E extracta mathematicae Vol. 32, Núm. 1, 83 – 103 (2017)

Resolutions of Cohomology Algebras and other
Struggles with Integer Coefficients

A. Jaleel, A. Percy

Villa College, Maldives

ahsan.jaleel@villacollege.edu.mv

CIAO, Federation University Australia, Australia

andrew.percy@federation.edu.au

Presented by Antonio M. Cegarra Received August 26, 2016

Abstract: There is a well known homotopy Π-algebra resolution of a space by wedges of
spheres. An attempt to construct the Eckmann-Hilton dual gives a nice resolution for Fp
coefficients which can then be used in a spectral sequence. For Z coefficients the dual con-
struction has several compounding problems illustrating that integral cohomology becomes
relatively problematic when we try to include primary operations.

Key words: primary cohomology operations, integer coefficients, free resolution, derived
functors.

AMS Subject Class. (2010): 55-02, 55S05.

1. Introduction

The cohomology groups of a topological space X are well defined over
coefficients in any abelian group, however, there is a richer, natural structure
that includes the primary cohomology operations. It is within this extra
structure that cohomology over integer coefficients becomes problematic when
compared to cohomology over coefficients in a finite field or the rationals. Over
the integers we may think of the primary structure as the integral version of
the Steenrod algebra, however, the viewpoint taken here is to consider these
cohomology algebras as the Eckmann-Hilton dual of the well known Π-algebras
of homotopy theory [20, 34, 2, 13].

To observe the Eckmann-Hilton duality we will consider the reduced spec-
tral cohomology over the Eilenberg-Mac Lane spectrum of a ring. This ordi-
nary, reduced theory is defined as the homotopy classes of maps into Eilenberg-
Mac Lane spaces (E-M spaces), H̃n(X; R) = [X, K(R, n)].

The ‘naturality’ of the operations increases the precision of calculations
by allowing a map between spaces to induce a morphism between cohomol-
ogy algebras rather than simply cohomology groups (see [33] for a survey of

83



84 a. jaleel, a. percy

applications of additional structure). Primary operations are operations that
are globally natural and are induced by a universal arrow between products
of E-M spaces, known as generalised E-M spaces (GEMs).

For coefficients in a field, the Künneth formula shows that cup product and
composition are the only primary operations and the Cartan formula tells us
how these distribute. For integer coefficients, there is no known analogue of
the Cartan formula. Moreover, there are binary operations with no known
formulation for the universal arrows representing them (see Example 3.5).
Consequently, for integer coefficients we have only a partial formulation for
the primary operation structure on cohomology groups.

With no explicit formulation for the primary structure, we might take
inspiration from the Eckmann-Hilton dual in which the primary homotopy
structure can be encoded in a category of operations Π. Functors from Π to
pointed sets are called Π-algebras and the image of a Π-algebra are homotopy
groups with the primary homotopy operations acting on them. We can en-
code the primary cohomology operations in the category H(Z), of products of
integral Eilenberg-Mac Lane spaces and homotopy classes of maps. We will
call the Eckmann-Hilton dual of a Π-algebra an H(Z)-algebra, which will be
a functor from H(Z) to the category of pointed sets. Generalizing we can de-
fine an H(R)-algebra to be a functor from the category H(R), of products of
Eilenberg-Mac Lane spaces over a ring R and homotopy classes of maps, to the
category of pointed sets. For R = Fp it is well known that an H(Fp)-algebra
is an unstable algebra over the mod p Steenrod algebra [8, 4].

Taking further inspiration from the theory of Π-algebras, we might attempt
to study the relation between the H(R)-algebra of X and X itself, using a free
cosimplicial resolution and a spectral sequence. In homotopy theory, Stover
[34] constructs a free simplicial resolution X• which is homotopy equivalent
to a wedge of spheres in each simplicial dimension. Taking the p-th homotopy
of this simplicial space gives a simplicial group and the homotopy groups
of that simplicial group fits into the E2 page of the Bousfield-Friedlander
spectral sequence [16]. The sequence commutes with primary operations and
by design of the Stover resolution, the sequence collapses on the third page,
to the Π-algebra of X.

A simplicial resolution of an H(R)-algebra is a simplicial H(R)-algebra
which is weakly equivalent to the constant simplicial H(R)-algebra. A free
simplicial resolution additionally has a free H(R)-algebra in each simplicial
dimension. A free cosimplicial resolution of a space is a cosimplicial space,
X•, with H∗(X•) a free simplicial resolution of the H(R)-algebra of X. The



resolutions of cohomology algebras 85

model category structure, defining the weak equivalences on simplicial H(R)-
algebras, is given by Blanc and Peschke for H(R) containing finite products
of E-M spaces [8].

Although a dual Stover construction will work fairly well for coefficients
in a finite field, there is a difficulty for integer coefficients. Unlike the finite
fields, Z is not algebraically compact, consequently, maps out of an infinite
product of E-M spaces over Z do not factor through a finite sub-product. In
order that the cosimplicial resolution be free, H(Z) needs to contain infinite
products and we need to allow infinitary primary operations. Fortunately, we
are still able to define a model category structure for H(Z)-algebras but the
free resolutions are larger than for field coefficients [26].

