MSTRUCTURES DETERMINE INTEGRAL
HOMOTOPY TYPE
JUSTIN R. SMITH
Abstract. This paper proves that the functor C(*) that sends
pointed, simplyconnected CWcomplexes to their chaincomplexes
equipped with diagonals and iterated higher diagonals, determines
their integral homotopy type _ even inducing an equivalence of
categories between the category of CWcomplexes up to homotopy
equivalence and a certain category of chaincomplexes equipped
with higher diagonals. Consequently, C(*) is an algebraic model
for integral homotopy types similar to Quillen's model of rational
homotopy types. For finite CW complexes, our model is finitely
generated.
Our result implies that the geometrically induced diagonal map
with all "higher diagonal" maps (like those used to define Steenrod
operations) collectively determine integral homotopy type.
1. Introduction
This paper forms a sequel to [11 ]. That paper developed the theory
of mcoalgebras and defined a functor C(*) that associated canonical
mcoalgebras to semisimplicial complexes.
Our main result is:
Corollary 3.6 on page 25: The functor (defined in 4.2 on page 30
of [11 ])
C(*): Homotop___0! M^
(see 2.15 on page 13 for the definition of M^) defines an equivalence of
categories, where Homotop___0is the category of pointed, simplyconnected
CWcomplexes and continuous maps, in which homotopy equivalences
have been inverted (i.e., it is the category of fractions by homotopy
equivalences).
This, of course, implies the claim made in the title _ that m
structures determine integral homotopy type.
____________
Date: June 9, 1998.
1991 Mathematics Subject Classification. 55R91; Secondary: 18G30.
Key words and phrases. homotopy type, minimal models.
1
2 JUSTIN R. SMITH
From the beginning, it has been a central goal of homotopy the
ory (and algebraic topology in general) to develop tractable models for
spaces and mappings. The early models were combinatorial, including
simplicial or semisimplicial complexes, chaincomplexes, DGA alge
bras and coalgebras and so on. These models tended to fall into two
classes:
o powerful, but computationally intractable (i.e., minimal models,
chaincomplexes of free rather than abelian groups, etc.)
o weak (chaincomplex) but wellbehaved.
The first major breakthrough came with the work of Quillen in [9],
in which he simplified the problem by focusing on rational homotopy
types. Rationalizing eliminates much of homotopy theory's complexity
by killing off cohomology operations, like Steenrod operations. Quillen
was able to create a complete and faithful model of rational homotopy
theory _ cocommutative DGAcoalgebras over Q.
This paper is the outcome of a research program of several years du
ration. One of the main goals of this program was to understand the
coproduct or cupproduct structure of the total space of a fibration. In
order to accomplish this, it was necessary to compute a topological co
product on the cobar construction and on the canonical acyclic twisted
tensor with fiber a cobar construction.
Although the cobar construction is defined for DGAcoalgebras, com
puting a "geometric" coproduct on the cobar construction requires
more than the mere coproduct. I quickly realized that various coho
mology operations entered into the cobar construction's coproduct. It
was necessary to equip the chain complex of a space with diagonals and
higher diagonals defined on the chain level (rather than on cohomology
with coefficients in a finite field).
These higher coproducts satisfy a complex web of relationships I call
coherence conditions. In [11 ], I developed an algebraic device called an
mcoalgebra over a formal coalgebra to encapsulate these relations. A
referee of [11 ] pointed out that formal coalgebras had been defined and
studied before under the name operad.
This research had a gratifying outcome: A coherent mcoalgebra's
cobar construction not only has a computable coproduct; it comes
equipped with a welldefined and geometrically valid mcoalgebra struc
ture (although the coherence condition must be weakened slightly).
It, consequently, becomes possible to iterate the cobar construction.
A sideeffect was an explicit procedure for computing geometric m
coalgebra structures on the total space of a fibration (represented by a
twisted tensor product).
MSTRUCTURES 3
This suggested to me a possibility of characterizing integral homo
topy theory: If one can compute coproducts (and higher coproducts)
on fibrations, one can in principal compute fibrations over fibrations,
and so on. This suggested the possibility of purely algebraic compu
tations of Postnikov towers _ possibly along the lines of Sullivan in
[13 ].
The present paper is the result.
In 1985, Smirnov proved a result similar to ours in [10 ] _ showing
that a functor whose value is a certain comodule over a certain operad
determines the integral homotopy type of a space. The operad and
comodule in question were uncountably generated in all dimensions
and in the simplest case.
In contrast, our functor is finitely generated in all dimensions for
finite simplicial complexes. Although it is considerably more complex
than the cocommutative coalgebras Quillen derived, it is highly un
likely one can get away with something much simpler: all of our functor
appears nontrivially in even the coproduct of a cobar construction.
At this point, I feel it is appropriate to compare and contrast my
results with work of Michael Mandell. In [7], he proved
Main Theorem.The singular cochain functor with coefficients in Zp
induces a contravariant equivalence from the homotopy category of con
nected nilpotent pcomplete spaces of finite ptype to a full subcategory
of the homotopy category of E1 Zpalgebras.
Here, p denotes a prime and Zp the algebraic closure of the finite
field of p elements. E1 algebras are defined in [4] _ they are modules
over a suitable operad.
At first glance, it would appear that his results are a kind of dual to
mine: He characterizes nilpotent pcomplete spaces in terms of E1 Zp
algebras. This is not the case, however. A complete characterization of
nilpotent pcomplete spaces does not lead to one of integral homotopy
types: One must somehow know that plocal homotopy equivalences
patch together. Consequently, his results do not imply mine.
The converse statement is also true: My results do not imply his.
My results in [11 ] imply that all the primes "mix" when one stud
ies algebraic properties of homotopy theory (for instance the plocal
structure of the cobar construction of a space depend on the qlocal
structure of the space for all primes q p). This is intuitively clear
when considers the composite (1 ) O (iterated coproducts) and
notes that Z2 acting on both copies of give rise to elements of the
symmetric group on 3 elements.
4 JUSTIN R. SMITH
Consequently, a characterization of integral homotopy does not lead
to a plocal homotopy theory: In killing off all primes other than p,
one also kills off crucial information needed to compute the cobar con
struction of a space.
