion C*(B; R) ff! C*(B; R) fi; these require the aug*
*mentation.
Hence C K is also fat, and C*(B; R) K is fibrant. So the remaining two homomo*
*rphisms
are objectwise equivalences of fibrant diagrams, and therefore induces weak equ*
*ivalences
of limits by [14].
Implicit in this proof is the remark that C*(Bn; R) is formal in dga +R, for *
*every n 1.
Lemma 4.5. The natural homomorphism g :C*(c(K); R) ! limC*(BK ; R) is a quasi-
isomorphism in dga +R.
Proof.The edge isomorphism hK of Corollary 3.14 factorises as
H(g) * K l * K
H(C*(c(K); R)) --! H(limC (B ; R)) -! limH (B ; R),
in dga +R, where l is induced by the compatible homomorphisms H(limC*(BK ; R)) !
H*(Bff; R).
Now let d be the differential on Cj(-; R) for every j 0, and define the cyc*
*le and
boundary functors Zj, Ij: top ! mod R as the kernel and image of d respectively*
*. They
determine diagram twins, and therefore fat functors ZK , IK :cat op(K) ! mod R.*
* Theo-
rem 3.12 then applies to confirm that limiZj(BK ; R) = limiIj(BK ; R) = 0 for a*
*ll i > 0
and j 0. It follows immediately that l is an isomorphism, and therefore that *
*H(g) is an
isomorphism, as sought.
We may now complete our analysis of C*(c(K); R).
Theorem 4.6. The differential graded R-algebras C*(c(K); R) is formal in dga +R.
12 DIETRICH NOTBOHM AND NIGEL RAY
Proof.Combining Corollary 3.14 with Lemmas 4.4 and 4.5 yields a zig-zag
g *
H*(c(K); R) -h! limH*(BK ; R) -! . .-. limC*(BK ; R) - C (c(K); R)
of quasi-isomorphisms, as required by (4.1).
Remark 4.7. The proof of Theorem 4.6 extends to exponential diagrams XK for wh*
*ich
C*(X; R) is formal in dga +Rand the Künneth isomorphism H*(XV ; R) ~= H*(X; R) V
holds. We replace _ in (4.2)by the corresponding zig-zag H*(X; R) -~! . .-.~ C**
*(X; R)
of quasi-isomorphisms, and repeat the remainder of the argument above.
5. Rational Formality
In our final section we turn to the rational case R = Q, and confirm the form*
*ality of
Sullivan's algebra of rational cochains on c(K) in the commutative setting. Thi*
*s involves
stricter conditions than those for general R, and has deeper topological conseq*
*uences;
in particular, it means that the rational homotopy type of c(K) is uniquely det*
*ermined
by K. In other words, the existence of an isomorphism H*(Y ; Q) ~= Q[K] implie*
*s that
there is a rational homotopy equivalence Y ' c(K), for any nilpotent space Y . *
*We refer
readers to Bousfield and Gugenheim [1] for details of the model category of dif*
*ferential
graded commutative Q-algebras, and to F'elix, Halperin and Thomas [10] for back*
*ground
information on rational homotopy theory.
We begin by replacing C*(X; R) with Sullivan's rational algebra APL(X) of pol*
*ynomial
forms[10]. The commutativity of the latter is crucial, and suggests we work in *
*the cate-
gory dgca +Qof differential graded commutative Q-algebras [1]. The existence of*
* a model
structure is assured by working over a field; as before, weak equivalences are *
*zig-zags
of quasi-isomorphisms, fibrations are epimorphisms, and cofibrations are define*
*d by the
appropriate lifting properties.
For each s 0, we write the differential algebra of rational polynomial form*
*s on the
standard s-simplex as rs(*). It is an object of dgca +Q. For each t 0, the *
*forms of
dimension t define a simplicial vector space ro(t) over Q, and ro(*) becomes a *
*simplicial
object in dgca +Q. So
A*(Yo) := sset(Yo, ro(*))
is also an object of dgca +Q, which is weakly equivalent to the normalised coch*
*ain complex
N*(Yo; Q). Then APL (X) is defined as A*(SoX), where So denotes the total sing*
*ular
complex functor sset ! top . The PL de Rham Theorem [1] asserts that the cohomo*
*logy
algebra H(APL (X)) is naturally isomorphic to H*(X; Q). As usual, we consider H*
**(X; Q)
to be an object of dgca +Qby investing it with the zero differential.