We would like to employ a second quadrant cohomology spectral sequence
E

−p,q
2 = πpH

q(X•; R) ⇒ Hq−p(X; R) which commutes with primary coho-
mology operations. The spectral sequence exists [18, 21, 5] although conver-
gence is not guaranteed. The spectral sequence would actually converge to
H∗(Tot(X•); R), where Tot(X•) is the total space associated to the cosimpli-
cial space X• [17]. Then, to meet our purposes, we also require that Tot(X•)
has the same cohomology algebra as X. This requires Tot(X•) to have the
same R-cohomology type as the R-completion of X, which, according to Bous-
field [15], will occur if H∗(X•; R) is acyclic over all R-module coefficients. For
integer coefficients, this requires an expansion of the category H(Z) to in-
clude all GEMs, that is, (countable) products of Eilenberg-Mac Lane spaces,
K(M, n), for arbitrary Z-modules M, whereby H(Z)-algebras will no longer
be well defined [29].

We expect integer cohomology to contain more topological information
than cohomology over a quotient ring of Z. When we look at primary op-
erations the additional information contained over integral coefficients is not
well understood. In fact, the research has focused, with great success, on
cohomology over finite fields and seems to have abandoned the integers since
Kochman’s work [27]. Perhaps the research has not been abandoned, it’s just
that no results have been achieved to report. One caveat for Eckmann-Hilton
duality should be “integers for homotopy, finite fields for cohomology”.

After some notation in Section 2, Section 3 looks at the problem of defin-
ing the universal arrows for primary integral operations. Section 4 develops
the encoding of operations in a category and defines H(R)-algebras. Section
5 gives the construction of free resolutions by simplicial H(R)-algebras and
cosimplicial spaces. The need to consider infinitary integral operations is also
explained in this section. Section 6 looks at the spectral sequence for calculat-



86 a. jaleel, a. percy

ing H(R)-algebras and the obstacles occurring for integer coefficients. Section
7 is a brief conclusion.

2. Notation

We assume that all statements and results are for some fixed but arbitrary
ring R. We will denote E-M spaces by Kn = K(R, n). Hence the reduced
cohomology groups of a space with coefficients in R are given by H̃n(X) =
[X, Kn]. For the various constructions given we will then discuss the obstacles
for R = Z compared to R = Fp.

For convenience we will consider the sub-category, T∗, of pointed topolog-
ical spaces which are simply-connected, CW-complexes. For spaces X, Y ∈
T∗, we write [X, Y ] for the homotopy classes of maps from X to Y . Pre-
composition by a map f : X → Y is denoted f∗(g) = gf, for g : Y → Z,
and post-composition denoted by f∗. The category of pointed sets is denoted
SET ∗ and graded pointed sets by GrSET ∗.

For a product
∏n

i=1 Xi, let pri denote the canonical projection onto the
factor Xi, and for maps xi : Y → X let {x1, x2, . . . , xn} : Y →

∏n
i=1 Xi denote

the canonical product map.

Let sC denote the category of simplicial objects over a category C and cC
that of cosimplicial objects.

Given functors u, v : C →C, an object X ∈C and natural transformations
µ : u → v, composition is defined as uµ : uu → uv, (uµ)X := u(µX) and
µu : uu → vu, (µu)X := µuX.

3. Primary operations and universal arrows

In this section we give a brief overview of primary cohomology operations
and explain where the integral theory differs from cohomology over Fp.

Definition 3.1. A cohomology operation of type (G, n, A, m) is a family
of functions θX : H

n(X; G) → Hm(X; A), one for each space X, satisfying
the naturality condition f∗θY = θXf

∗ for any map f : X → Y .

Operations exist for any abelian group coefficients, G and A, whether
finitely generated or not and any integers n ≤ m. The set of all (G, n, A, m)
operations is denoted θ(G, n, A, m) and it follows from the naturality condition
that θ(G, n, A, m) ∼= Hm(K(G, n); A) so that the cohomology classes of E-M
spaces are often called primary cohomology operations [33, 22].



resolutions of cohomology algebras 87

The cohomology operations are generally functions on the underlying sets
of the cohomology groups, however, they can be semi-additive or additive.
Additive operations induce homomorphisms on cohomology groups and, for
any spectral cohomology theory, there is a range called the stable range in
which operations form families of additive operations. For G = A = Z/2, the
stable range are called Steenrod squares,

Sqi : Hn(X; Z/2) −→ Hn+1(X; Z/2) , 0 ≤ i ≤ n ,

which, under composition form the mod-2 Steenrod algebra.

Similarly, if G = A = Z/p for an odd prime p, there are stable operations
called the reduced powers,

P i : Hn(X; Z/p) −→ Hn+2i(p−1)(X; Z/p) , 0 ≤ i ≤
n

2
,

which, under composition form the mod-p Steenrod algebra.

If G = A = Z/p for any prime p there are the Bockstein operations,
β : Hn(X; Z/p) → Hn+1(X; Z/p) and if G = Z/pk, for p prime and k ≥ 1,
and A = Z/pk+1, there are the Pontrjagin p-th powers, βp : H2n(X; Z/pk) →
H2np(X; Z/pk+1).