In [7], Dr. Mandell proved that one must pass to the algebraic clo
sure of Zp to get a characterication of pcomplete homtopy theory. I
conjecture that, in passing to the algebraic closure, one kills off addi
tional data within the homotopy type _ namely the data that depends
on larger primes. Consequently, one restores algebraic consistence to
the theory, regaining the ability to characterize local homotopy types.
I am indebted to Jim Stasheff for his encouragement and to Michael
Mandell for pointing out errors and inconsistencies in an earlier version
of this paper.
2. Definitions and preliminaries
We recall a few relevant facts from [11 ].
Definition 2.1. If f: C1 ! D1, g: C2 ! D2 are maps, and a b 2
C1 C2 (where a is a homogeneous element), then (f g)(a b) is
defined to be (1)deg(g).deg(a)f(a) g(b).
Remarks. 2.1.1. This convention simplifies many of the common
expressions that occur in homological algebra _ in particular it elim
inates complicated signs that occur in these expressions. For instance
the differential, @ , of the tensor product C D is just @C 1 + 1 @D .
2.1.2. Throughout this entire paper we will follow the convention
that groupelements act on the left. Multiplication of elements
of symmetric groups will be carried out accordingly _ i.e.
1 2 3 4 1 2 3 4 1 2 3 4
2 3 1 4 * 4 3 2 1 = result of applying 2 3 1 4 first
1 2 3 4 1 2 3 4
and then 4 3 2 1 . The product is thus 4 3 1 2 .
2.1.3. Let fi, gi be maps. It isn't hard to verify that the Koszul
convention implies that (f1 g1) O (f2 g2) = (1)deg(f2).deg(g1)(f1O f2
g1 O g2).
2.1.4. We will also follow the convention that, if f is a map between
chaincomplexes, @f = @ O f  (1)deg(f)f O @. The compositions of
a map with boundary operations will be denoted by @ O f and f O @
_ see [1]. This convention clearly implies that @(f O g) = (@f) O g +
(1)deg(f)f O (@g). We will call any map f with @f = 0 a chainmap.
We will also follow the convention that if C is a chaincomplex and
MSTRUCTURES 5
": C ! C and #: C ! 1C are, respectively, the suspension and
desuspension maps, then " and # are both chainmaps. This implies
that the boundary of C is " O @C O # and the boundary of 1C is
 #O@C O ".
2.1.5. We will use the symbol T to denote the transposition operator
for tensor products of chaincomplexes T : C D ! D C, where
T (c d) = (1)dim(c).dim(d)d c, and c 2 C, d 2 D.
Definition 2.2. Let {Un} denote a sequence of differential graded Z
chaincomplexes with preferred Zbases, {bff}, with n running from 1
to 1. This sequence will be said to constitute an operad with Un being
the component of rank n if:
given Zbasis elements, S1 and S2, the following (possibly
distinct) composites are defined: {S1 Ok S2}, where 1
k rank (S2) and all are defined to have rank equal to
rank(S1) + rank(S2)  1 and degree equal to dim (S1) +
dim (S2). These composition operators are subject to the
following identities:
1. (S1 OiS2) Oj S3 = S1 Oi+j1 S2 Oj S3;
2. if j < i then S1 Oi+rank(S2)1S2 Oj S3 = S2 Oj S1 OiS3
The differential @: U ! U:
1. preserves rank;
2. imposes the following additional condition on composition opera
tions @(S1 OiS2) = @S1 OiS2 + (1)dim(S1)S1 Oi@S2.
Remarks. 2.2.1. Multiple compositions are assumed to be right
associative unless otherwise stated _ i.e. S1OiS2OjS3 = S1Oi(S2OjS3).
2.2.2. An operad will be called unitary if it contains an identity ele
ment with respect to the compositionoperations {Oi}. This will clearly
have to be an element of rank 1 and degree 0.
2.2.3. Our definition of an operad in the category of DGA algebras
is slightly different from the standard one given in [4]. It is a simple
exercise to see that the two definitions are equivalent: The fundamental
degreen operation of an operad, Z, (in the standard definition) is a
n + 1linear map
Zi1 Zi2 . . .Zin Zn ! Zi1+...+in
which is simply an nfold iteration of our "higher" compositions:
z1 O1 z2 O2 . .z.nOn b
where b 2 Zn and zj 2 Zij.
6 JUSTIN R. SMITH
Our notation lends itself to the kinds of computations we want to
do.
Definition 2.3. Let A and B be operads. A morphism f: A ! B
is a morphism of the underlying chaincomplexes, that preserves the
composition operations.
Now we give a few examples of operads:
Definition 2.4. The trivial operad, denoted I, is defined to have one
basis element {bi} for all integers i 0. Here the rank of bi is i and
the degree is 0 and the these elements satisfy the compositionlaw:
biOffbj = bi+j1, regardless of the value of ff, which can run from 1 to
j. The differential of this formal coalgebra is identically zero.
Remark. 2.4.1 This is clearly a unitary operad _ the identity ele
ment is b1.
Definition 2.5. Let C1 and C2 be operads. Then C1 C2 is defined
to have:
1. component of rank i = (C1)i (C2)i, where (C1)i and (C2)i are,
respectively, the components of rank i of of C1 and C2;
2. composition operations defined via (a b) Oi (c d) =
(1)dim(b) dim(c)(a Oic b Oid), for a; c 2 C1; b; d 2 C2.
Definition 2.6. Let C be a DGAmodule with augmentation ffl: C !
Z, and with the property that C0 = Z. Then the endomorphism operad
of C, denoted P(C) is defined to be the operad with:
1. component of rank i = Hom Z(C; Ci), with the differential induced
by that of C and Ci. The dimension of an element of Hom Z(C; Ci)
(for some i) is defined to be its degree as a map.
