homology and cohomology operations johnrognes 1 ... · unital) algebra f generated by the steenrod...

HOMOLOGY AND COHOMOLOGY OPERATIONS

John Rognes

1. Based spaces

1.1 Homology and cohomology of spaces.

1.1.1. We will work in the category of cofibrantly based compactly generated weakHausdorff spaces.

1.1.2. Let all chains, cochains, homology and cohomology groups have implicitcoefficients Z/2. Hence we briefly write H∗(X) = H∗(X;Z/2) and H∗(X) =H∗(X;Z/2).

1.1.3. The evaluation of cochains on chains S∗(X)⊗S∗(X) → Z/2 induces a perfectpairing 〈 , 〉 : H∗(X)⊗H∗(X) → Z/2, or equivalently a natural isomorphism

H∗(X) ∼= Hom(H∗(X),Z/2).

This is a case of the universal coefficient theorem.

1.1.4. The Eilenberg–Zilber shuffle homomorphism S∗(X) ⊗ S∗(Y ) → S∗(X × Y )provides an associative and commutative chain homotopy inverse to the Alexander–Whitney map S∗(X × Y ) → S∗(X) ⊗ S∗(Y ) . Both maps are chain homotopyequivalences, and induce the Kunneth isomorphisms

H∗(X)⊗H∗(Y ) ∼= H∗(X × Y )

andH∗(X)⊗H∗(Y ) ∼= H∗(X × Y ).

We write x × y for the image of x ⊗ y under either of these isomorphisms. See[McL].

1.1.5. The diagonal map ∆: X → X ×X and the collapse map c : X → ∗ inducea coproduct

ψ : H∗(X)∆∗−−→ H∗(X ×X) ∼= H∗(X)⊗H∗(X)

and counitǫ : H∗(X)

c∗−→ H∗(∗) = Z/2

which give H∗(X) the structure of a coalgebra. It is cocommutative, and coaug-mented by η : Z/2 = H∗(∗) → H∗(X) induced by the inclusion i : ∗ → X of thebase point.

1.1.6. Dually H∗(X) ∼= Hom(H∗(X),Z/2) is an algebra, with cup product

φ : H∗(X)⊗H∗(X) → H∗(X)

Typeset by AMS-TEX

1

2 JOHN ROGNES

derived from ∆∗ and unit η : Z/2 → H∗(X) derived from c∗ . It is commutative,and augmented by ǫ : H∗(X) → Z/2 derived from i∗ . We write φ(x ⊗ y) = x y = xy for the cup product.

1.1.7. Our algebras, coalgebras and Hopf algebras are always associative and unital(or coassociative and counital, etc.).

1.2. The Steenrod squares.

1.2.1. Let C2 = 1, T be the group of order 2, and let EC2 be a free contractibleC2 -space. Its orbit space BC2 = EC2/C2 is the classifying space for principalC2 -bundles. One model for EC2 is S∞ =

⋃

n Sn with the antipodal action.

Let ǫ : W∗ → Z/2 be the periodic resolution of Z/2 by free Z/2[C2] -modules.Specifically, let Wi = Z/2[C2]ei be the free module of rank 1 on a generator eiand let ∂(ei+1) = 1 · ei + T · ei , for all i ≥ 0. Then Hi(BC2) ∼= Hi(W∗ ⊗C2

Z/2) =Z/2ei for all i ≥ 0.

Dually H∗(BC2) = Z/2[t] , with t ∈ H1(BC2) dual to e1 . Then tn is dual to en .

1.2.2. There is a chain homotopy equivalence H∗(X) → S∗(X) , where H∗(X) isviewed as a chain complex with zero boundary maps, given by choosing a basis forH∗(X) and choosing representing cycles in Z∗(X) ⊂ S∗(X) for these homologyclasses.

Likewise there is a C2 -equivariant chain homotopy equivalence W∗ → S∗(EC2) .Viewing EC2 as S∞ as the union of the Sn ’s with Sn = Sn−1 ∪ Dn

+ ∪ Dn− , one

model takes en and Ten to chains representing Dn+ and Dn

− , respectively.