Remark 3.2. For the case where G is a finitely generated abelian group
and A = Z/p for any prime, from the results of Cartan [19] all cohomology
operations are generated (by compositioins) from the Steenrod squares, the
reduced powers, the Bockstein operations and the Pontrjagin p-th powers [33].

This can equivalently be shown using the Serre spectral sequence [30].
However, if A = Z, the process is much more difficult because of mixed torsion
occuring on the E2 page, and no complete calculation has been given for the
operations Hm(K(Z, n); Z) (see Remark 3.4).

We should also note that if G = A = Q, rational cohomology is an algebra.
Hence, there are only the Q-vector space structure for n = m and a power
operation x 7→ x2 in even degrees, satisfying Definition 3.1 [28].

Now, if G = A = R, for R a commutative ring, there exists the binary
operation of cup product

∪X : Hn(X; R)×Hm(X; R) −→ Hn+m(X; R) ,

satisfying the naturality condition, f∗∪Y = ∪X(f∗×f∗) for any map f : X →
Y (see diagram (1)), which gives the graded cohomology groups a graded



88 a. jaleel, a. percy

ring structure and the mapf induces a graded ring homomorphism. This
introduces the concept of an n-ary operation and we now use Fp for the finite
field rather than the cyclic group Z/p, so cup products will exist.

Definition 3.3. An n-ary operation θ : H̃m1(X) × ··· × H̃mn(X) →
H̃q(X), n ∈ N, is primary if, given any spaces X and Y and any map f : X →
Y , the following naturality diagram commutes

H̃m1(X)×···× H̃mn(X) θ // H̃q(X)

H̃m1(Y )×···× H̃mn(Y )

f∗×···×f∗
OO

θ // H̃q(Y )

f∗
OO

.

(1)

There may be other operations (such as secondary or higher) that satisfy
diagram (1) for some X, Y and mi, but only primary operations satisfy this
diagram universally. Since all spaces are simply connected there is no action
of π1(X) on the cohomology groups of X. It follows directly from (1) that the
homotopy class

θ(pr1, . . . , prn) : K
m1 ×···×Kmn −→ Kq

is a universal arrow and the operation θ(x1, x2, . . . , xn) is given by the com-
position

X
{x1,...,xn}−−−−−−−−→ Km1 ×···×Kmn

θ(pr1,...,prn)−−−−−−−−−→ Kq

The proof [31] is Eckmann-Hilton dual to that for homotopy operations [36].
Composition with a representative element θ ∈ H̃m(Kn) are the only

possible unary operations (for a fixed coefficient ring R). The universal arrow
is θ and the operation θ∗ : H̃

n(X) → H̃m(X).
The group addition is given by identifying Kn with the loop space ΩKn+1,

up to homotopy. Then addition + : Kn×Kn → Kn is given by concatenation
of loops.

Other binary operations are given by the Künneth formula applied to a
product of two Eilenberg-Mac Lane spaces. Iteration of the Künneth formula
shows all finitary operations are generated by the binary and unary operations.
The Künneth formula is derived from a short exact sequence which splits non-
naturally giving

H̃n (Kr ×Ks) ∼=
⊕

i+j=n

H̃i(Kr)⊗H̃j(Ks)
⊕

i+j=n+1

Tor
(
H̃i(Kr), H̃j(Ks)

)
(2)



resolutions of cohomology algebras 89

Elements α ⊗ β ∈ H̃i(Kr) ⊗ H̃j(Ks) are included into H̃n (Kr ×Ks) by
the “external cup product” given by α∗(prr)∪β∗(prs) where ∪ is the standard
cup product (see [23, p. 210 and p. 278]). If we denote the universal arrow for
cup product by h∪ then the universal arrow for the binary operation α⊗β is
h∪∗({α∗(prr), β∗(prs)}).

To give all binary operations and hence, by iteration, all n-ary operations,
it remains to describe elements of the summands⊕

i+j=n+1

Tor
(
H̃i(Kr), H̃j(Ks)

)
.

Here it should be noted that integer coefficients differ from coefficients in Fp.
For coefficients in Fp there are no Tor terms [23, Theorem 3.16] so that all
primary operations are combinations of compositions and cup products as for
α ⊗ β above. In fact, as a refinement to Remark 3.2, by explicitely listing
generators of the groups Hn(K(Z/p, m); Z/q), Cartan showed that the only
operations over Fp are freely generated by the stable reduced powers and cup
product [19].

In [22, Section 12] Eilenberg and Mac Lane defined a certain type of sec-
ondary operation, called cross-cap products, and showed these were in bijec-
tive correspondence with the Tor groups of the homology Künneth formula.
The construction dualises from chain to cochain complexes and we can define
a cross-cap product, α×̄β on elements α ∈ H̃i(Kr) and β ∈ H̃j(Ks) of co-
homology groups with torsion (for details see [31]). Cross-cap products are
generalized by secondary Massey products [35]. These cross-caps correspond
to elements of ⊕

i+j=n+1

Tor(H̃i(Kr), H̃j(Ks))

by considering these Tor groups as cokernels of the external cup products,
hence cosets

H̃i+j−1(Kr ×Ks)

/ ⊕
i+j=n+1

[
H̃i(Kr)⊗ H̃j−1(Ks) + H̃i−1(Kr)⊗ H̃j(Ks)

]
.