A differential graded commutative Q-algebras A* is formal in dgca +Qwhenever *
*there is
a zig-zag of quasi-isomorphisms
(5.1) H(A*) -~! . .-.~ A*
in dgca +Q. A topological space X is rationally formal whenever APL (X)) is fo*
*rmal in
dgca +Q. One of the basic results of rational homotopy theory states that a min*
*imal model
for a rationally formal space X may be constructed directly from its rational c*
*ohomology
HOMOTOPY TYPES, FORMALITY AND RATIONALISATION 13
algebra, and therefore that the rational homotopy type of X is uniquely determi*
*ned. Our
final goal in this section is to explain why c(K) is rationally formal. The pro*
*of is parallel to
that for general R, but the need to respect commutativity forces several change*
*s of detail.
We choose D to be APL in (3.1), creating diagram twins (APL (BK ), APL(BK )) *
*in dgca +Q.
As before, we form limits by working in dgmod +Q, and superimposing the induce*
*d multi-
plicative structure. Applying cohomology yields the twins (H*(BK ; Q), H*(BK ; *
*Q)), whose
value on each face oe is SQ(oe) in dgmod +Q. Both APL (BK ) and H*(BK ; Q) ar*
*e fat, by
Lemma 3.8.
Using the fact that H(APL (BV )) is isomorphic to SQ(V ), we choose cocycles *
*OEj in
APL (BV ) representing vj for every 1 j m. We may then define a homomorphism
(5.2) OE: H*(BV ; Q) -! APL(BV )
by OE(vj) = OEj, because APL (BV ) is commutative. Moreover, OE is a quasi-iso*
*morphism,
reflecting the well-known rational formality of the Eilenberg-Mac Lane space H(*
*QV ; 2).
By restriction, we interpret the OEj as cocycles in APL (Bff) for every face oe*
*. They then
represent vj in SQ(oe) when oe contains vj, and 0 otherwise. We obtain compatib*
*le quasi-
isomorphisms on each Bff, which combine to create a map
OE: H*(BK ; Q) -! APL(BK )
of catop(K)-diagrams in dgca +Q. It is an objectwise weak equivalence. Taking l*
*imits yields
a homomorphism
l(OE): lim(H*(BK ; Q) -! limAPL (BK )
of differential graded commutative algebras over Q.
Lemma 5.3. The homomorphism l(OE) is a quasi-isomorphism in dgca +Q.
Proof.Both diagrams are fat, and therefore fibrant by (2.5). So OE induces a we*
*ak equiva-
lence of limits.
Because the Bffare Eilenberg-Mac Lane spaces, it is convenient to complete ou*
*r proof of
Theorem 5.5 in terms of simplicial sets. We may then take advantage of the fact*
* that A*
converts colimits in sset to limits in dgca +Q, for reasons which are purely se*
*t-theoretic.
We denote the realisation functor sset ! top by | |. Given an arbitrary simpl*
*icial set
Yo, we have
(5.4) APL (|Yo|) := A*(So|Yo|) ~=A*(Yo),
where the second isomorphism is induced by the natural equivalence Yo ! So|Yo|)*
*. For
each face oe of K, we choose |Ho(Zff; 2)| as our model for Bff; it is well-poin*
*ted, by the
cofibration induced by the inclusion of the trivial subgroup {0} ! Zff. We writ*
*e HKo for
the corresponding diagram of simplicial sets, which takes the value Ho(Qff; 2) *
*on oe.
Theorem 5.5. The space c(K) is rationally formal.
Proof.Since realisation is left adjoint to So, it commutes with colimits. So we*
* may write
c(K) = colim|HKo| ~=| colimHKo|,
14 DIETRICH NOTBOHM AND NIGEL RAY
where the second colimit is taken in sset. Applying (5.4)yields
limAPL (BK ) ~=limA*(HKo) ~=A*(colimHKo) ~=APL (c(K)),
where the first and third isomorphisms define APL, and the second follows from *
*the property
of A* described above. Combining with Corollary 3.14 and Lemma 5.3 yields homom*
*or-
phisms
l(ffi) K ~=
H*(c(K); Q) -h! limH*(BK ; Q) --! limAPL (B ) -! APL(c(K)),
whose composition is a quasi-isomorphism. The result follows from (5.1).
