MODULI SPACES OF HOMOTOPY THEORY
DAVID BLANC
Abstract. The moduli spaces refered to are topological spaces whose path*
* compo-
nents parametrize homotopy types. Such objects have been studied in two s*
*eparate
contexts: rational homotopy types, in the work of several authors in the *
*late 1970's;
and general homotopy types, in the work of Dwyer-Kan and their collaborat*
*ors. We
here explain the two approaches, and show how they may be related to each*
* other.
1.Introduction
The concept of "moduli" for a mathematical object goes back to Riemann, who u*
*sed
it to describe a set of parameters determining the isomorphism class of a Riema*
*nn
surface of a given topological type. He also recognized that the set of all con*
*formal
equivalence classes of such surfaces can itself be given a complex structure, a*
*lthough
this was only made precise by Teichm"uller, Ahlfors, Bers, and others in the tw*
*entieth
century. Similarly, the family of birational equivalence classes of algebraic *
*curves
of each genus (over C, say) are parametrized by an algebraic variety of moduli,*
* as
Mumford showed, and there are analogues in higher dimensions.
If we fix an oriented surface S of genus g, a marked Riemann surface is a cho*
*ice
of an orientation-perserving diffeomorphism of S into a compact Riemann surface*
* X,
and the equivalence classes of such marked surfaces form Teichm"uller space Tg*
*, a
complex analytic variety. The mapping class group g of orientation preservi*
*ng
diffeomorphisms of S acts on Tg, and the quotient is isomorphic to the moduli
space Mg of isomorphism classes of complex structures on S - or equivalentl*
*y, of
smooth projective curves over C. Since the action of g is virtually free, an*
*d Tg is
contractible, Mg has the same rational homology as the classifying space B g.
Note that the set of all Riemann surfaces of a given genus can be obtained by
deforming a given conformal (or equivalently, complex) structure on a fixed ssa*
*mple
surface g - or the corresponding Fuchsian group . Such deformations are typic*
*ally
governed by an appropriate cohomology group.
Note also that there are several levels of possible structures on a compact s*
*urface:
topological, differentiable, real or complex analytic, hyperbolic, conformal, m*
*etric,
and so on, which define different types of moduli space. See [IT ] or [Su ] fo*
*r more
details and references on the classical theory.
1.1. Moduli spaces. Thus, there are a number of common themes in the moduli
problems arising in various areas of mathematics:
___________
Date: January 25, 2004. Revised: October 3, 2004.
1991 Mathematics Subject Classification. Primary 55P15; Secondary 55Q99, 55P*
*62, 18G30,
14J10.
This article is dedicated to the memory of Bob Brooks, colleague and friend.
1
2 DAVID BLANC
a) Finding a set of parameters to describe appropriate equivalence classes *
*of
objects in a certain geometric category. These may occur at varous level*
*s, and
we distinguish between discrete parameters (such as the genus of a surfa*
*ce)
and the finer (continuous) moduli.
b) Giving the set of such equivalence classes the structure of an object in*
* the
same category, which we may call a moduli space.
c) The deformation of a given structure, controlled by cohomology, as a mea*
*ns
of obtaining all other possible structures.
d) To pass between levels of structure, one can often quotient by a group a*
*ction.
As we shall see, all these themes will find expression in the context of homo*
*topy
theory. One aspect of the usual moduli spaces for which we have no analogue in *
*our
setting is the important role played by compactifications.
1.2. Moduli spaces in homotopy theory. The homotopy theory of the classical
moduli spaces of Riemann surfaces and the corresponding mapping class groups has
been studied extensively - see, for example, [Hr , MW , Ti]. However, this i*
*s not our
subject; we shall be concerned here rather with the analogues of such moduli sp*
*aces in
the category of topological spaces, answering to the general description given *
*above.
The simplest examples of this approach are provided by various mapping spaces,
starting with the loop space X of all pointed maps from the circle into a top*
*ological
space X. Here the set ss0 X of components is isomorphic to the fundamental gr*
*oup
of X, and the higher homotopy groups are again those of X (re-indexed). More
significantly, the components of space of maps from X to BO or BU, the classify*
*ing
spaces of the infinite orthogonal or unitary groups, correspond to equivalence *
*classes of
(real or complex) vector bundles over X, and the higher homotopy groups corresp*
*ond
to various reductions of structure, by Bott periodicity. Further examples of s*
*imple
"moduli spaces" of topological spaces are provided by the configuration spaces *
*of n
distinct points in a manifold M, which have also been studied extensively (see *
*[FH ]
for a comprehensive survey).
1.3. Moduli spaces and homotopy types. Our focus here will be more specifi-
cally on moduli spaces of all homotopy types (suitably interpreted), which have*
* been
investigated from at least two points of view over the past twenty-five years:
o In work of Lemaire-Sigrist, F'elix and Halperin-Stasheff, simply-connect*
*ed ra-
tional homotopy types correspond to the components of a certain (infinite
dimensional) algebraic variety over Q - or alternatively, the quotient*
* of such
a variety by appropriate group actions (Sec. 2).
o The "classification complex" approach of Dwyer-Kan yields a space whose
components parametrize the homotopy types of all CW complexes (Sec. 6).
Of course, neither statement has much content, as stated; the point is that b*
*oth
approaches offer (surprisingly similar) tools for inductively analyzing the col*
*lection
of all relevant homotopy types:
In each case we begin with a coarse classification of homotopy types by algeb*
*raic
invariants, such as the cohomology algebra H of a space, or the corresponding s*
*truc-
ture on homotopy groups, called a -algebra (which reduces rationally to a grad*
*ed
Lie algebra over Q - see Sec. 3). One might also refine this initial classifica*
*tion using
both homotopy and cohomology groups (Sec. 5).
MODULI SPACES OF HOMOTOPY THEORY 3
(a) For each H, we have homological classification of spaces with this cohom*
*ology
algebra. This is usually presented as an obstruction theory, stated in *
*terms
of (algebraically defined) cohomology groups of H. It can be reinterpret*
*ed as
expressing the moduli space as the limit of a tower of fibrations (Sec. *
*7), with
successive fibers described in terms of these cohomology groups (Sec. 8).
(b) In the rational case, we have a deformation theory approach, which descr*
*ibes
all homotopy types with H*(X, Q) ~= H by perturbing a canonical model.
In fact, this was the main focus of the rational homotopy theory approach
mentioned above.
Our main goal here is to bring out the connection between the integral and ra*
*tio-
nal cases. From the first point of view, this can be made explicit at the alge*
*braic
level, comparing the two cohomology theories through appropriate spectral seque*
*nces
(Sec. 9).
On the other hand, the rational deformation theory has no integral analogue, *
*since
it uses differential graded models, which exist only for simply-connected ratio*
*nal
homotopy types. However, it turns out that the deformations can be described ge*
*o-
metrically in terms of higher homotopy operations (Sec. 4), and these do have i*
*ntegral
versions.
1.4. Remark. One could - with some justice - argue that the analogy between
the rational and integral "spaces of homotopy types" we describe here and class*
*ical
moduli spaces is rather tenuous (although it is merely incidental to the point *
*we want
to make). One aspect in particular for which we can offer no real analogue is *
*the
fact that the classical moduli spaces have the same structure (algebraic or ana*
*lytic
variety, etc.) as the objects being classified.