This means the cross-cap product α×̄β is not uniquely defined and conse-
quently cannot be identified with a unique universal arrow. Moreover, since
the Künneth short exact sequence splits non-naturally, there is no natural
way to identify a representative arrow in each equivalence class corresponding
to the cross-cap products. That these secondary operations correspond to



90 a. jaleel, a. percy

cosets of H̃n (Kr ×Ks) in equation (2) is the closest description we have of
an unknown type of primary operation for integer coefficients.

Remark 3.4. Appendix 8 contains a table of primary, unary cohomology
operations over Z coefficients and a short discussion of differences with oper-
ations over Fp coefficients. In particular, integer operations do not generate
freely and compositions can be unstable and multiple.

The calculations shown in Table 1 can be used to demonstrate the existence
of the unknown primary operations in

Example 3.5. The binary operations (for Z coefficients)

H̃14
(
K3 ×K3

) ∼= H̃3(K3)⊗ H̃11(K3)⊕Tor (H̃6(K3), H̃9(K3))
∼= Z/3⊕Z/2 (3)

are generated by two types of universal arrows. The first is an external cup
product universal arrow of order 3 and the second, of unknown formulation
and order 2, corresponding to the generator of Z/2.

4. H(R)-algebras

Since being able to list all integral cohomology operations and relations
between them is not possible, we may wish to take another approach in which
this information is encoded in a category and does not need to be known
explicitly. This was the point of view, adopted in the 1990’s, for the study of
primary homotopy operations and their relations acting on homotopy groups
to give a Π-algebra [20, 34].

Let H(R) be the category of finite products of Eilenberg-Mac Lane spaces
over R, including the trivial product ∗, with pointed homotopy classes of
maps. For any X ∈ T∗ consider the functor of homotopy classes of pointed
maps [X, ] : H(R) → SET ∗. The image of these functors (with all copies
of H̃n(X), n ≥ 0, identified as one isomorphism class) give reduced cohomol-
ogy groups and the morphisms of H(R) induce primary operations on those
groups. Such functors, as well as their image in SET ∗ will be called topo-
logical H(R)-algebras. Figure 1 indicates the identification of functor with
graded cohomology algebra.



resolutions of cohomology algebras 91

H(R)
SET

∗

K4

K1 × K2

K3

H̃4(X)

H̃1(X)

H̃2(X)

...
[X, ]h+

+

∗
0

H̃3(X)

h∪
∪

α

α∗

Figure 1: Topological H(R)-algebras

Similarly to Π-algebras we can abstract the notion of topological H(R)-
algebras by

Definition 4.1. An H(R)-algebra is a functor Z : H(R) → SET ∗ send-
ing the point to the singleton set, 0, and with the property that the map

Z
(∏N

i=1 K
ni
)
→

∏N
i=1 Z(K

ni) is a natural isomorphism of abelian groups.

The category of H(R)-algebras will be denoted SET ∗H(R). There are sev-
eral definitions of algebraic structures describing H(R)-algebras. They could
be considered a variety of algebras [29], universal graded algebras [11], models
of the algebraic theory H(R) [12] or models of the finite product sketch H(R)
[8]. Mac Lane states that SET ∗H(R) contains all limits and colimits and that
there exists a free functor, left adjoint to the underlying (forgetful) functor to
SET ∗ [29].

Remark 4.2. Borceux [12] shows that the cohomology H(R)-algebras of
the objects of H(R) are free H(R)-algebras.

In addition, Blanc and Stover show that simplicial objects over any cate-
gory of universal graded algebras, such as SET ∗H(R), has a closed simplicial
model category structure [11]. Blanc and Peschke give a resolution model
category structure on simplicial models of a finite product sketch arising from
an adjunction of free and forgetful functors. Therefore we have a definition of
weak equivalence between simplicial H(R)-algebras.

Remark 4.3. In addition, the model category structure is enriched, form-
ing a simplicial model category with tensor and cotensor products [8].



92 a. jaleel, a. percy

5. Standard constructions and resolutions

A standard construction of simplicial and cosimplicial functors from a
comonad and a monad respectively, is given by Huber in [24].

Definition 5.1. A comonad, ⟨T, µ, η⟩, in a category C consists of an
endofunctor T : C →C and natural transformations, the counit µ : T → 1 and
comultiplication η : T → T 2 such that the following diagrams commute [29]:

T
η

−−−−→ T 2

η

y yTη
T 2 −−−−→

ηT
T 3 ,

T T T∥∥∥ yη ∥∥∥
1T

µT
←−−−− T 2 −−−−→

Tµ
T1 .

Any comonad generates a simplicial functor V• = (Vn, d
i
n, s

i
n), n ≥−1 with

a sequence of functors Vn : C →C, face maps din : Vn → Vn−1 and degeneracy
maps sin : Vn → Vn+1, 0 ≤ i ≤ n. This is achieved by letting T 0 = 1 and
T n+1 = TT n then letting Vn = T

n+1, din = T
iµT n−i and sin = T

iηT n−i. One
may also consider the simplicial functors V• = (Vn, d

i
n, s

i
n), n ≥ 0 with an

augmentation µ : V0 → 1 with µd00 = µd
1
0.