1.2.3. We can now construct the Steenrod squaring operations. The diagonal ∆induces a C2 -equivariant map 1 × ∆: EC2 × X → EC2 × X2 . Here C2 acts byT (e, x) = (Te, x) on the left, and by T (e, x, y) = (Te, y, x) on the right. Hencethere is an induced map of orbit spaces

1×C2∆: BC2 ×X → EC2 ×C2

X2.

We can compute the cohomology of the target. There are isomorphisms

H∗(EC2 ×C2X2) ∼= H∗(S∗(EC2)⊗C2

S∗(X)⊗2) ∼= H∗(W∗ ⊗C2H∗(X)⊗2)

induced by shuffle homomorphisms, by the chain equivalence W∗ ≃ S∗(EC2) , andthe chain equivalence S∗(X) ≃ H∗(X) .

Hence we get a homomorphism

θ∗ : H∗(W∗ ⊗C2H∗(X)⊗2) −→ H∗(BC2 ×X)

by composing the isomorphisms above with (1×C2∆)∗ .

1.2.4. Now suppose we are given a class x ∈ Hn(X) . View it as an n -cocycleξ : Hn(X) → Z/2. Also view the augmentation ǫ : W∗ → Z/2 as a 0-cocycle. Theproduct ǫ ⊗ ξ ⊗ ξ : W∗ ⊗ H∗(X)⊗2 → Z/2 is C2 -equivariant, hence descends todefine a cohomology class

[ǫ⊗ ξ ⊗ ξ] ∈ H∗(W∗ ⊗C2H∗(X)⊗2).

HOMOLOGY AND COHOMOLOGY OPERATIONS 3

By the isomorphism above this gives a cohomology class in H∗(EC2 ×C2X2) .

Applying (1×C2∆)∗ we obtain a cohomology class

θ∗[ǫ⊗ ξ ⊗ ξ] ∈ H∗(BC2 ×X) ∼= H∗(BC2)⊗H∗(X).

This class has degree 2n , and can be uniquely written as a sumn∑

i=0

tn−i ⊗ Sqi(x) ∈n

⊕

i=0

Hn−i(BC2)⊗Hn+i(X)

where the terms Sqi(x) ∈ Hn+i(X) are determined by this formula. This definesthe Steenrod squaring operation

Sqi : Hn(X) → Hn+i(X).

1.2.5. We write (i, j) =(

i+ji

)

for the binomial coefficient. Thus (i, j) = (j, i) . It is(i + j)!/i!j! if i ≥ 0 and j ≥ 0, and 0 if i < 0 or j < 0. In the formulas below,the binomial coefficient needs only be considered mod 2.

1.2.6. Theorem. There are homomorphisms Sqi : H∗(X) → H∗(X) for all n, i ≥0 , which satisfy:

(1) The Sqi are natural with respect to maps f : X → Y of spaces.

(2) Sqi raises degrees by i .

(3) Sqi(x) = 0 if i > deg(x) .

(4) Sqi(x) = x2 if i = deg(x) .

(5) Sq0(x) = x for all x .

(6) Sq1(x) = β(x) is the Bockstein homomorphism in cohomology associated to theshort exact sequence of coefficients 0 → Z/2 → Z/4 → Z/2 → 0 .

(7) The external and internal Cartan formulas hold:

Sqk(x× y) =∑

i+j=k

Sqi(x)× Sqj(y)

in H∗(X × Y ) for x ∈ H∗(X) , y ∈ H∗(Y ) , and

Sqk(xy) =∑

i+j=k

Sqi(x)Sqj(y)

in H∗(X) for x, y ∈ H∗(X) .

(8) The Adem relations hold: If a < 2b then

SqaSqb =∑

i

(a− 2i, b− a+ i− 1)Sqa+b−iSqi.

The binomial coefficient can also be written(

b−i−1a−2i

)

. The sum runs over i with

a− b+ 1 ≤ i ≤ a/2 .