In some sense, however, a homotopy type is just the collection of all its hom*
*otopy
invariants, suitably interpreted (see [Q1 , I,4.9]). In fact, one goal of homot*
*opy the-
ory is to produce a manageable set of such invariants, sufficient for distingui*
*shing
between type. A starting point for such a set of invariants is the Postnikov sy*
*stem of
a space (and its invariants); so it is perhaps fitting that our analysis of the*
* space of
all homotopy types is carried out by means of its Postnikov system, and the ass*
*oci-
ated collections of homotopy invariants (higher homotopy operations, or cohomol*
*ogy
classes) which appear in our obstruction theory. Thus one could choose to view *
*these
collections of invariants themselves - rather than the space of homotopy type*
*s -
as the true "moduli object". This is certainly the best way to understand the *
*con-
nections between the rational and integral cases, and it also might make (somew*
*hat
far-fetched) sense of the claim that the moduli object is itself of the same ki*
*nd as the
things it classifies. We leave the philosophically-inclined reader to pursue th*
*is line of
thought at his or her discretion.
1.5. Acknowledgements. I would like to thank the referee for his or her comment*
*s,
and Bill Dwyer, Paul Goerss, Jim Stasheff, and Yves F'elix for helpful conversa*
*tions
and remarks on various aspects of the subject.
2. Rational homotopy and deformation theory
We first describe the deformation-theoretic approach to rational homotopy typ*
*es,
starting with some basic background material:
4 DAVID BLANC
2.1. Rational homotopy. Rational homotopy theory deals with rational algebraic
invariants of homotopy types - that is, it disregards all torsion in the homo*
*logy and
homotopy groups. Quillen and Sullivan, respectively, proposed two main algebra*
*ic
models for the rational homotopy type of a simply-connected space X:
(i)A differential graded-commutative algebra, or DGA (A*X, d), where (Ai*
*X)1i=0
is a graded-commutative algebra over Q, equipped with a differential d of
degree +1 (which is a graded derivation with respect to the product),
and H*(AX , d) ~=H*(X, Q). Since X is simply-connected, we may assume
(A*X, d), is, too - that is, A0X= Q, A1X= 0.
(ii)A differential graded Lie algebra, or DGL (LX , @), with LX = (LXi)1*
*i=1a
positively-graded Lie algebra over Q, and @ a graded Lie derivation of d*
*egree
-1, such that Hi(LX , @) ~=ssi X Q for i 1, with the Samelson products
as Lie brackets.
For more information see the survey [FHT ], and the original sources [Q2 , S*
*].
2.2. Definition. A DGA of the form ( V, d), where V is the free graded-
commutative algebra generated by the graded vector space V = (V i)1i=0 is cal*
*led
cofibrant. A cofibrant DGA ( V, d) such that Im (d) + V . + V (where *
*+ V
is the sub-algebra of elements in positive degree), is called a minimal model.
A DGA or DGL map which induces an isomorphism in (co)homology is called a
quasi-isomorphism.
2.3. Proposition (see [BL ]). Any simply-connected DGA A has a minimal model wi*
*th
a quasi-isomorphism q : ( V, d) ! (A, dA ), and this is unique up to isomorph*
*ism.
One can also define cofibrant and minimal models for DGLs, and show that they
have similar properties (cf. [FHT , x21]).
2.4. Remark. Both the category DGA 1 of simply-connected DGAs, and the catego*
*ry
DGL 0 of connected DGLs have model category structures (see [Q1 ]); thus each *
*has a
concept of homotopy, and as Quillen showed, the corresponding homotopy categori*
*es
hoDGA 1 and ho DGL 0 are both equivalent to the rational homotopy category
hoT1Q of simply-connected topological spaces (cf. [Q2 ]). The weak equivalenc*
*es in
T1Q correspond of course to quasi-isomorphisms for DGAs or DGLs.
Note that every simply-connected graded-commutative algebra over Q is realiza*
*ble
as the rational cohomology ring of some (1-connected) space, and similarly, eve*
*ry
positively graded Lie algebra is realizable as the rational homotopy groups of *
*a space.
2.5. The space of rational homotopy types. We may therefore identify the collec-
Q
tion M = MT1 of all rational weak homotopy types with the set of weak homotopy
types of 1-connected DGAs (over Q) - i.e., equivalence classes under the equiva*
*lence
relation generated by the quasi-isomorphisms - and similarly for DGLs. Moreov*
*er,
there is a coarse classification of such types, with the discrete parametrizati*
*on pro-
vided by the cohomology rings; we denote the sub-collection of M consisting of *
*all
DGAs with a given graded cohomology ring H by MH . This has a distinguished
element:
MODULI SPACES OF HOMOTOPY THEORY 5
2.6. Definition. A DGA A is called formal if it is quasi-isomorphic to (H*A, 0)*
* (the
cohomology ring of A thought of as a DGA with zero differential). Dually, a DGL*
* L
is coformal if it is quasi-isomorphic to (H*L, 0).
MH was studied by Jean-Michel Lemaire and Fran,cois Sigrist in [LS ], by Yv*
*es
F'elix in [F2 ], and independently by Steve Halperin and Jim Stasheff in [HS ].*
* They
showed that every rational homotopy type in MH can be obtained by suitable
deformations of a fixed model of the unique formal object:
2.7. Definition. The bigraded model (B, d) for H is the minimal model B = Z
for (H, 0) of Proposition 2.3, where Z = Z**, the graded vector space of genera*
*tors,
is equipped with an additional homological grading (indicated by the lower inde*
*x).
The differential takes Zin to ( n-1k=1Zk)i+1, so one still has a homologi*
*cal grad-
ing on the cohomology groups of B. The vector spaces Z*n are defined (almost
canonically) by induction on n, so that at the n-th stage we kill the cohomolog*
*y in
homological dimension k for 0 < k < n. Z*, d) is essentially the Tate-Joze*
*fiak
resolution of a graded commutative algebra H (cf. [Ta , J]) - i.e., a minimal "*
*cellular"
resolution. See [HS , x3] for the details.
The deformation consists of a perturbation of the differential d to D = d+D0 *
*- of
course, this is done in such a way that the cohomology is unchanged, and D O D*
* = 0.
We can no longer expect D to respect both gradings, but if we think of B as bei*
*ng
only filtered by FnB := nk-0( Z*)k, then the perturbation is such that D0 t*
*akes
Zn into Fn-2B.
2.8. Theorem (cf. [HS , Theorem 4.4]). If (A, dA ) is any DGA with H*(A, dA ) ~*
*=H,
one can deform the bigraded model (B, d) for (H, 0) into a filtered model *
*(B, D)
quasi-isomorphic to (A, dA ), and this is unique up to isomorphism.
2.9. The variety of deformations. The deformations D of d are determined by
successive choices of linear transformations D0n: Zn ! Fn-2 Z* x (tn, . .,.t1) in n+1(u, v*
*) x I ;
then the relation ~ is generated by
f1 Ot1f2. .f.nOtnfn+1 ~ f1 Ot1. .O.ti-1(fifi+1) Oti+1. .O.tnOfn+1 if ti*
*= 0
for 1 i n, where (fifi+1) denotes fi composed with fi+1.
The categorial composition in W is given by the concatenation:
(f1 Ot1. .O.tlfl+1) O (g1 Ou1. . .Ouk gk+1) := (f1 Ot1. .O.tlfl+1O1 g1 Ou1. . *
*.Ouk gk+1).