2. The Zsummand is generated by one element, e, of rank 0.
Let s1 2 Hom Z(C; Ci) and s2 2 Hom Z(C; Cj) be elements of rank i
and j, respectively, where i; j 1. Then the composition s1Oks2, where
1 k j, is defined by: s1 Ok s2 = 1__._._.s1_.z._.1_____"Os2: C !
kthposition
Ci+j1. The composition e Ok s2 is defined in a similar way, by iden
tifying e with the augmentation map of C _ it follows that e Ok s2 2
Hom Z(C; Cj1), as one might expect.
The canonical subcomplex Hom Z(C; Ci) of elements of rank i, is
equipped with a natural Siaction _ it is defined by permutation of
the factors of the target, Ci.
Remarks. 2.6.1. This is a unitary operad _ its identity element is
the identity map id 2 Hom Z(C; C).
MSTRUCTURES 7
2.6.2. In general, operads model structures like the iterated coprod
ucts that occur in the endomorphism operad. We will use operads as
an convenient algebraic framework for defining other constructs that
have topological applications.
Proposition 2.7. Let C be a DGAmodule. Coassociative coalgebra
structures on C can be identified with morphisms f: I ! P(C), the the
trivial operad to the endomorphism of C.
We now define a very important operad _ the symmetric construct.
It models the formal behavior of {Hom Z(C; Cn )} in which each Cn is
equipped with an action of Sn that permutes the factors of C.
The symmetric construct will be denoted S. Its components are
{R (Sn)}n2Z+ , where:
1. Sn denotes the symmetric group on n objects;
2. R (Sn) denotes the barresolution of Z over ZSn;
Here we follow the convention that R(S0) = R (S1) = Z, concentrated
in dimension 0. Pure elements of S are canonical basis elements of
R(Sn) for all values of n, or the generator 1 of the Zsummand (by
canonical basis elements, we mean elements of the form [g1 : ::gk] 2
R(Sn)).
See x 2 of [11 ] for a detailed description of the composition operations
of S.
We are now in a position to define mstructures
Definition 2.8. Let C be a chaincomplex with H0(C) = Z. Then:
1. An mstructure on C is defined to be a sequence of chain maps
f[C]n: C ! Hom ZSn(R[C] n; Cn ), where R = {R[C] n} is some f
resolution, and n is an integer that satisfies 0 n < 1. We
assume that:
(a) the composite e1 O f1: C ! C1, is the identity map of C;
(b) and the composite e0 O f0: C ! C0 = Z coincides with the
augmentation of C;
(c) For any c 2 C, at most a finite number of the {f[C]n(c)}
are nonzero. Here Cn is equipped with the Snaction that
permutes the factors.
(d) The adjoint will be denoted fg[C]n: R[C] n C ! Cn , and
is defined by gf[C]n(r c) = (1)dim(r).dim(c)f[C]n(c)(r), where
r 2 R[C] n and c 2 C. With this definition in mind, we require
gf[C] n
n(R[C] n C(k)) C(k) , where C(k) is the kskeleton of
C.
8 JUSTIN R. SMITH
2. An mstructure will be called weaklycoherent if the adjoint maps
fit into commutative diagrams:
1R[C]nR[C]mC
R[C] n R[C] m C ______________/R[C]/n_ R[C] m C
1gf[C]m Oi
fflffl fflffl
R[C] n Cm R[C] n+m1 C
Vi1 gf[C]n+m1
fflffl fflffl
Ci1 R[C] n C Cmi _________________/Cn+m1/_
1...gf[C]n...1
for all n; m 1 and 1 i m. Here V : R[C] n Cm ! Ci1
R[C] n C Cmi is the map that shuffles the factor R[C] n to
the right of i  1 factors of C.
3. An mstructure {f[C]n: C ! Hom ZSn(R[C] n; Cn )}, will be called
strongly coherent (or just coherent) if it is weakly coherent, and
R[C] = S.
A chaincomplex, C, equipped with an mstructure will be called an
mcoalgebra. The maps f[C]n: C ! Hom ZSn(R[C] n; Cn ), where n is an
integer such that 0 n < 1, will be called the structure maps of C.
Remarks. 2.8.1. If C is an incoherent mcoalgebra we may, with
out loss of generality, assume that R[C] = S, since the contracting
homotopy, , that is packaged with R[C] , allows us to construct a
unique sequence of chainmap Sn = R (Sn) ! R[C] n, for n an in
teger such that 0 n < 1. We then compose the structure maps
of the original mcoalgebra with the induced natural transformation
Hom ZSn(R[C] ; *) ! Hom ZSn(S; *), to get the structure maps of the
modified mcoalgebra.
2.8.2. An mcoalgebra can be given the following interpretation: The
adjoint isomorphism allows us to regard the structure maps as a fam
ily of Snequivariant chainmaps gf[C]n: R(Sn) C ! Cn . The map
gf[C] 2
2: R(S2) C ! C , restricted to [ ] C, defines a kind of coprod
uct on C, called the underlying coproduct of the mcoalgebra. Define
Da = fg[C]i(a *): C ! Ci. These maps will be called the higher
coproducts associated with the mcoalgebra. The map D[(1;2)]: C ! C2
defines a chainhomotopy between = D[ ]and T O , where T is the
transposition map defined in 2.1.5 on page 5.
MSTRUCTURES 9
2.8.3. The basic definitions can be stated in terms of operads in the
category of graded differential modules. Operads were originally de
fined in terms of topological spaces by May in [8] and this concept was
extended to DGmodules by Smirnov in [10 ]. Essentially:
1. the operad S, constitutes an operad, and
2. a coherent mcoalgebra is a comodule over this operad, in the sense
of x 3 of [10 ].
2.8.4. My original definition of an mcoalgebra regarded a coher
ent mstructure as a morphism of operads S ! P(C), and a weakly
coherent mstructure as a morphism R[C] ! P(C). Although this
definition has the advantage of being much more elegant than the one
given above it doesn't lend itself to effective computation unless C is
finitely generated as a Zmodule _ this means:
1. Ci 6= 0 for at most a finite number of values of i;
2. each of these nonzero Ciis, itself, finitely generated as a Zmodule.