Remark 5.6. By analogy with Remark 4.7, the proof of Theorem 5.5 extends to exp*
*o-
nential diagrams XK for which X is rationally formal; the Künneth isomorphism *
*holds
automatically, because we are working over Q. The product APL (X) V ! APL (XV*
* ) of
the projection maps is a quasi-isomorphism, and is natural with respect to proj*
*ection and
inclusion of coordinates. So we may replace OE in (5.2)by the corresponding zig*
*-zag
~= * V ~ ~ V ~= V
H*(XV ; Q) -! H (X; Q) - ! . .-. APL (X) - ! APL (X )
of quasi-isomorphisms, and proceed with the remainder of the argument above.
Theorem 5.5 confirms that a minimal Sullivan model for c(K) may be constructe*
*d di-
rectly from the Stanley-Reisner algebra of K. It consists of an acyclic fibrati*
*on
jK :(SQ(W K), d) -! H*(BK ; Q),
where W K is an appropriately graded set of generators (necessarily exterior in*
* odd dimen-
sions), and provides a cofibrant replacement for H*(BK ; Q) in dgca +Q. In gene*
*ral, W K is
not easy to describe, although special cases such as Example 5.8 below are stra*
*ightforward.
The properties of W K are linked to those of the loop space c(K), whose study *
*was begun
in [21]; we expect to return to this relationship elsewhere.
As described by Bousfield and Gugenheim [1], the fundamental result of ration*
*al homo-
topy theory is the Sullivan-de Rham equivalence
ho ssetQ ------!ho dgca +Q
of homotopy categories, which identifies homotopy classes of maps [c(K)0, c(L)0*
*] with ho-
motopy classes of morphisms [SQ(W L), SQ(W K)]. Since every object of dgca +Qis*
* fibrant,
it actually suffices to consider the homotopy classes [SQ(W L), H*(c(K); Q)]; o*
*f course
SQ(W L) cannot be substituted similarly, because H*(c(L); Q) is not usually cof*
*ibrant.
Nevertheless, the function
(5.7) [SQ(W L), H*(c(K); Q)] -! dgca +Q(H*(c(L); Q), H*(c(K); Q))
induced by taking cohomology is always a surjection, and it would be of interes*
*t to under-
stand its kernel.
Example 5.8. Let (~(k) : 1 k t) be a sequence of disjoint subsets of V , wh*
*ere ~(k)
has cardinality n(k), and define L to be the subcomplex of (V ) obtained by de*
*leting
all faces containing one or more of the ~(k). We write ~ := [k~(k), and |~| = *
*n. The
generating set W Lconsists of V in dimension 2, and elements w(k) in dimension *
*2n(k)-1,
HOMOTOPY TYPES, FORMALITY AND RATIONALISATION 15
for 1 k t; the differential is given by dvj = 0 for all j, and dw(k) = v~(k*
*). The fibration
jL identifies the vertices V in dimension 2, and annihilates every w(k). In thi*
*s situation,
any dgca +Q-morphism SQ(W L) ! H*(c(K); Q) is determined by its effect on V , b*
*ecause
the w(k) are odd dimensional. So the function (5.7)is bijective.
It follows that Aut ho(c(K)0) is isomorphic to the group of automorphisms Aut*
* (Q[K])
of the algebra Q[K] under pcomposition, and is therefore a subgroup of GL (Q, m*
*). It
contains all matrices of the form (LM 0), where L 2 GL (Q, m - n) acts on QV \~*
*, and
permutes the elements of ~. The permutations act on the elements of each indivi*
*dual ~(k),
and interchange those ~(k) which are of common cardinality.
We conjecture that every element of Aut(Q[K]) has the form (LM0 ).
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Department of Mathematics and Computer Science, University of Leicester, Univ*
*ersity
Road, Leicester LE1 7RH, England
E-mail address: dn8@mcs.le.ac.uk
Department of Mathematics, University of Manchester, Oxford Road, Manchester
M13 9PL, England
E-mail address: nige@ma.man.ac.uk