We write P := W (vinit, vf) .
4.3. Definition. The basis category bW for a lattice is defined to be the*
* cu-
bical subcategory of W with the same objects, and with morphisms given by
bW (u, v) := W (u, v) if (u, v) 6= (vinit, vfin), while
[
bW (vinit, vfin) := {ff O fi | fi 2 W (vinit, w), ff 2 W (w, vfin), vini*
*t6=,w 6= vfin}
so that bP := bW (vinit, vfin) consists of all decomposable morphisms.
4.4. Fact ([BM , Proposition 2.15]). For any lattice , W (vinit, vfin) is *
*isomorphic
to the cone on its basis, with vertex corresponding to the unique maximal 1-cha*
*in.
We can use the W -construction to define a higher homotopy operations, as fol*
*lows:
4.5. Definition. Given A : ! ho T* for a lattice as above, the correspond*
*ing
higher order homotopy operation is the subset <> [A(vinit) o bP, A(vfin)*
*]hoT* of
the homotopy equivalence classes of maps
CA|bW (vinit,vfin): bW (vinit, vfin) = bP -! T*(A(vinit), A(vfin))
induced by all possible continuous functors CA : bW ! T* such that ss O CA*
* =
AO("|bW ), where the half-smash is defined X oK := (X xK)=({*}xK) = X ^K+ .
<> is said to vanish if it contains the homotopy class of a constant map *
* bP -!
T*(A(vinit), A(vfin)).
Given a lattice , a rectification of a functor A : ! ho T* is a functor F *
*: ! T*
with ssOF naturally isomorphic to A (where ss : T* ! ho T* is the obvious proj*
*ection
functor).
4.6. Proposition ([BM , Thm. 3.8]). A has a rectification if and only if the h*
*omotopy
operation <> vanishes (and in particular, is defined)
With this at hand, we can now reformulate the deformation theory for the bigr*
*aded
DGL model for (P, 0) in the following form:
10 DAVID BLANC
4.7. Theorem ([B6 , Thm. 7.14]). For each connected graded Lie algebra P of fin*
*ite
type, there is a tree TP , with each node ff indexed by a cofibrant DGL Lff *
*such
that H*Lff~= P , starting with L0 ' (P, 0) at the root 0. For each node ff*
* there
is an integer nff> 0 such that Lff agrees with its successors in degrees n*
*ff,
with n0 = 1, so that the sequential colimit L(1) = colimkL(k) along any bran*
*ch
is well defined, and for any rational homotopy type in MP there exists such a*
* tree.
Furthermore, for each two immediate successors fi, fl of a node ff, there is *
*a higher
homotopy operation <> [Sn, L] which vanishes for L = Lfi, but not*
* for
L = Lfl (or conversely).
Thus we have reformulated the deformation of the bigraded model as an obstruc*
*tion
theory for distinguishing between the various realizations of a given rational *
*graded
Lie algebra P in terms of higher homotopy operations. This can also be done int*
*egrally
(cf. [B3 ]), and one could identify these operations with appropriate cohomolog*
*y classes
in the dual version of the Halperin-Stasheff obstructions (see [B4 ]).
4.8. Example of a DGL deformation. We shall not explain how the higher oper-
ations <> are defined, since the construction is rather technical; se*
*e [B6 , x4] for
the details. Instead, the reader might find the following example instructive:
Consider the graded Lie algebra P = L=I, where L is the f*
*ree
graded Lie algebra on a graded set of generators T*, (with the subscripts ind*
*icating
the dimension), and I is here the Lie ideal generated by [a, a] and [[c, a],*
* [b, a]].
Step I: The minimal model for the coformal DGL (P, 0) 2 DGL 0 is (B, @B *
*),
where B in dimensions 7 is Lz7, with @B (x) = [a*
*, a],
@B (y) = 3[x, a], @B (w) = [[c, a], [b, a]], and @B (z) = 4[y, a] + 3[x, x].*
* The bigraded
model A** is obtained from B by introducing an additional (homological) gradin*
*g:
a, b 2 A0,1, c 2 A0,2, x 2 A1,3, w 2 A1,6, y 2 A2,5, z 2 A3,7, and so on.
Step II: The free simplicial DGL resolution Co ! L = (L, 0) may be described
in homological dimensions 3 as follows:
(1) C(0)0is the DGL coproduct of S1(a(0))= S1(), S1(b(0))= S1(**), an*
*d S2(c(0))=
S2().
(2) C(0)1= S2(x(0))qS6(w(0)), where x(0)= <[****, ]> and w(0)= <[[, *
*], [****,>****]].
(3) C(0)2consists of S5(y(0)), where y(0)= <3[<[, ]>, <>]>.
(4) C(0)3consists of S6(z(0)), where
z(0)= <4[<3[<[, ]>, <>]>, <<>>] + 6[<[<>, <>]>, <<[<*
*a>,>]>>].
In analogy with the DGL sphere of x3.3, the DGL n-disk Dn(x)is the free grad*
*ed
Lie algebra on two generators: x in dimension n and @(x) in dimension n - 1. *
*With
this notation, for C(1)owe need in addition:
(1) D3(x(1)),! C(1)0with @W (x(1)) = d1(x(0)) = <[a, a]>.
(2) D6(y(1)),! C(1)1with y(1)= <3[, ]> and @W (y(1)) = d2(y(0)) = <3[<*
*[a, a]>, ]>.
MODULI SPACES OF HOMOTOPY THEORY 11
(3) D7(z(1)),! C(1)2with z(1)= <4[<3[, ]>, <>] + 6[<[, ]>,><*
*>]and
@W (z(1)) = d3(z(0)) = <4[<3[<[a, a]>, ]>, <>] + 6[<[, ]>,><*
*<[a,.a]>>]
For C(2)owe need in addition
(1) D7(y(2)),! C(2)0with @W (y(2)) = d1(y(1)) = <3[x, a]>.
(2) D8(z(2)),! C(2)1 with z(2) = <4[, ] + 3[, ]> and @W *
*(z(2)) =
d2(z(1)) =
<4[<3[x, a]>, ] + 6[<[a, a]>,>].
For C(3)owe must add D9(z(3)),! C(3)0with @W (z(1)) = d1(z(2)) = <4[y, a] + 3*
*[x,>x].
Step III: So far, we have constructed the DGL simplicial resolution of the co*
*formal
DGL (B, @B ) ' (P, 0). Now we consider a different DGL having the same homolo*
*gy
(i.e., a non-weakly equivalent topological space with the same rational homotopy
groups):
Consider the DGL (B0, @B0) where B0 := L.,. .is a*
* free Lie
algebra with @B0(x) = [a, a], @B0(y) = 3[x, a] - [[b, a], c], @B0(z) = 4[y, *
*a] + 3[x, x],
and so on.
Here P ~= H*(B) = L=<[a, a], [[c, a],>[b,,a]]so the bigraded mod*
*el for
(P, 0) is (A**, @B ) of Step II above, and the filtered model is obtained f*
*rom it by
setting D(y) = 3[x, a] - [[b, a], c] and D(z) = 4[y, a] + 3[x, x] - 4w + 2[[*
*x, b], c].