2.8.5. The definition of weak coherence of an mstructure can be
restated in terms of the maps {f[C]n} themselves, rather than their
adjoints {gf[C]n}. An mstructure is weakly coherent if and only if the
diagram in figure 2.2.2 on page 22 of [11 ] commutes for all integers n
such that 0 n < 1. In this diagram, the map Vi0represents the
composite
(2.1) Hom ZSn(R[C] n; Ci1 Hom ZSm (R[C] m ; Cm ) Cni)
__i1_// i1 m ni
Hom ZSn(R[C] n; C Hom ZSm (R[C] m ; C ) C )
Hom_ZSn(1;i2)//_
Hom ZSn(R[C] n; Ci1 Hom ZSm (R[C] m ; Cm ) Cni)
____//_HomZ(R[C] n R[C] m; Cn+m1 )
where i1 and i2 are inclusion mappings of the Hom ZSnfunctors in
the respective Hom Zgroups. We are also including Hom ZSi(*; *) in
10 JUSTIN R. SMITH
Hom Z(*; *), by simply forgetting that the elements are ZSi linear.
f[C]n n
C____________________________//_HomZSn(R[C]n; C )
 
 HomZ(1;1...fm ...1)
 fflffl
fn+m1 G
  0
 Vi
fflffl fflffl
Hom ZSn+m1(R[C]n+m1 ; Cn+m1H)om_____//HomZ(R[C]n R[C]m; Cn+m1 )
Z(Oi;1)
where G = Hom ZSn(R[C] n; Ci1 Hom ZSm (R[C] m ; Cm ) Cni). This
diagram means that the compositionoperations in the coordinate coal
gebra correspond to actual compositions of the adjoint maps.
Coherence of an mstructure implies a number of identities involv
ing compositions of higher coproducts. For instance, D[(1;2)] 1 O
D[(1;2)]= D[(1;2)]"T2;1[(1;2)]= D[(1;3;2)]"[(1;2)]= D[(1;3;2)(1;2)][(1;2)(1;*
*2;3)]=
D[(1;3;2)(1;2)]D[(1;2)(1;2;3)]. In fact, we can translate any formula involv
ing compositions of highercoproducts into one without compositions
involving elements of the {R (Sn)}.
Proposition 2.9. Let R1 = {R1;n} and R2 = {R2;n}
be fresolutions, and let C1 and C2 be chaincomplexes.
Then there exists a natural transformation of func
tors En: Hom ZSn(R1;n; Cn1) Hom ZSn(R2;n; Cn2) !
Hom ZSn(R1;n R2;n; (C1 C2) n), for all n.
Remark. 2.9.1 If u 2 Hom ZSn(R1;n; Cn1), v 2 Hom ZSn(R2;n; Cn2),
then En sends u v to (c1 c2 ! Vn((u v)(c1 c2))), where c1 2 C1,
c2 2 C2 and Vn: Cn1 Cn2! (C1 C2) n is the map that shuffles the
factors of together.
Now we recall how morphisms of mcoalgebras were defined in [11 ]:
Definition 2.10. Let C1 and C2 be mcoalgebras with sets of structure
maps {f[Ci]n: Ci ! Hom ZSn(R[Ci] n; Cni)}, i = 1; 2, and all 0 n < 1.
A strict morphism {g; h}: C1 ! C2 consists of:
1. a chainmap from g: C1 ! C2;
2. a morphism of fresolutions, h: R[C2] ! R[C1] such that the dia
gram
MSTRUCTURES 11
f[C1]n n
C1 ____//_HomZSn(R[C1] n; C1 )
g HomZSn(h;gn)
fflffl fflffl
C2 f[C_//_HomZSn(R[C2] n; Cn2)
2]n
commutes for all n.
Definition 2.11. A contraction of chaincomplexes
(f0; p; '): C ! D
is a pair of maps f0: C ! D, f: D ! C and a chainhomotopy ': C !
C such that:
1. f0 O f = 1D
2. f O f0  1C = @'.
3. '2=0, ' O f = 0, and f0 O ' = 0
The map f0 is called the projection of the contraction and f is called
its injection _ see [2].
Remark. 2.11.1 In his thesis ([6]), Martin Majewsky called contrac
tions EilenbergZilber maps.
Definition 2.12. Let C and D be weaklycoherent mcoalgebras. A
contraction
(f0; f; '): C ! D
with the injection, f, a strict morphism of mcoalgebras, will be called
an elementary equivalence from C ! D. We will use the notation
_f___
C o//oooo/o/o/D
f0
to denote an elementary equivalence.
Remark. 2.12.1 It is wellknown (for instance, see the discussion of
Schanuel's Lemma in [5]) that any chainhomotopy equivalence of two
chaincomplexes can be decomposed into two iterated contractions.
This implies that contractions are of limited interest when one is
studying chaincomplexes. This is no longer true when the chain
complexes have additional structure _ that of an mcoalgebra, for
instance. In this case, the injection of a contraction induces a condi
tion on mstructures somewhat similar equivalence of quadratic forms.
12 JUSTIN R. SMITH
Definition 2.13. The category of weaklycoherent mcoalgebras, de
noted M, is defined to be the localization of M0 by the set of strict
morphisms whose associated chainmaps of underlying chaincomplexes
are injections of contractions of chaincomplexes.
Remarks. 2.13.1. The objects of this category are weaklycoherent
mcoalgebras as before, but a morphism from A to B (say) is a formal
composite:
m1 //si_ _sj_oo_
A _____//. . . Ai ooooo/o/o/. ././/Aj///o/oo.B. .
s0i s0j
where the {mj} are strict morphisms and the {sk} are elementary
equivalences defined in 2.12 on the page before _ which may go to
the left or right. We have weakened the definition of morphism consid
erably in going from M0 to M. Since projections of contractions are
chainmaps, we can still regard a morphism as having an underlying
chain map of chaincomplexes.
We will also identify morphisms with the same underlying chain map.
A morphism will be an equivalence if all of its constituents are ele
mentary equivalences or their formal inverses.
2.13.2. The definition is essentially set up so that the maps in the
EilenbergZilber theorem on page 31 of [11 ] are morphisms. Neither
map is a strict morphism, but they both turn out to be equivalences.