The construction of a cosimplicial functor from a monad is categorically
dual to the construction above [24]. A monad ⟨T, ϵ, δ⟩ with T : C → C,
ϵ : 1 → T (the unit) and δ : T 2 → T (the multiplication), is used to build a
cosimplicial functor V • = (V n, dni , s

n
i ), n ≥−1 as a sequence of functors V

n :
C →C, coface maps dni : V

n−1 → V n and codegeneracy maps sni : V
n+1 → V n,

0 ≤ i ≤ n. This is achieved by letting T 0 = 1 and T n+1 = TT n then
letting V n = T n+1, dni = T

iϵT n−i and sni = T
iδT n−i. One may also consider

the cosimplicial functors F • = (V n, dni , s
n
i ), n ≥ 0 with an augmentation

ϵ : 1 → V 0 with d00ϵ = d
0
1ϵ.

5.1. Simplicial H(R)-algebras. Every adjunction between categories
gives rise to both a comonad and a monad [29]. Let F : GrSET ∗ →SET ∗H(R)
and U be the free and underlying functors for H(R)-algebras with unit of
adjunction α : 1 → UF and counit of adjunction β : FU → 1. A comonad,
⟨T, ϵ, δ⟩, is formed by letting the endofunctor be T = FU, the counit ϵ = β
and the comultiplication δ = FαU.

From this comonad we would like to construct a simplicial H(R)-algebra,
T•Z, for any given H(R)-algebra, Z, and would hope that T•Z can be used as
a free algebraic resolution of any Z in the sense of homotopical algebra. This



resolutions of cohomology algebras 93

requires a free object in each simplicial dimension and a homotopy equivalence
with the constant simplicial H(R)-algebra over Z.

The underlying graded set of the H(R)-algebra, Z, may contain sets of
infinite cardinality. Consequently, the first simplicial dimension of the reso-
lution, that is, TZ, will be the H(R)-algebra of an infinite product of E-M
spaces. By Remark 4.2, this H(R)-algebra will be free if it is an object of
H(R).

For R = Fp coefficients we have that

H∗(P ; Fp) ∼= lim←
α

H∗(Pα; Fp) (4)

where Pα are finite subproducts of the infinite product P [25, Proposition 2.1].
This means we can replace TZ with the H(Fp)-algebra of a finite subproduct
and H(Fp) need only contain finite products of E-M spaces. In addition,
by Remark 3.2, all finitary operations in H(Fp) are iterations of the Künneth
formula on the Steenrod operations, reduced powers and Bocksteins, so H(Fp)
is well understood.

Because Z is not algebraically compact property (4) does not hold and
maps out of infinite products of E-M spaces over Z do not factor through a
finite subproduct and we must extend our definition of primary operations to
cover “infinary” operations and hence let H(Z)∞ contain arbitrary products
of integral E-M spaces. This then brings up the question of whether the model
category structure on SET ∗H(R) for H(R) containing finite products is still
valid for Z coefficients.

Sketch categories containing infinite products have many of the properties
of finite sketch categories under certain restrictions. A good overview is given
in [1]. It turns out that even when a sketch category contains infinite products
the category of models will be locally presentable [12, 5.6.8]. As such, the
category of H(Z)∞-algebras has all limits and colimits [12, 5.5.8]. Then the
model category structure of [8] can be extended to simplicial H(Z)∞-algebras.

According to this model category structure the homotopy groups πn(T•Z),
n ≥ 0, of a simplicial H(R)-algebra are the n-th homotopy groups of the
underlying simplicial graded group. It can then be shown [26] that

πq (T•Z) ∼=

{
0 if q ̸= 0
Z if q = 0

so that T•Z has the property of a free simplicial resolution of Z.



94 a. jaleel, a. percy

5.2. Resolution by cosimplicial space. We would also like to con-
struct a simplicial resolution of a topological H(R)-algebra as the H(R)-
algebra of a cosimplicial space. This will allow the use of spectral sequences
in the calculation of topological H(R)-algebras. That is, for a given space X
we would like to construct an augmented cosimplicial space, X• ← X such
that the augmented simplicial group H̃q(X•) → H̃q(X) satisfies

πpH
q(X•) =

{
Hq(X) if p = 0

0 otherwise
(5)

for all q, with the homomorphism π0H̃
q(X•) ∼= H̃q(X) induced by the aug-

mentation. We would also require H̃∗(X•) to have a free H(R)-algebra in
each simplicial dimension which, by Remark 4.2, requires X• to have an ob-
ject of H(R) in each cosimplicial dimension. Figure 2 shows the simplicial
H(R)-algebra, H̃∗(X•) where each column in SET ∗ is an H(R)-algebra and
each row a simplicial (abelian) group.

H

K4

K1 × K2

K3

[X•, ]h+
+

∗

h∪
∪

α α
∗

X•

cosimplicial space
X0 X1 X2

s
0

d
0

s
1

s
0

s
1

s
2

d
0

d
1

H̃1(X0)

H̃2(X0)

0

H̃3(X0)

H̃4(X0)

d0

s0

s1

...