The corresponding free simplicial DGL resolution is obtained from Co of Ste*
*p II
by making the following changes:
(1) Set y(1):= <3[, ] - [[****, ****],>], with
@W (y(1)) = d2(y(0)) = <3[<[a, a]>, ]>
as before (but now @W (y(2)) = <3[x, a] - [[b, a],>c], of course).
(2) Set
z(1) :=<4[<3[, ]>, <>] + 6[<[, ]>, <>]
- 4<[[, ], [****,>****]]+ 2[[<[, ]>, <****>], <>*
*]>
(with @W (z(1)) unchanged).
(3) Set z(2):= <4[, ****] + 6[, ] - 4 + 2[[, ****], ]>, wi*
*th
@W (z(2)) =d3(z(1)) = <4[<3[x, a] - [[b, a],>c], ****]
+ 6[<[a, a]>, ] - 4<[[c, a], [b,>a]]+ 2[[<[a, a]>, ****],*
* ]>
(4) Finally, @W (z(3)) = <4[y, a] + 3[x, x] - 4w + 2[[x, b],>c].
Step IV: In order to define the higher homotopy operations which distinguish *
* B0
from B, observe that the required half-smashed are defined as follows:
S2(x)o D[1] = (L, @0), where X2 = {(x, (d0)), (x, (d1))}, X3 = {(x, (*
*Id))},
and @0(x, (Id)) = (x, (d0)) - (x, (d0)).
12 DAVID BLANC
Similarly, S3(y)oD[2] = (L, @0), where Y3 = {(y, (d0d1)), (y, (d0d2)), (y*
*, (d1d2))},
Y4 = {(y, (d0)), (y, (d1)), (y, (d2))}, and Y5 = {(y, (Id))}, with @0*
*(y, (Id)) =
-(y, (d0)) + (y, (d1)) - (y, (d2)), @0(y, (d0)) = -(y, (d0d1)) + (y, (d0d2)), *
* @0(y, (d1)) =
-(y, (d0d1)) + (y, (d1d2)), and @0(y, (d2)) = -(y, (d0d2)) + (y, (d1d2)).
Step V: For the DGL B0 of Step III, with Co as in Step II, we define h0 =*
* f0 :
C0,*! B0 by setting f0(****) = a, f0(****) = b, f0() = c, and f0 = 0 fo*
*r all
other disks in C0,* (so for example f0(<[a, a]>) = 0).
Thus on S2(x(0))o D[1] we have h1(x(0), (d0)) = [a, a] and h1(x(0), (d1)*
*) = 0, (see
[B6 , Def. 4.7]), so we must choose h1(x(0), (Id)) = x 2 B03.
Now on S3(y(0))oD[2] we have h2(y(0), (d0d1)) = h2(y(0), (d0d2)) = h2(y(0), (*
*d1d2)) =
0, and h2(y(0), (d0)) = (h1Od0)(y(0), (d0) = 3[x, a], while h2(y(0), (d1)) = h2*
*(y(0), (d2)) =
0.
Thus in Definition 4.5 we shall be interested in a lattice corresponding to*
* the face
maps of a simplicial object, with bW ~=@D[2]. We have just defined a contin*
*uous
functor CA : bW ! T* for Co, as required there, and the resulting seconda*
*ry
operation is <<2, y(0)>> = {<3[x, a]>]} H4B0 (remember that the homology of a*
* DGL
corresponds to the rational homotopy groups of the associated space). Since 3*
*[x, a]
does not bound in B0, <<2>> does not vanish, and we have found an obstructio*
*n to
the coformality of B0.
Unfortunately, DGL higher homotopy operations such as <<2, y(0)>> are unsat*
*is-
factory, in as much as one cannot translate them canonically into integral homo*
*topy
operations. One way to avoid this difficulty is to use more general non-associ*
*ative
algebras, rather than Lie algebras, as our basic models. Thus, consider the cat*
*egory
DGN of non-associative differential graded algebras. A DGN whose homology ha*
*p-
pens to be a graded Lie algebra will be called a Jacobi algebra. Every DGL has *
*a free
Jacobi model, and these can be used to resolve any DGL in sDGN (see [B6 , x*
*7]).
Step VI: Consider a Jacobi model G for the coformal DGL P of Step I. We
construct the minimal Jacobi resolution Jo ! G corresponding to the bigraded
model for L (embedded in the canonical Jacobi resolution Uo ! G) by modifying
the free resolution Co ! L of Step II, as follows:
For each n 0 define the Jacobi algebras J(0)n(n = 0, 1, . .).to be the copr*
*oducts
J(0)0= S1(****)q S1(****)q S2(), J(0)1= S2(x(0))q S6(w(0)), J(0)2= S5(y(0)),*
* J(0)3= S6(z(0)), and
so on.
The face maps di (i = 0, 1, . .,.n) are defined as follows: write 2 F*
* (B) for
the generator corresponding to an element x 2 B0*, then recursively a typical*
* DGL
generator for Wn = Wn,* (in the canonical Stover resolution Wo(B), defined in*
* [B6 ,
x5.3]) is , for ff 2 Wn-1, so an element of Wn is a sum of iterated Lie pr*
*oducts
of elements of B0*, arranged within n + 1 nested pairs of brackets <<. .>.*
*>. With
this notation, the i-th face map of Wo is "omit i-th pair of brackets", and *
*the j-th
degeneracy map is "repeat j-th pair of brackets". We assume the bracket operat*
*ion
<- > is linear - i.e., that = ff + fi for ff, fi 2 Q a*
*nd x, y 2 B.
MODULI SPACES OF HOMOTOPY THEORY 13
1. For y(0)= <3[<[****, ]>, <>]> 2 J(0)23, we have d0(y(0)) = 3[<[, ]>, <>] =
3[x(0), a(0)] and d2(y(0)) = <3[<[a, a]>, ]>, while d1(y(0)) = <3[[, *
*], ]>. This
no longer vanishes as in Step II, since the Jacobi identity does not hold in D*
*GN ,
but we have an element y(1;1):= <~3( )> 2 J(1)14 (in the notatio*
*n of
[B6 , Example 7.8], with @(y(1;1)) = d1(y(0)).
On the other hand, we also have an element y(1;2):= <3[, ]> 2 J(1)14 *
*(which
we denoted simply by y(1) in Step II, with @(y(1;2)) = d2(y(0)). The simpl*
*icial
identity d1d2 = d1d1 implies that d1(y(1;2)) - d1(y(1;1)) = <3[x, a] - ~3(a *
* a a)>
is a @J-cycle, so we have y(2;1,2)2 J(2)05 with @J(y(2;1,2)) = <3[x, a] - ~*
*3(a a a)>.
2. For z(0)= <4[<3[<[, ]>, <>]>, <<>>] + 6[<[<>, <>]>, <<[*
*,>]>>]in
J(0)34:
(a) z(1;1):= <~3(x(0) <> <>)>, with
d1(z(0)) = <6(2[[x(0), <>], <>], +[[<>, <>],>x(0)])= @J(z(1*
*;1)),
(b) z(1;2):= <4[~3( ), <>]>, with
d2(z(0)) = <4[[[a(0), a(0)], a(0)],><>]= @J(z(1;2)) since [x(0), x(0*
*)] = 0).
(c) z(1;3)= <4[<3[<[a, a]>, ]>, <>] + 6[<[, ]>,><<[a,2a]>>]J(1)2*
*5, with
d3(z(0)) = @J(z(1;3)).