2.13.3. Morphisms preserve mstructures up to a chainhomotopy.
Definition 2.14. Let C = (C; {f[C]n: C ! Hom ZSn(R[C] n; Cn )}) be
a weaklycoherent mcoalgebra. Then C will be called strictly cellular
if there exist strict morphisms of formal coalgebras
gk: R[C] ! S
supporting strict isomorphisms of mcoalgebras
nk !
_
fk: Sk;nk= C Sk1 ! C(k  1)
i=1
such that
nk !
_ [
C(k) = C Dk C(k  1)
i=1 fk
for all k 0. Here, C(k) denotes the kskeleton of C, Sk;nk is the
canonical coherent mcoalgebra of the singular complex of a wedge
MSTRUCTURES 13
of spheres (see 4.2 on page 30 of [11 ]), and the Dk are disks whose
boundaries are the Sk1.
We will call a weakly coherent mcoalgebra cellular if it is equivalent
(in M) to a strictly cellular mcoalgebra.
Remarks. 2.14.1. If X is a CW complex, C(X) = C( _(X)), where
_ (*) is the singular semisimplicial complex functor.
2.14.2. Note that cellularity requires the mstructure of an
mcoalgebra to be an iterated extension of mstructures of spheres.
2.14.3. Clearly, the canonical mcoalgebra of any CWcomplex is
cellular. The converse also turns out to be true _ see 3.5 on page 25.
It is not hard to find noncellular mcoalgebras: Consider the m
coalgebra, B, concentrated in dimensions 0 and 3 (say), where under
lying chain groups are equal to Z. Equip this with a trivial coprod
uct and higher coproducts (subject to the defining conditions in 3.3
on page 19 of [11 ]). Let {ei} be the generator R (S2) with boundary
@ei = (1 + (1)it)ei1, where t 2 Z2 is the generator. We define a map
: R(S2) B ! B B
where
1. B0 = Z,
2. B3 = Z, generated by x,
3. The "higher coproducts" are defined by
8
><1 x + x 1 if i = 0
(ei x) = 0 if i = 2
>:
x x if i = 3
(the last condition is required by 3.3 on page 19 of [11 ] and implies
that the Steenrod operation Sq0 is the identity). Here, we assume that
t 2 Z2 acts trivially on B and multiplies B3 B3 = Z by 1.
This is (trivially) coherent  indeed, it is the mcoalgebra induced
on the homology of the 3sphere. It cannot possibly be cellular because
the Hopf invariant of any map from it to a 2sphere is identically 0.
Definition 2.15. Define M^ be the full subcategory of cellular objects
of M.
We conclude this section with two algebraic results used in the next
section:
14 JUSTIN R. SMITH
Lemma 2.16. Suppose we have a commutative diagram of weakly
coherent mcoalgebras:
f
__________________________________________________*
*______________________________________________________________@
_______________________________________________________*
*______________________________________________________________@
___________________________________________________________*
*______________________________________________________________@
_____________________________________________________________*
*______________________________________________________________@
_______________________________________%%_______________________*
*___________________________//si_oosj_//sk_
(2.2) A . .U.iooo0oo/o/o/./.0./Uj////o/oo.U.k.0ooooo/o/o/B
 si sj sk 
a  b
fflffl fflffl
C ________________________________________C
where the top row is an equivalence from A to B (whose composite is
f), and the downwardmaps are strict morphisms.
Then we can expand diagram 2.2 to the diagram
f
__________________________________________________*
*______________________________________________________________@
_______________________________________________________*
*______________________________________________________________@
___________________________________________________________*
*______________________________________________________________@
_____________________________________________________________*
*______________________________________________________________@
_______________________________________%%_______________________*
*__________________________//si_oosj_//sk_
(2.3) A . .U.iooo0oo/o/o/./.0./Uj////o/oo.U.k.0ooooo/o/o/B
  si sj   sk 
a  pi  pj pk b
    
fflfflfflffl//ti_ _tj_oofflfflfflffl_fflffl//tk_
C . .Z.iooooo/o/o/. ././/Zj///o/oo.Z.k.ooooo/o/o/C
t0i t0j t0k
where
1. The maps from the first row to the second are all strict morphisms
(see 2.10 on page 10).
2. For all 0 i k, the following diagram commutes
'Ui
Ui _____//Ui
pi pi
fflffl fflffl
Zi _'Z__//Zi
i
where 'Ui and 'Zi are the contracting homotopies used in the ele
mentary equivalences _ see 2.11 on page 11 and 2.12 on page 11.
MSTRUCTURES 15
Proof. We will actually construct the more complicated diagram:
_si__ _sj_oo_ //sk_
(2.4) A . .U.io//0oooo/o/o/.0./.Uj////o/o/o.U.k.0ooooo/o/o/B
  si sj   sk 
a  pi  pj pk b
    
fflfflfflffl//ti_ _tj_oofflfflfflffl_fflffl//tk_
C . .Z.iooooo/o/o/. ././/Zj///o/oo.Z.k.ooooo/o/o/C
 fflt0iO t0j fflO fflt0kO
 vifv0iflOfflO vj v0jfflOfflOvkv0kfflOfflO
 fflO fflO fflO 
 OOfflfflfflffl OOfflfflfflfflOOfflfflfflffl
C . . .C______ . . ._____C_ . . .C _______C
We construct the lower rows by scanning the upper, from left to
right, and:
1. Whenever we encounter a subdiagram of the form
__si_
Ui //ooooo/o/o/Ui+1
s0i _____
pi _____
fflffl fflffl____
Zi _______//____________?
fflO
viv0ifflO
OOfflfflfflfflfflO
C
We replace the `?' with the pushout _ Zi+1 = Zi Ui+1=Ui
(embedded via (si; pi)) _ and the appropriate maps. This results
in the subdiagram
_si__
Ui //ooooo/o/o/Ui+1
s0i
pi pi+1
fflffltifflffl
Zi //___ooooo/o/o/Zi+1
fflOt0i fflO
vi v0ifflOvi+1v0i+1fflO
OOfflfflfflfflfflOOOfflfflfflfflfflO
C ________C
where
(a) pi+1 and ti are defined by the canonical property of a push
out and are strict morphisms of mcoalgebras (see 2.10 on
page 10).