H̃1(X1)

H̃2(X1)

...

0

H̃3(X1)

H̃4(X1)

∪

+

α∗

d0

s0

s1

d1

s2

SET
∗

Figure 2: Simplicial H(R)-algebra of a cosimplicial space

A cosimplical space can indeed be constructed, homotopic to a product of
E-M spaces in each cosimplicial dimension, dual to Stover’s pushout construc-
tion on wedges of spheres and discs [34].

Let PKn be the path space over Kn and e be the evaluation map sending
a based path to it’s end point. If a map f : X → Kn is null-homotopic there
are maps g : X → PKn such that f ∼= e∗(g) [3, Proposition 1.4.9]. Dually to
Stover’s construction, we define an endofunctor T : T∗ → T∗ such that, for a



resolutions of cohomology algebras 95

space X, T(X) is the pullback of diagram (6). ϕ is the projection onto the
subfactors Kn

e∗(g)
such that f = e∗(g), followed by indentifying the indices

e∗(g) and g :

TX //

��

∏
n∈N

∏
g:X→PKn

PKng

∏
e

��∏
n∈N

∏
f:X→Kn

Knf
ϕ

//
∏
n∈N

∏
g:X→PKn

Kng

(6)

Effectively TX has a factor of Kn for each map f : X → Kn. If f = e∗(g)
is a null-homotopic map then the points of Knf are identified with the end
point of the paths in PKng . Contracting any path-space leaves the set of loops
on Kn. Since ΩKn ∼= Kn−1 it follows that TX is homotopy equivalent to
a product of Eilenberg-Mac Lane spaces and hence will have a free H(R)-
algebra.

Remark 5.2. The space TX will be less connected than X because of the
loop spaces in the product. Consequently, at some cosimplicial dimension the
product space will cease to be simply connected.

The unit for the monad is the natural transformation µ : 1 → T defined on
any space X as the unique map given by the universal property of a pullback:

X

{g}

,,YYYYY
YYYYYY

YYYYYY
YYYYYY

YYYYYY
YYYYYY

YYYYYY

{f}

��?
??

??
??

??
??

??
??

??
??

??
??

?

µX ((QQ
QQQ

QQQ
QQQ

QQQ
QQQ

QQ

T(X) //

��

∏
n∈N

∏
g:X→PKn

PKng

∏
e

��∏
n∈N

∏
f:X→Kn

Knf
ϕ

//
∏
n∈N

∏
g:X→PKn

Kng

The multiplication is a natural transformation η : T 2 → T defined for any
X ∈T∗ as follows:



96 a. jaleel, a. percy

T 2(X)

φ2

,,YYYYY
YYYYYY

YYYYYY
YYYYYY

YYYYYY
YYYYYY

YYYYYY

φ1

  A
AA

AA
AA

AA
AA

AA
AA

AA
AA

AA
AA

AA

ηX
((QQ

QQQ
QQQ

QQQ
QQQ

QQQ
QQ

T(X) //

��

∏
n∈N

∏
g:X→PKn

PKng

∏
e

��∏
n∈N

∏
f:X→Kn

Knf
ϕ

//
∏
n∈N

∏
g:X→PKn

Kng

The map φ1 is defined by projecting the factors K
n
prf

onto the factor Knf and
similarly, φ2 is defined by projecting the factors PK

n
prg

onto the factor PKng .
From the monad, ⟨T, µ, η⟩ we can then form a cosimplicial functor by

the standard construction and apply this to a space X to give a cosimplicial
space X•.

For the simplicial H(R)-algebra, H̃∗(X•), dually to Stover [34], we can
show that defining πpH

q(X•) as the p-th homotopy group of the underlying
simplicial group H̃q(X•) we have formed a free resolution [26]. That is, each
simplicial dimension is a free H(R)-algebra and equations (5) hold.

Remark 5.3. It is in the proof that the simplicial groups are acyclic that
the path spaces must be included in the pullback construction T in diagram
(6), to provide a null-homotopy for dn+1i prf, 1 ≤ i ≤ n + 1, so that for every
f in the n-th Moore chain complex prf is in the (n + 1)-th Moore complex
with dn+10 prf = f.

6. Calculating with the cosimplicial resolution

If we had a convergent spectral sequence for a cosimplical space

E
−p,q
2 = πpH

q(X•) ⇒ Hq−p(Tot(X•)) ,

then we would be able to calculate the cohomology of more complicated spaces,
such as map(Y, X). This would be achieved by identifying the E2 page as
derived functors of map(Y, ) applied dimensionwise to a free cosimplicial
resolution X• [7].

Bökstedt and Ottosen [5] were similarly motivated to construct spectral se-
quences for calculating the cohomology of Fp-completions of the string space



resolutions of cohomology algebras 97

(ΛX)hT of a simply-connected space, X of finite type. Using a free cosim-
plicial resolution RX• → X by simplicial Fp-modules [17], they dualize the
construction of Bousfield’s homology spectral sequence [14]. Using (Λ )hT
applied dimensionwise to RX• gives a spectral sequence

E
−p,q
2 =

(
πpH

∗((ΛRX•)hT; Fp
)q ⇒ H∗(Tot(((ΛRX•)hT); Fp)

and associated convergence criteria. For a resolution by Fp-modules, Tot(RX•)
is the Fp-completion of a space X [17], so the spectral sequence converges to
the Fp-cohomology algebra H∗((ΛX)hT); Fp).