3. Next, in J(2)o:
(a) The simplicial identity d1d1 = d1d2 implies that
d1(z(1;1)) - d1(z(1;2)) = <4[~3( ), ] - 6~3([, ] *
* )>
is a @J-cycle - and indeed we have z(2;1,2):= <~4( )> 2
J(2)26 with @(z(2;1,2)) = d1(z(1;1)) - d1(z(1;2)).
(b) Similarly,
d1(z(1;3)) - d2(z(1;1)) = <12[[, ], ] + 6[[, ], ] - 6~3(<[a, *
*a]> > )
is a @J-cycle, so we have z(2;1,3):= <6~3( )> 2 J(2)26 w*
*ith
@(z(2;1,3)) = d1(z(1;3)) - d2(z(1;1)).
(c) d2(z(1;3))-d2(z(1;2)) = <4[<3[x, a]>, ] + 6[<[a, a]>, ] - 4[<~3(a *
* a >a)>, ]
is a @J-cycle, hit by z(2;2,3):= <4[, ] + 3[, ]> 2 J(2)26.
4. Finally, we have
(a) For d1z(2;1,2)= <~4(a a a a)> we have
@J(d1z(2;1,2)) = d1d2(z(1;2)) - d1d2(z(1;1)) = <4[~3(a a a), a] - 6~3([a, a*
*] a >a)
(b) For d1z(2;1,3)= <6~3(x a a)> we have
@J(d1z(2;1,3)) = d1d2(z(1;1))-d1d1(z(1;3)) = <6~3([a, a] a a) - 12[[x, a], a*
*] - 6[[a,>a], x]
(c) For d1z(2;2,3)= <4[y, a] + 3[x,>a]we have
@J(d1z(2;2,3)) = d1d2(z(1;2))-d1d1(z(1;3)) = <4[~3(a a a), a] - 12[[x, a], *
*a] - 6[[a,>a],,x]
14 DAVID BLANC
So there is an element z(3;1,2,3)2 J(3)07 with
@J(z(3;1,2,3)) = <~4(a a a a) - 6~3(x a a) + 4[y, a] + 3[x,>a].
This defines the second-order homotopy operation we are interested in (which al*
*so
has an integral version).
5.Refined moduli spaces
Since neither the cohomolgy algebra nor the homotopy Lie algebra of a rational
space determine the other, we can refine the coarse partition of the rational m*
*oduli
space M by specifying both:
5.1. Definition. For each 1-connected graded-commutative algebra H and graded L*
*ie
algebra P over Q, let MH,P denote the collection of all simply-connected rat*
*ional
homotopy types of spaces X with H*(X; Q) ~=H and ss*(X) Q ~=P .
Note that we cannot simply identify this as a quotient variety with MH \ MP *
* in
any natural way, since points in MH are represented by filtered DGA models, w*
*hile
those in MP are represented by filtered DGA models.
The set MH,P may, of course, be empty; but Lemaire and Sigrist have shown that
it can also be infinite (see [LS , x3]). In order to analyze this refined modul*
*i space, we
shall need one additional ingredient.
5.2. The Quillen equivalences. The fact that ho DGL 0 and ho DGA 1 are equiv-
alent to each other, leads us to expect a direct algebraic relationship between*
* the cor-
responding model categories, which in fact exists, and may be described as foll*
*ows:
For simplicity, restrict attention to simply-connected spaces of finite type.*
* We may
therefore assume that AiX is finite dimensional for each i 0 (and of cour*
*se
A0X= Q, A1X= 0). Taking the vector space dual of A*X, we obtain a 1-connect*
*ed
differential graded-cocommutative coalgebra (CX*, ffi), whose homology is H*(X;*
* Q),
with the usual coalgebra structure (dual to the ring structure in cohomology). *
* Let
DGC denote the category of such coalgebras.
Quillen defined a pair of adjoint functors
C
DGL 0 L DGC
as follows:
I: Given a DGC (C*, d), let L(C*) := (Prim ( C*), @) denote the graded L*
*ie
algebra of primitives in the cobar construction of C*, constructed as follows:
If C* ~= Q ~C* (where C~*= C 2, in our case), and -1V is the graded
vector space V shifted downwards (so that ( -1V )i:= Vi+1, with oe-1v $ v),*
* let
C* := T ( -1C~*) denote the tensor algebra on -1C~* with
1 X |c0| -1 0 -1 00
@(oe-1x) := -oe-1(dx) + __ (-1) i[oe ci, oe ci] ,
2 i
P
where ~ c := ic0i c00iis the (reduced) comultiplication in ~C*, and [ , *
*] is the
commutator in T ( -1C~*).
MODULI SPACES OF HOMOTOPY THEORY 15
II: Given a DGL (L*, @), let C(L*, d) be the DGC ( ( L*), d), where *
*L*
is L* shifted upwards, V denotes the cofree graded coalgebra cogenerated by*
* the
graded vector space V , and the coderivation @ is defined by d = d0- d00, wh*
*ere
(5.1)
Xn P
d0(oex1 ^ . .^.oexn) := (-1)i+ j***, we hav*
*e the
variety V***of all DGC maps OE : (B^, ^@) ! (A", "d) between them. Sinc*
*e both
are cofree, these are determined by maps of graded vector spaces. The condition
that OE be a quasi-isomorphism translates into a series of rank conditions, so *
*that
we get a semi-algebraic set parametrizing all such pairs which are equipped wit*
*h a
quasi-isomorphism between them - and in particular, the requisite homology H and
homotopy groups P .
6. Nerves and moduli spaces
Since ordinary (integral) homotopy types do not have any known differential g*
*raded
models, there is no hope of generalizing the deformation approach of sections 2*
*-5 to
cover them, too. However, in [DK3 ], Dwyer and Kan suggested an approach to
such "moduli problems" based on the concept of nerves, which has proved useful
conceptually in a number of contexts.
6.1. Definition. The nerve N C of a small category C is a simplicial set whose
k-simplices are the sequences of k composable arrows in C, with di="delete i-th
object and compose", sj="insert identity arrow after j-th object". The geomet*
*ric
realization of the nerve is called the classifying space of C, written BC := |*
*N C|.
The nerve was originally defined by Segal in [Se], based on ideas of Grothend*
*ieck;
Quillen, in [Q3 ] helped clarify the close connection between nerves and homoto*
*py
theory, as evinced in the following properties:
1) The functor N : Cat ! S takes natural transformations to homotopies (c*
*f.
[Q3 , x2, Prop. 2]), so that
16 DAVID BLANC
2) if a functor F has a left or right adjoint, then N F is a homotopy equi*
*valence;
more general conditions are provided by [Q3 , x2, Thm. A]). If C has an *
*initial
or final object, then N C ' *.
3) The nerve of a functor is not often a fibration; conditions when N F f*
*its into
a quasi-fibration sequence are provided by [Q3 , x2, Thm. B].
4) Similarly, N C is not usually a Kan complex, unless C is a groupoid (*
*cf.
[GJ , I, Lemma 3.5]).
6.2. Definition. A classification complex in a model category C (see [DK3 , x2*
*.1]) is
the nerve of some subcategory D of C, all of whose maps are weak equivalences, *
*and
which includes all weak equivalences whose source or target is in D. The catego*
*ry D
is only required to be homotopically small (cf. [DK1 , x2.2]) - that is, N *
*D has a
set of components, and its homotopy groups (at each vertex) are small.