(b) t0i= (1; pi O s0i): Zi Ui+1=Ui ! Zi. This map is surjective
since s0iis, and we have made explicit use of the fact that s0i
is a leftinverse of si.
We define a contracting homotopy 'Zi+1 = (0; 'Ui+1): Zi+1 !
Zi+1, where 'Ui+1 is the contracting homotopy of the upper
16 JUSTIN R. SMITH
row (which exists because it is an elementary equivalence _
see 2.12 on page 11). This makes the lower row an elementary
equivalence.
(c) vi+1 = (vi; 0): H ! Zi+1 = Zi Ui+1=Ui and v0i+1= v0iO t0i
2. Whenever we encounter a subdiagram of the form
__si_
Ui //ooUi+1o/////oo
s0i _____
pi _____
fflffl fflffl____
Zi oo_____?______________
fflO
viv0ifflO
OOfflfflfflfflfflO
C
we simply pull back Zi to form the diagram
si
Ui oo___Ui+1
pi piOsi
fflffl fflffl
Zi oo1___Zi
fflO fflO
viv0ifflOvi+1v0i+1fflO
OOfflfflfflfflfflOOOfflfflfflfflfflO
C ________C
where vi+1 = vi.
This procedure works until we come to the end (i.e., the right end
of diagram 2.4 on the page before).
(2.5) B
_b

fflffl
Zt
fflO
vtv0tfflO
OOfflfflfflfflfflO
C
_
where b is induced by b _ its target is the embedded copy of C.
MSTRUCTURES 17
The commutativity of diagram 2.2 on page 14 implies that we can
splice an extra column onto diagram 2.5 on the preceding page to get
(2.6) B ______B
_b 
 b
fflfflvfflfflt
Zt ____ooC_/////o/o/o
fflv0tO
vtfv0tflO
OOfflfflfflfflfflO
C ______C
__
__
Corollary 2.17. Suppose we have a commutative diagram of weakly
coherent mcoalgebras:
f
(2.7) A _____//B
a  b
fflfflfflffl
C __=__//C
where the top row is an equivalence, and the downwardmaps are strict
morphisms.
Then there exists an equivalence of weaklycoherent mcoalgebras
^f: A ffOaFC ! B ffObFC
where F(*) denotes the cobar construction, ff: C ! FC is the canonical
twisting cochain, and the twisted tensor products are equipped with the
canonical weaklycoherent mstructures described in Proposition 1.19
on page 84 of [11 ].
In addition, the following diagram commutes:
f1
(2.8) A ffOaFC _____//B ffObFC
1ffl 1ffl
fflffl fflffl
A ______f______//_B
Remark. 2.17.1 We will use this and the results of [11 ] to show that
the equivalence C(X1) ! C(X2) implies the existence of an equivalence
between the next stages of Postnikov towers of X1 and X2.
18 JUSTIN R. SMITH
Proof. This follows by taking diagram 2.3 on page 14 and putting a
third row of cobar constructions and twisting cochains
__si__ __sj_oo_ //sk___
(2.9) A . . .Ui //o0oooo/o/o/./.0.Uj/////o/oo.U.k.0ooooo/o/o/o/B
  si sj   sk 
a pi pj pk b
    
fflffl fflffl//ti_ __tj_oofflffl_ fflffl//tfflfflk_
C . . .Zi ooo0oo/o/o/. ./.0/Zj////o/oo.Z.k.o0oooo/o/o/o/C
  ti tj   tk 
ff ffi ffj ffk ff
    
fflffl fflffl//F(ti)_F(toojfflffl)_fflffl//Ffflffl(tk)_
FC . . .FZi ooooo/o/o/. ././FZj////oo/o.F.Z.kooooo/o/o/FC
F(ti)0 F(tj)0 F(tk)0
where ffi: Zi ! FZi are the canonical twisting cochains.
The elementary equivalences on the bottom row are the result of __
applying Proposition 2.32 on page 58 of [11 ]. __
3. Topological realization of morphisms
In this section, we will prove the main results involving the topo
logical realization of mcoalgebras and morphisms. We begin with a
proof that equivalences topologically realizable mcoalgebras are topo
logically realizable.
Theorem 3.1. Let X1 and X2 be pointed, simplyconnected,
locallyfinite, simisimplicial sets, with associated canonical
mcoalgebras, C(Xi), i = 1; 2.
In addition, suppose there exists an equivalence of mcoalgebras
f: C(X1) ! C(X2)
as defined in [11 ] or in 2.13 on page 12 and the surrounding discussion.
Then there exist refinements (simplicial subdivisions) X0i, i = 1; 2, of
Xi, respectively and a simplicial map
^f: X01! X02
such that
f0 = C(f^): C(X01) ! C(X02)
Consequently, any mcoalgebra equivalence is topologically realizable
up to a chainhomotopy.
Remarks. 3.1.1. We work in the simplicial category because the
functors C(*) were originally defined over it.
MSTRUCTURES 19
It is wellknown that the category of locallyfinite simplicial sets
coincides with the category of CW complexes. We could also have
worked with the functors C(*), computed from singular complexes.
3.1.2. The refinement is a barycentric subdivision whose degree is
finite within a neighborhood of each vertex of the Xi, if they are finite
dimensional. If the Xi are finite, we can bound this degree by a finite
number.
In any case, however, there are canonical equivalences
C(Xi) ~=C(X0i)
for i = 1; 2.
Proof. The hypothesis implies that the chaincomplexes are
chainhomotopy equivalent, hence that the Xi, i = 1; 2, have the same
homology. This implies that the lowestdimensional nonvanishing
homology groups _ say M in dimension k _ are isomorphic. We get
a diagram
f
(3.1) C(X1) ___________//C(X2)
C(c1) C(c2)
fflffl fflffl
C(K(M; k)) __=__//C(K(M; k))
Here, the maps are defined as follows:
1. The maps {C(ci)}, i = 1; 2, are induced by geometric classifying
maps;
2. f is the composite of rightward arrows in the equivalence between
the C(Xi), i = 1; 2:
_si__ _sj_oo_ //sk_
(3.2) C(X1) . . .Ui//ooooo/o/o/././.Uj////oo/o.U.k.ooooo/o/o/C(X2)
s0i s0j s0k
where the {Uff} are all weaklycoherent mcoalgebras and the {s*}
all define elementary equivalences (see 2.12 on page 11).