To explain the E2 terms, (πpH
∗((ΛRX•)hT; Fp)q are the collection of Fp-

modules πpH
q((ΛRXq)hT; Fp) with the mod p Steenrod operations acting

columnwise. Bökstedt and Ottosen [5] used the known endofunctor l, on
the category of unstable algebras over the Steenrod algebra, to show that
E

−p,q
2 is the derived functor Hp(H

∗(X); l)q where the homology of H∗(X)
with coeficients in the functor l is defined by

Hp(H
∗(X), l) = πpl(RX

•).

When dualizing Bousfield’s homology spectral sequence for use with in-
teger coefficients we must remember that we would want convergence to the
cohomology H(Z)-algebra of the space X or its Z-completion. To do so, it
is important that the cosimplicial space X• be fibrant so Tot(X•) is the in-
verse limit of a tower of fibrations from which a convergent spectral sequence
could be defined [17, §6]. Fibrancy is ensured by the group object property of
GEM’s [17, X §4.9]. However, unfortunately for integer coefficients Tot(X•)
need not have the same cohomology as X itself.

In order to have the same cohomology, Tot(X•) must be homotopy equiv-
alent to the R-completion of X and this is known to occur only when H̃∗(X•)
is an acyclic simplicial group for cohomology in any R-module coefficients [15,
§7]. We have seen in Section 5 that by constructing X• with GEM’s over
Z, H̃∗(X•) is acyclic over Z coefficients. However, by Remark 5.3, H̃∗(X•)
is not acyclic for cohomology in other Z-modules, for instance Z/2, because
the pullback construction (6) supplies null homotopies only for Z coefficients.
On the other hand, for Fp coefficients the construction (6) will give H̃∗(X•)
acyclic for cohomology in any Fp-module since any Fp-module is a direct sum
of copies of Fp. Hence, for Fp coefficients Tot(X•) is the Fp-completion of X.

To ensure H̃∗(X•) is acyclic over all Z-modules we could expand construc-
tion (6) to be a pullback over indexed products of E-M spaces and path spaces



98 a. jaleel, a. percy

over all Z-module coefficients. This requires the construction to contain arbi-
trary products over a proper class of E-M spaces. As such, the construction
does not exist in T∗, since topological spaces must in particular be sets, not
proper classes. Furthermore, by Remark 4.2, the GEM in each cosimplicial
dimension must be an object in H(Z). Functors from the resulting sketch
category H(Z) may not have a well defined image in SET ∗ [29].

Remark 6.1. It is possible, however, to restrict the GEM in each cosim-
plicial dimension, and hence the category H(Z), to products indexed by a
set of bounded (infinite) cardinality, λ, and still achieve acyclicity over all
Z-modules. The construction of Blanc and Sen [9] is complex and requires
the topological H(Z)-algebra H̃∗(X; Z), that is, the cardinality λ depends on
the space X. Primarily the construction relies on the enrichment of H(Z) of
Remark 4.3, which also enriches the resultant H(Z)-algebras. It turns out that
this enrichment can identify sufficient information, whilst putting an upper
bound on λ, to ensure Tot(X•) is the Z-completion of X [9, Corollary 4.28].

Finally then, it becomes a question of whether the spectral sequence

E
−p,q
2 = πpH

q(X•) ⇒ Hq−p(Tot(X•))

can be shown to converge. A general result does not exist in the literature.
What can be said is that wherever the well known Bousfield-Kan resolution
RX• → X by R-modules [17] can be used in a spectral sequence, the restricted
resolution by GEMs of Remark 6.1 can be used.

One advantage of resolutions by GEMs is recognizing the E2 page of
spectral sequences as derived functors. For coefficients in a general R, the
Yoneda Lemma gives an embedding of the free representable H(R)-algebras
into H(R)op [31]. Applying a functor dimensionwise to a cosimplical resolu-
tion by GEMs is equivalent to a simplicial functor from H(R)-algebras and
then taking (co)homotopy gives the E2 page of spectral sequences as a derived
functor.

7. Conclusion

In their introduction to part I of [17], Bousfield and Kan justify the impor-
tance of completions and localizations as being able to decompose a homotopy
type into Fp type and, together with coherence information over the rationals,
be able to reconstruct it. They point out that many problems can be solved
with Fp information alone. In this review of properties of integral cohomology



resolutions of cohomology algebras 99

we have seen that often more can be obtained with Fp cohomology, since inte-
gral cohomology is both overly complex (Section 3) and unsuited to many of
the powerful tools available for calculation (Section 5). These problems with
the primary integral operations do not seem to get a mention in homology
texts which usually switch from Z to Z/2 coefficients, without explanation,
when introducing the higher structure of operations.