This construction has two main properies:
a) The components of N D are in one-to-one correspondence with the weak
homotopy types of D in C.
b) The component of N D corresponding to an object X 2 C is weakly
homotopy equivalent to the classifying space of the monoid haut X of s*
*elf
weak equivalences of X (cf. [DK3 , Prop. 2.3]).
6.3. Remark. Taking the model category C := T0 of connected pointed spaces,
with D = WC the subcategory of all weak homotopy equivalences in C, MC :=
N WC is the candidate suggested by the Dwyer-Kan approach for the moduli space *
*of
pointed homotopy types. There is also an unpointed version, of course; on the o*
*ther
hand, allowing for non-connected spaces merely complicates the combinatorics of*
* M,
without adding any new information.
One might ask in what sense this qualifies as a moduli space (aside from havi*
*ng the
right components). Even though the analogy with the classical examples mentioned
in the introduction is not clear-cut, note that MT0 as defined using the nerve *
*(which
Q
is natural choice for a space to associate to a category) is related to MT1 of*
* x2.5, and
the latter does exhibit many of the attributes listed in x1.1. Moreover, as we *
*shall see
in the next section, this construction can be used to interpret the obstruction*
* theory
of x2.10, as well as its integral analogue, and also to describe M as the limit*
* of a
tower of fibrations, which give increasingly accurate approximations to M.
Finally, the monoid hautX of self equivalences corresponds to the mapping c*
*lass
group of self diffeomorphisms of a surface (Sec. 1), whose classifying space is*
* closely
related (or even homotopy equivalent) to the moduli space.
6.4. The model categories. In order to provide a uniform treatment in different
model categories, and in particular to allow for the comparisons mentioned in x*
*3.4,
it is useful to consider a resolution model category structure (x3.2) on a cate*
*gory
C = sE of simplicial objects over another category E. In fact, one can do this*
* at two
different levels, and in some sense the comparison between these is the heart o*
*f this
approach:
I. The topological level - where E can take several forms:
(a) The category T0 of connected pointed spaces.
MODULI SPACES OF HOMOTOPY THEORY 17
(b) The subcategory T0Q of pointed connected spaces having rational univer*
*sal
covers (but arbitrary fundamental group).
Note that one can approximate any pointed connected space X 2 T0 by
its fibrewise rationalization X0Q (cf. [BK , I, 8.2]), which lies in *
*T0Q: if
"X! X ! B(ss1X) is the universal covering space fibration for X, then *
*X0Q
fits into a functorial fibration sequence fXQ! X0Q! B(ss1X), in which *
* fXQ
is the usual rationalization of the universal cover. However, algebraic*
* DGL,
DGA, or DGC models do not generally extend to this case (unless ss1X is
finite and acts nilpotently on the higher groups - see, e.g., [Tr]).
(c) The subcategory T1Q of 1-connected pointed rational spaces (in T0Q). *
*This
can be replaced by the Quillen equivalent model categories DGL 0 or DG*
*A 1.
(d) Other variants are possible - for instance, we could consider functor ca*
*tegories
over E - that is, diagrams D ! E for a fixed small category D (see [*
*BJT ]).
II. The algebraic level - where E is correspondingly:
(a) The category -Alg of -algebras - that is, positively graded groups*
* G*,
abelian in dimensions 2, equipped with an action of the primary homot*
*opy
operations (Whitehead products, compositions, and G1-action) satisfying *
*the
usual identities (see [B1 ] for a more explicit definition).
(b) The subcategory -AlgQ of rational -algebras, which are (positively) g*
*raded
Lie algebras equipped with a "fundamental group action" of an arbitrary *
*group
ss, satisfying the usual identities (see [Hi]).
(c) When ss = 0, we obtain the subcategory -AlgQ1 ~= grLie of simply-
connected rational -algebras, which are just graded Lie algebras over Q.
(d) There is also a concept of -algebras for arbitrary diagrams (again, see*
* [BJT ]).
7.Approximating classification complexes
The classification complex of a model category C, as defined in x6.2, may app*
*ear to
be a somewhat artificial marriage of a traditional moduli space, whose set of c*
*ompo-
nents correspond to the (weak) homotopy types of C, and the individual componen*
*ts,
which are classifying spaces of the form B hautX. It is not clear at first g*
*lance
why such complexes might be useful. To understand this, we show how MC can
be approximated by a tower of fibrations, in a way that elucidates the obstruct*
*ion
theory of x2.10:
7.1. Postnikov systems and Eilenberg-Mac Lane objects. Recall from [DK2 ,
x1.2] that functorial Postnikov towers
X ! . .!.PnX ! Pn-1X ! . .!.P0X
may be defined in categories of the form C = sE using the matching space con-
struction of [BK , X,x4.5]. By considering the fibers of successive Postnikov *
*sections,
we can define the analogue of homotopy groups in each such category, and find t*
*hat
they are corepresented, as one might expect, by appropriate suspensions of the *
*"good
objects" in E. For both E = T0 and E = -Alg, we find that the natural "hom*
*o-
topy groups" ^ssnX of any X 2 sE (the bigraded groups of [DKS2 ]) take va*
*lues in
-Alg, with the obvious modifications for the rational variants.
18 DAVID BLANC
Note that for a simplicial space X 2 sT0, applying ss* in each simplicial dim*
*ension
yields a simplicial -algebra ss*X, and the two sequences of -algebras (^ss*
*nX)1n=0
and (ssnss*X)1n=0 fit into a "spiral long exact sequence":
@?n+1 sn hn @?n h0
(7.2) . .s.sn+1ss*X ---! ^ssn-1X -! ^ssnX -! ssnss*X -! . .^.ss0X -! ss0s*
*s*X ! 0
(see [DKS2 , 8.1]), in which each term is not only a -algebra, but a module *
*over
^ss0X ~=ss0ss*X under a "fundamental group action", for n 1. Here den*
*otes
the abelian -algebra obtained from a -algebra by re-indexing and suspending
the operations (so that in particular ss*X ~=ss* X for any space X).
Furthermore, one can construct classifying objects B 2 sT0 for any -algebra
(with ^ss0B = and ^ssiB= 0 otherwise); Eilenberg-Mac Lane objects B( , n*
*);
and twisted Eilenberg-Mac Lane objects B (M, n) for any -algebra , -module
M, and n 1, with
8
>< if i = 0
^ssiB (M, n) ~= M (as a -module) ifi = n
>:
0 otherwise.
This is true more generally for resolution model categories, under reasonable*
* as-
sumptions. To simplify the notation we denote by K the analogous simplicial
-algebra (with ssnK ~= for n = 0, and 0 otherwise), and similarly the Eilenb*
*erg-
Mac Lane objects K( , n), and twisted Eilenberg-Mac Lane objects K (M, n) in
s -Alg. We shall use boldface in general to indicate constructions in sT0, as o*
*pposed
to s -Alg.
Finally, one can define natural k-invariants for the Postinkov system of a si*
*mpli-
cial object X 2 sE, as in [BDG , Prop. 6.4], and these take values in approp*
*riate
Andr'e-Quillen cohomolgy groups (represented by the twisted Eilenberg-Mac Lane *
*ob-
jects). These groups are denoted respectively by Hn(X= ; M) := [X, B (M, n)]B *
* ,
for a simplicial space X equipped with a map to B , and Hn(G= ; M) :=
[G, K (M, n)]K , for a simplicial -algebra G equipped with a map to K .