Claim: If we forget simplicial structures (i.e., regard the simpli
cial sets in 3.1 as CWcomplexes), we may assume that diagram 3.1
commutes exactly. to be precise:
1. The cellular chain complexes of the Xi are naturally isomorphic
to the underlying chaincomplexes of the C(Xi).
2. We construct the map c1 by finding a topological realization of the
composite C(c2) O f. That this can be done follows by elementary
obstruction theory and the fact that all the spaces in question
are simplyconnected _ see [14 ], for instance. We replace the
20 JUSTIN R. SMITH
simplicial map, c1, by a cellular map, c01, homotopic to it.The
result is a map of pairs
((X1)k; (X1)k1) ! (X2)k; (X2)k1)
(where (X1)k denotes the kskeleton) for all k 0, such that the
induced map of cellular chain modules
ssk((X1)k; (X1)k1) = C(X1)k ! ssk((X2)k; (X2)k1) = C(X2)k
exactly coincides with f (regarded only as a map of chain com
plexes).
3. Now, we refine the simplicial sets until we can replace c01by a
simplicial approximation. The image of each simplex of X1 lies
in a finite subcomplex of K(M; 1) and X2, so we can simplicially
approximate the restriction of c01to this simplex. Consequently, a
finite (but, possibly, unbounded) number of subdivisions of each
simplex suffices.
In the following discussion, we will assume that this subdivision and
simplicial approximation has been carried out _ and we will suppress
the extra notation (i.e., the prime) for the subdivided complexes and
induced maps.
All of the maps in 3.1 on the page before are strict morphisms of
mcoalgebras (see 2.10 on page 10), except for the map f: The vertical
maps and the lower horizontal map are strict because they were induced
by geometric maps.
Corollary 2.17 on page 17 implies that there exists an equivalence
f^: C(X1) ffOC(g1)FC(K(M; k)) ! C(X2) ffOC(g1)FC(K(M; k))
such that the following diagram commutes:
(3.3)
^f
C(X1) ffOC(g1)FC(K(M; k)) _____//C(X2)ffOC(g1)FC(K(M; k))
1ffl 1ffl
fflffl fflffl
C(X1) _____________f_____________//C(X2)
Lemma 3.1 of page 93 and Corollary 3.5 on page 96 of [11 ] imply the
existence of equivalences (of weaklycoherent mcoalgebras)
C(Xix^ffOgiK(M; k)) ! C(Xi) ffOC(gi)FC(K(M; k))
for i = 1; 2
We conclude that there is an equivalence
^F: C(X1 x^ffOg1K(M; k)) ! C(X2 x^ffOg2K(M; k))
MSTRUCTURES 21
where (*) denotes the loop space functor and
^ff: K(M; k) ! K(M; k) is the canonical twisting func
tion (defining a fibration as twisted Cartesian product _ see
[3]).
In addition, the commutativity of 3.3 on the facing page implies that
f*(2) = 1 2 Hk+1 (X1; M)
where 1 and 2 are the kinvariants of the fibrations X1x^ffOg1K(M; k)
and X2 x^ffOg2K(M; k), respectively.
Since the Xi x^ffOgiK(M; k) are homotopy fibers of the gi maps
for i = 1; 2, respectively, we conclude that the second stage of the
Postnikov towers of X1 and X2 are equivalent.
A straightforward induction implies that all finite stages of the Post
nikov tower of X1 are equivalent to corresponding finite stages of the
Postnikov tower of X2. It follows that all finitedimensional obstruc
tions to realizing the underlying chainmap of f by a geometric map of
CWcomplexes vanish.
It is necessary to make one last remark regarding our simplicial ap
proximations to maps in diagrams like 3.1 on page 19 that arise during
inductive steps. Clearly, after any finite number of inductive steps, we
are still dealing with finite subdivisions of the simplicial sets from the
hypothesis. If the original spaces were finite dimensional, we only need
a finite number of inductive steps. __
The conclusion follows. __
Next, we prove a similar result for wellbehaved morphisms that
aren't a priori equivalences. We are heading toward a proof that arbi
trary morphisms are topologically realizable.
Proposition 3.2. Let X1 and X2 be pointed, simplyconnected
simisimplicial complexes complexes, with associated canonical
mcoalgebras, C(Xi), i = 1; 2.
In addition, suppose there exists a strict morphism of weakly coherent
mcoalgebras that induces homology isomorphisms in all dimensions
f: C(X1) ! C(X2)
as defined in [11 ] or in 2.13 on page 12 and the surrounding discussion.
Then there exists a map of CWcomplexes (i.e, we forget the semi
simplicial structure of the spaces and regard them as CWcomplexes _
or pass to suitable simplicial refinements, as in 3.1 on page 18):
^f: X1 ! X2
22 JUSTIN R. SMITH
such that
f = C(f^)
Consequently, f is an equivalence.
Remarks. 3.2.1. This is interesting because strict morphisms don't
generally define mcoalgebra equivalences _ even when they are ho
mology equivalences. The topological realizability of the mcoalgebras
in question is crucial here.
3.2.2. We could actually have stated that the map f is a composite
e1 O f0 O e2, where e1 and e2 are equivalences of mcoalgebras and f0 is
a strict morphism inducing homology isomorphisms.
Proof. We follow an argument exactly like that used in 3.2 on the
preceding page above. In each inductive step we have a morphism of
the form e1OfiOe2, where e1 and e2 are equivalences of mcoalgebras and
fiis a strict morphism inducing homology isomorphisms. the only thing
we must do differently, here, is to invoke the Serre Spectral Sequence
of a fibration to verify that the fi+1 will be a homology equivalence, __
given that fi is. __
Corollary 3.3. Suppose C1 and C2 are weakly coherent mcoalgebras
that are topologically realizable _ i.e., they are equivalent in M
(see 2.13 on page 12) to C(Xi), respectively, for two pointed,
simplyconnected semisimplicial complexes, Xi, i = 1; 2.