Since Eckmann-Hilton duality is not a categorical duality it should not be
expected to hold universally. The construction of a free simplicial resolution of
a space using wedges of spheres and discs has been of great benefit to homotopy
theory and algebraic topology generally however, the dual resolution is more
complex and has not yet delivered the same benefits when working with integer
coefficients.

It is true that, in comparison, the cosimplicial resolution by simplicial R-
modules of Bousfield and Kan [17, §4] is smaller and remains simply connected
for a simply connected space (see Remark 5.2), even for Z coefficients. How-
ever, the resolution by products of GEMs contains the additional information
of higher order operations [9], which can be used to identify enough informa-
tion to ensure Tot(X•) is the R-completion of X, for any commutative ring
R. The additional information has been sufficient to reconstruct X, up to
R-completion from H̃∗(X; R), but, once again, this important result has only
been achieved for R = Fp or Q [9, 10].

Resolutions by products of E-M spaces over Fp coefficients have been used
to solve the realisation problem of which unstable coalgebras over the Steen-
rod algebra can be realised as the cohomology H(Fp)-algebras of a space,
H̃∗(X; Fp) [6]. This work is also dual to work done in homotopy using Stover’s
resolution but does not extend to Z as it requires (co)homology to consist of
vector spaces rather than modules.

The attempt to Eckmann-Hilton dualise Stover’s seminal construction has
been achieved with considerable effort for integer coefficients. However, for Fp
coefficients, construction (6) is quite straightforward and new results, such as
the realisation problem, are commencing to be attained. This survey should
suffice to convince the reader that any extension of results for Fp coefficients
to Z coefficients are likely to be difficult to achieve.



100 a. jaleel, a. percy

8. Appendix: Unary integral cohomology operations

Table 1 shows results of the (Leray)-Serre spectral sequence calculation of
the groups

H̃m(Kn) , 2 ≤ m ≤ 14 , 2 ≤ n ≤ 7

(extended from [32] to include m = 14). Angle brackets ⟨x⟩ indicate that the
class x generates the summand. The cup product structure is given by the
s.s. and written as juxtaposition so b2 = bb = b ∪ b. The generators of other
summands have been tested for decomposability into compositions or cross-
cap products using dimensional arguments or known relations. The generator

...................................................................................

n
m

2 3 4 5 6 7

14 Z⟨a7⟩ 0 Z/2⟨g2⟩
Z/3⟨n⟩
⊕

Z/5⟨o⟩
0

Z/2⟨t2⟩
⊕

Z/2⟨u⟩

13 0 Z/2⟨bd⟩
Z/5⟨j⟩
⊕

Z/3⟨fh⟩
Z/2⟨kΩ−1g⟩ Z/2⟨s⟩ 0

12 Z⟨a6⟩
Z/2⟨b4⟩
⊕

Z/5⟨e⟩
Z⟨f3⟩ Z/2⟨m⟩ Z⟨p2⟩

11 0 Z/3⟨bc⟩
Z/2⟨fg⟩
⊕

Z/2⟨?⟩
0

Z/2⟨q⟩
⊕

Z/3⟨r⟩

10 Z⟨a5⟩ Z/2⟨d⟩ 0
Z/2⟨k2⟩
⊕

Z/3⟨l⟩
9 0 Z/2⟨b3⟩ Z/3⟨h⟩ 0
8 Z⟨a4⟩ Z/3⟨c⟩ Z⟨f2⟩
7 0 0 Z/2⟨g⟩ ⟨t⟩
6 Z⟨a3⟩ Z/2⟨b2⟩ ⟨p⟩
5 0 0 ⟨k⟩
4 Z⟨a2⟩ ⟨f⟩
3 0 ⟨b⟩
2 Z⟨a⟩

Table 1: The groups H̃m(K(Z, n); Z)



resolutions of cohomology algebras 101

of one summand of H̃11(K4) remains undetermined as either indecomposable
or f2 ◦Ω−4g, hence is given as ⟨?⟩. The stable range, where

⟨Ωx⟩ = H̃m(Kn) ∼= H̃m+1(Kn+1) = ⟨x⟩ , n ≤ m ≤ 2n−1 ,

is left blank, although the fundamental class is denoted by a new letter rather
than as Ω−ia, where Ω−1 is the transgression.

There are some notable differences to cohomology groups,

H̃m(K(Fp, n); Fp) ,

of Eilengerg-Mac Lane spaces over Fp. We see that there are non-stable com-
positions for integer coefficients, moreover, H11(K6) shows that there can be
more than one stable composition of a given degree, possibly of different order.

H̃14(K3) ∼= 0 would be a surprising result if we had coefficients in a field
since in those coefficients cup product generates freely [19] whereas with the
integers we have b2c = 0. Some relation is involved here forcing a cup product
of non-trivial generators to be trivial. We may also expect elements created
by compositions Ω−10g◦(bc), Ω−3r◦(b3) and Ω−3q◦(b3) within H̃14(K3) ∼= 0.
There is also the external cup product and “unknown” primary operation of
Example (3.5) which could act by universal example on two copies of the
fundamental class.

Acknowledgements

The authors would like to thank David Blanc for generously giving
advise during their visits. We would also like to thank the referee for
helpful comments to improve the paper.

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