It turns out that for any simplicial space X there is a natural isomorphism
~= n
(7.3) Hn(X= ; M) -! H (ss*X= ; M)
for every n 1 (cf. [BDG , Prop. 8.7]).
7.4. Relating classification complexes. We shall now show how when E = T0
- and more generally - the classification complex MT0 can be exhibited as
the homotopy limit of a tower of fibrations, where the successive fibers have a*
* co-
homological description showing the relationship between ss0MT0 and the higher
homotopy groups. The tower in question is constructed essentially by taking suc*
*ces-
sive Postnikov sections. The idea is an old one, and is useful even in analyzin*
*g the
self-equivalences of a single space X (see, for example, [W ]).
7.5. Definition. For a given -algebra , we denote by D( ) = DsT0( ) the
category of simplicial spaces X 2 sT0 such that ss*X ' B (in s -Alg) (th*
*at is,
ssnss*X ~= for n = 0, and ssnss*X - 0 otherwise). The nerve of DsT0( ) *
* will be
denoted by M .
MODULI SPACES OF HOMOTOPY THEORY 19
The "pointed" version is the nerve of the category R( ) of pairs (X, ae), *
* where
X 2 sT0 and ae : B ! ss*X is a specified weak equivalence in s -Alg (again*
* with
weak equivalences as morphisms).
Although M is the more natural object of interest in our context, we actua*
*lly
study R( ). As noted in [BDG , x1.1], there is a fibration sequence
(7.6) N R( ) ! M ! B Aut( ) ,
where Aut( ) is the group of automorphisms of the -algebra .
7.7. Definition. For each n 1, let Rn( ) denote the category of n-Postnik*
*ov
sections under K - that is, the objects of Rn( ) are pairs (X, ae), *
*where
X 2 sT0 is a simplicial space such that PnX ' X (Postnikov sections*
* in
sT0), and ae : PnK ! Pnss*X is a weak equivalence. The morphisms of Rn(*
* )
are weak equivalences of simplicial spaces compatible with the maps ae up to we*
*ak
equivalence of simplicial -algebras.
7.8. A tower of realization spaces. Given , the Postnikov section functors Pn*
* :
sT0 ! sT0 of x7.1 induce compatible functors n : R( ) ! Rn( ) and Fn :
Rn+1( ) ! Rn( ); and as in [BDG , Thm. 9.4] and [DK3 , Thm. 3.4], these in t*
*urn
induce a weak equivalence
(7.9) N R( ) ! holimn N Rn( ).
Combining (7.6) and (7.9), we may try to obtain information about the space of
realizations M by studying the successive stages in the tower
NFn-1
(7.10) . .N.Rn+1( ) NFn--!N Rn( ) ----! . .!.N R1( ).
8. Analyzing the tower
The first step in analyzing the tower (7.10)is to understand when the success*
*ive
fibers are non-empty, and if so, to count their components. In the rational cas*
*e, too,
our main task was identifying the components of the space of rational homotopy *
*types,
and a partial ordering on the components was induced by the successive deformat*
*ions.
The problem of empty fibers did not arise there, since all DGLs (or DGAs) are
realizable by rational spaces. However, the fact that we have a ordered the suc*
*cessive
choices in a tower, rather than a tree as in Theorem 4.7, suggests that we can *
*describe
them by means of an obstruction theory, as follows:
Assume given a point (X, ae) in N Rn( ), so that X 2 sT0 is a simp*
*licial
space which is an n-Postnikov stage (for some putative simplicial space Y real*
*izing
the given -algebra ), and ae : K ! Pnss*X is a choice of a weak equiva*
*lence.
We can then reinterpret [BDG , Prop. 9.11] as saying:
8.1. Proposition. X 2 Rn( ) extends to an (n+1)-Postnikov stage in Rn+1( )
if an only if the n + 1-st k-invariant for the simplicial -algebra ss*X van*
*ishes in
Hn+3( ; n+1 ).
20 DAVID BLANC
The spiral long exact sequence (7.2)actually determines the homotopy "groups"
of the simplicial -algebra ss*X completely:
8
>< for k = 0,
(8.2) sskss*X ~= n+1 for k = n + 2
>:
0 otherwise.
However, (8.2) in itself does not imply that ss*X is an twisted Eilenber*
*g-
Mac Lane object K ( n+1 , n + 2) in s -Alg - for that to happen, the map
ss*X ! K must have a section (whose existence is equivalent to the vanishi*
*ng
of the k-invariant in Proposition 8.1).
8.3. Remark. It may help to understand why if one considers a simplicial group *
*K 2 G
(as a model for a connected topological space). The fundamental group then appe*
*ars
as = ss0K, the set of path components of K (all homotopy equivalent to each o*
*ther)
- which happens to have a group structure. When K = K (M, n) = K(M, n) n
is a twisted Eilenberg-Mac Lane object, the choice of the section s : B ! K *
*is
what distinguishes K from a disjoint union (of cardinality` | |) of copies of *
*ordinary
Eilenberg-Mac Lane objects K(M, n) - that is, L := | |K(M, n). Even thou*
*gh
we can put a group structure on ss0L so as to make it abstractly isomorphic t*
*o ,
we will not get the right action of on M (which, in the case of K, appears in*
* the
usual way by conjugation with any representative of fl 2 = ss0K).
8.4. Distinguishing between liftings. Since the weak homotopy types of the re-
alizations of a given -algebra (that is, components of MT0) are in one-to-o*
*ne
correspondence with the weak homotopy types of the simplicial spaces Y realizi*
*ng
K in sT0 (if any), we should be able to describe them inductively in terms *
*of the
simplicial-space k-invariants of Y , and thus of the successive Postnikov appro*
*xima-
tions X.
So, one naturally expects that there will be a correspondence between the sec*
*tions
sn and the k-invariants kn in sT0. However, by [BDG , Prop 9.11] (using *
*the
identification of (7.3)) we know that the map OE : K ( n+1 , n+2) ! K ( n+1 , n+
2) corresponding to kn must be a weak equivalence, in our case - that is, it is*
* within
the indeterminacy of the k-invariants, which is certainly contained in Aut ( n*
*+1 ).
Thus, the relevant information for constructing successive stages in a Postni*
*kov
tower for the simplicial space Y with ss*Y ' K (if it exists), aside from *
*the -
algebra , is not its k-invariants, nor the k-invariants for the simplicial -a*
*lgebras
ss*PnB (n = 1, 2, . .).. Instead, it is the seemingly innocuous choice of th*
*e section
sn : K ! K ( n+1 , n + 2), with which any twisted Eilenberg-Mac Lane object (*
*of
simplicial spaces or -algebras) is equipped:
8.5. Proposition. If X 2 Rn( ) can be lifted to Rn( ), the possible choi*
*ces for
the (n + 1)-st Postnikov stage X are determined by the choices of sec*
*tions
sn : K ! ss*X, which correspond to elements in Hn+2( ; n+1 ).
8.6. The components. As noted above, the Dwyer-Kan concept of classification
complexes has one advantage over the traditional moduli spaces, in that the top*
*ology
of each component encodes further information about the corresponding homotopy
type: namely, the topological group of self-equivalences.