Then a morphism
f: C1 ! C2
is an equivalence if and only if it induces isomorphisms in homology.
Theorem 3.4. Let X1 and X2 be pointed, simplyconnected,
locallyfinite, simisimplicial sets complexes, with associated canonical
mcoalgebras, C(Xi), i = 1; 2.
In addition, suppose there exists a morphism of mcoalgebras
(see 2.10 on page 10):
f: C(X1) ! C(X2)
Then there exists a map of CWcomplexes (i.e, we forget the semi
simplicial structure of the spaces and regard them as CWcomplexes _
or form simplicial refinements, as in 3.1 on page 18):
^f: X1 ! X2
such that
f = C(f^)
MSTRUCTURES 23
Consequently, any morphism of mcoalgebras is topologically realiz
able up to a chainhomotopy.
Proof. We prove this result by an inductive argument somewhat differ
ent from that used in theorem 3.1 on page 18.
We build a sequence of fibrations
Fi
pi
fflffl
X2
over X2 in such a way that
1. the morphism f: C(X1) ! C(X2) lifts to C(Fi) _ i.e., we have
commutative diagrams
C(Fi)::
fi uuu
uuuu C(pi)
uu fflffl
C(X1) _f__//_C(X2)
For all i > 0, Fiwill be a fibration over Fi1 with fiber a suitable
EilenbergMacLane space.
2. The map fi is iconnected in homology.
If the morphism f were geometric, we would be building its Postnikov
tower.
Assuming that this inductive procedure can be carried out, we note
that it forms a convergent sequence of fibrations (see [12 ], chapter 8,
x 3). This implies that we may pass to the inverse limit and get a
commutative diagram
C(F1:):
f1 uuu
uuuu C(p1)
uu fflffl
C(X1) __f__//C(X2)
where f"1 is a morphism of weaklycoherent mcoalgebras that is a
homology equivalence. Now 3.3 on the preceding page implies that f1
is an equivalence of mcoalgebras, and 3.1 on page 18 implies that it is
topologically realizable.
It follows that we get a (geometric) map
f1 : X1 ! F1
and the composite of this with the projection p1 : F1 ! X2 is a topo
logical realization of the original map f: C(X1) ! C(X2).
24 JUSTIN R. SMITH
It only remains to verify the inductive step:
Suppose we are in the kth iteration of this inductive procedure. Then
the mapping cone, A(f) is acyclic below dimension k. Suppose that
Hk(A(fk)) = M. Then we get a long exact sequence in cohomology:
(3.4) : : :! Hk+1 (X1; M) ! Hk(A(fk); M) = Hom Z(M; M)
! Hk(Fk; M) ! Hk(X1; M) ! 0
Let 2 Hk(Fk; M) be the image of 1M 2 Hk(A(fk); M) =
Hom Z(M; M) and consider the map
h : X2 ! K(M; k)
classified by . We pull back the contractible fibration
K(M; k) xffK(M; k)
over h to get a fibration
Fk+1 = Fk xffOh K(M; k)
where, as before, (*) represents the loop space.
Claim: The morphism fk lifts to a morphism fk+1: C(X1) ! C(Fk+1)
in such a way that the following diagram commutes:
fk+1
C(X1) _____//C(Fk+1)
JJJ 
JJJ p
fk JJJ%%fflffl
C(Fk)
where p0k+1: Fk+1 ! Fk is that fibration's projection map.
Proof of Claim: We begin by using Lemma 3.1 of page 93 and
Corollary 3.5 on page 96 of [11 ] to conclude the existence of a commu
tative diagram:
(3.5) C(Fk xffOh K(M; k)) ___e//_C(Fk) ffOh FC(K(M; k))
VVVVVV 
VVVVVV 1ffl
p0k+1VVVVVVVV fflffl
VV++
C(Fk)
where e is an mcoalgebra equivalence.
If we pull back this twisted tensor product over the map fk, we get
a trivial twisted tensor product (i.e., an untwisted tensor product),
because the image of f*() = 0 2 Hk(X1; M), by the exactness of 3.4.
MSTRUCTURES 25
Theorem 1.20 on page 85 of [11 ] implies the existence of a morphism
(3.6) C(X1) ! C(X1) 1 C(X1) FC(K(M; k))
! C(Fk) ffOh FC(K(M; k))
The composition of this map with e in 3.5 on the facing page is the
required map
C(X1) ! C(Fk+1)
To see that Hk(A(fk+1)) = 0, note that:
1. 2 Hk(Fk; M) = Hk(C(Fk) ; M) is the pullback of the class in
Hk(A(fk); M) inducing a homology isomorphism
: Hk(A(fk)) ! Hk(K(M; k))
(by abuse of notation, we identify with a cochain) or
: Hk(A(fk)) ! Hk(C(K(M; k)) )
2. in the stable range, C(Fk) ffOh FC(K(M; k)) is nothing but the
algebraic mapping cone of the chainmap, , above. But the al
gebraic mapping cone of clearly has vanishing homology in di
mension k since induces homology isomorphisms.
__
__
Corollary 3.5. A weaklycoherent mcoalgebra is topologically realiz
able if and only if it is cellular (see 2.14 on page 12).
Proof. Clearly, topologically realizable mcoalgebras are cellular.
Theorem 3.4 on page 22 implies the converse, because all of the __
attaching morphisms in 2.14 on page 12 are topologically realizable. __
Corollary 3.6. The functor
C(*): Homotop___0! M^
(see 2.15 on page 13 for the definition of M^) defines an equivalence of
categories, where Homotop___0is the category of pointed, simplyconnected
CWcomplexes and continuous maps, in which homotopy equivalences
have been inverted (i.e., it is the category of fractions by homotopy
equivalences).
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26 JUSTIN R. SMITH
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Department of Mathematics and Computer Science
Drexel University
Philadelphia, PA 19104
Email: jsmith@mcs.drexel.edu
Home page: http://www.mcs.drexel.edu/"jsmith