MODULI SPACES OF HOMOTOPY THEORY 21
Such groups are generally hard to analyze, as one might expect from the topol*
*ogical
analogue (see, e.g., [R ]). One advantage of the tower (7.10)is that the succes*
*sive fibers
are generalized Eilenberg Mac-Lane spaces:
8.7. Proposition ([BDG , Prop. 9.6ff.]). For any -algebra and n 0, the *
*fiber
ofQ Rn+1( ) ! Rn( ) is either empty, or has all components weakly equivalent to
n+1 n+3-i n+1
i=1 K(H ( , ), i).
9.Comparing the rational and integral versions
The constructions of the rational moduli space as an infinite-dimensional alg*
*ebraic
variety over Q, and of the integral moduli space as a classification complex, a*
*re
not related in any evident way. However, given a -algebra 2 -Alg and its
rationalization Q 2 -AlgQ (defined Q1:= 1 and Qi:= i Q for i 2),
there is a connection between the set of components of M Q and that of M , *
*and
also between the respective towers (7.10).
9.1. The preimage of rationalization. Clearly, is realizable only if Q is (*
*which
is automatic in the simply-connected case), but the converse need not hold (as *
*the
example of a non-realizable torsion -algebra in [B2 , Prop. 4.3.6] shows). The*
*refore,
the number of components of M Q need have no relation to that of M , since there
seems to be no known method for determining all the (integral) homotopy types X
whose fibrewise rationalizations X0Q (x6.4I(b)) are homotopy equivalent. Howev*
*er,
the obstruction theories of x2.10 and Propositions 8.1-8.5, respectively, do pr*
*ovide a
framework for studying the preimage of the (fibrewise) rationalization:
Given a rationalized -algebra A, we have
(i) an algebraic question: which -algebras have Q ~=A?
(ii) a homotopy-theoretic question: what are the components of M (though
different components can map to the same component of MA under the
fibrewise rationalization functor)?
For the first question, restrict attention to the simply-connected case, so t*
*hat A
is just a (connected) graded Lie algebra over Q. In this case we must first con*
*sider
which connected graded Lie algebras over Z have L Q ~=A. Given such an L, we
would like to classify all possible -algebra structures on L; one possible app*
*roach
to this problem is given by the obstruction theory of [BP , x12], in terms of c*
*ertain
relative cohomology groups (where "relative" involves comparing the same object*
* in
different categories via a forgetful functor).
Theoretically, the obstruction theory of Section 8 provides an answer to the *
*second
question, although it does not tell us how to identify all integral homotopy ty*
*pes
having weakly equivalent rationalizations.
9.2. Comparing cohomology theories. In view of the obstruction theories of Sec-
tions 2 and 8, a first step towards understanding the relation between the inte*
*gral
and rational classifications is to compare the cohomology theories that house t*
*he
respective obstructions. There are two main cases to consider:
22 DAVID BLANC
I. We can compare the cohomology of a -algebra , with coefficients in some *
* -
module M (say, M = n ) with the cohomology of the rationalization Q, with
rationalized coefficients in MQ (which is a -module, so in particular a Q-mo*
*dule).
q Q
In this case the short exact sequence of -modules 0 ! K -i!M -! M ! 0 (for
K := Ker (q)) yields a long exact sequence in cohomology:
q* n Q ffi* n+1
. .H.n( , K) i*-!Hn( , M) -! H ( , M ) -! H ( , K) . . .
in the usual way. Moreover, the functors
-Alg T-! -AlgQ S-!AbGp ,
where T (-) := (-)Q and S(-) := Hom (-, MQ ), satisfy the conditions of [*
*BS ,
Thm. 4.4], since a rationalized free -algebra is free (and so H-acyclic) in *
*-AlgQ .
Thus we have a generalized Grothendieck spectral sequence with
(9.3) E2s,t= (Ls~St)(L*T ) ) Hs+t( Q; MQ )
converging to the cohomology in the category -AlgQ ). Here Ls denotes the
s-th left derived functor, and (for simply-connected ) ~S: bg Lie! grAbGp i*
*s the
functor induced by S, which exists because the homotopy groups of any simplicial
graded Lie algebra over Q actually take value in the category bgLie of bigrad*
*ed Lie
algebras.
For general 2 -Alg, by [BS , Prop. 3.2.3] we have instead a functor S~ :
( -AlgQ )- -Alg ! gr AbGp whose domain is the analogue for -AlgQ of the
category of -algebras for spaces. Its objects are bigraded groups, endowed wit*
*h an
action of all primary homotopy operations which exist for the homotopy groups of
a simplicial rational -algebra. As in the simply-connected case, these includ*
*e the
bigraded Lie bracket mentioned above, and presumably others.
II. Alternatively, one could start with a rationalized -algebra A - for si*
*mplicity,
a graded Lie algebra - and try to compare its cohomology (with coefficients i*
*n an
A-module M) taken in the category of Lie algebras with that obtained by thinkin*
*g of
it as an ordinary -algebra. This means taking the derived functors of the comp*
*osite
of
-AlgQ -I! -Alg S-!AbGp ,
where I is the inclusion, and S(-) := Hom -Alg=A(-, M) = Hom -Alg=I(A)(-, M).
These do not satisfy the conditions of [BS , Thm. 4.4], since a rationalized *
*free -
algebra is not a free -algebra. However, we can take a simplicial resolution *
*V ! A
in the category s -AlgQ of simplicial graded Lie algebras, and then resolve e*
*ach Vn
functorially in s -Alg to get a bisimplicial free -algebra Woo, with Xd := dia*
*gWoo
a free -algebra resolution of I(A). If Ab(-) denotes the abelianzation in *
*the over
category -Alg=A, then Ab Woo is a bisimplicial abelian object, or equivale*
*ntly,
a double complex in ( -Alg=A)ab, and the bicosimplicial abelian group SWoo *
*is
Hom -Alg=A(Ab Woo, M). Since
Hom -Alg=A(X, M) = Hom ( -Alg=A)ab(diag AbWoo, M)
' Tot Hom ( -Alg=A)ab(Ab Woo, M) = Tot Hom -Alg=A(Woo, M)
MODULI SPACES OF HOMOTOPY THEORY 23
by the generalized Eilenberg-Zilber Theorem, we see that both the spectral sequ*
*ences
for the bicomplex SWoo converge to the cohomology of I(A). Since by definiti*
*on
sstvSWno ~=Ht(IQn, M) = Ht(LnI(-), M)(A),
we obtain a cohomological spectral sequence with
(9.4) Es,t2= (sssHt(-, M))(L*I)A ) Hs+t(I(A), M).
This is less useful than (9.3), since we cannot identify the E2-term explici*
*tly as a
derived functor.
9.5. Remark. As was pointed out in Section 4, another way to relate the integra*
*l and
rational moduli spaces is geometrically, using higher homotopy operations. This*
* was
the motivations behind [B6 ] (in conjunction with [B3 ]). The main difficulty i*
*n using
homotopy operations in any systematic way for such a purpose is the lack of an *
*appro-
priate taxonomy. The most promising way to overcome this would be by establishi*
*ng
a clear correspondence between higher operations and suitable cohomology groups.
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Dept. of Mathematics, Univ. of Haifa, 31905 Haifa, Israel
E-mail address: blanc@math.haifa.ac.il
**