(9) The Sqi are stable and the Kudo transgression theorem holds: Sqiσ∗(x) =

σ∗Sqi(x) where σ∗ : Hn+1(X) → Hn(ΩX) is the cohomology suspension. Hence

also SqiΣ(x) = ΣSqi(x) where Σ: Hn(X) → Hn+1(ΣX) is the suspension iso-morphism.

If X is simply-connected and if x ∈ Hn(ΩX) transgresses to y ∈ Hn+1(X) inthe Serre spectral sequence of the path space fibration ΩX → PX → X , thenSqi(x) ∈ Hn+i(ΩX) transgresses to Sqi(y) ∈ Hn+i+1(X) .

For proofs, see [ES] and [May2].

4 JOHN ROGNES

1.3. The Steenrod algebra A .

1.3.1. Consider sequences I = (i1, . . . , ik) of integers with is > 0. Define thedegree, length and excess of I by

d(I) =k

∑

s=1

is, ℓ(I) = k and

e(I) =k−1∑

s=1

(is − 2is+1) + ik = i1 −k

∑

s=2

is.

The sequence I determines the cohomology operation

SqI = Sqi1 · · · Sqik : H∗(X) → H∗(X)

which increases degrees by d(I) . I is said to be admissible if is ≥ 2is+1 for1 ≤ s < k . The empty sequence I = () is admissible and satisfies d(I) = 0,ℓ(I) = 0 and e(I) = ∞ ; it determines the identity cohomology operation Sq() = 1.

1.3.2. Define the mod 2 Steenrod algebra A as the quotient of the free (associative,unital) algebra F generated by the Steenrod squaring operations Sqi for i > 0, bythe two-sided ideal J generated by the Adem relations 1.2.6(8). Here Sq0 = 1.

Then J coincides with the two-sided ideal K of elements in F that annihilateH∗(X) for every space X . Hence A equals the algebra generated by the Steenrodsquares of cohomology operations on spaces. In turn, every stable mod 2 cohomo-logy operation Hn(X) → Hn+i(X) is of this form. Hence we may refer to A asprecisely the algebra of stable mod 2 cohomology operations.

1.3.3. The admissible monomials SqI form a Z/2-module basis for A . These begin:

1;Sq1;Sq2;Sq3, Sq2Sq1;Sq4, Sq3Sq1;Sq5, Sq4Sq1;

Sq6, Sq5Sq1, Sq4Sq2;Sq7, Sq6Sq1, Sq5Sq2, Sq4Sq2Sq1; . . .

1.3.4. The cohomology of any space X is naturally a left module over the Steenrodalgebra A : The pairing

λ : A⊗H∗(X) → H∗(X)

takes Sqi ⊗ x to the value Sqi(x) , and more generally SqI ⊗ x to SqI(x) .

The cohomology of a space is an unstable left A -algebra, in the sense that Sqi(x) =x2 for i = deg(x) and Sqi(x) = 0 for i > deg(x) .

1.3.5. The Steenrod algebra A admits a unique structure of Hopf algebra, withcoproduct ψ : A → A ⊗ A determined by ψ(Sqk) =

∑

i+j Sqi ⊗ Sqj , and counit

determined by ǫ(Sqi) = 0 for i > 0. The coproduct is cocommutative.

1.3.6. The cohomology algebra of any space X is naturally a (commutative aug-mented) left A -module algebra. This means that the cup product map

φ : H∗(X)⊗H∗(X) → H∗(X)

taking x ⊗ y to xy is a left A -module homomorphism. (And similarly for theunit map η .) Here the left A -module structure on H∗(X)⊗H∗(X) uses the Hopfalgebra coproduct on A as follows:

a · (x⊗ y) =∑

a′x⊗ a′′y


where ψ(a) =∑

a′ ⊗ a′′ . With a = Sqk and ψ(Sqk) =∑

i+j=k Sqi ⊗ Sqj the

assertion that the cup product map is a left A -module homomorphism amounts tothe internal Cartan formula 1.2.6(7).

1.3.7. Let X = K(Z/2, n) be the Eilenberg–MacLane space with πn(X) = Z/2 asits only nonzero homotopy group. For any set S let PS and ES denote thepolynomial and exterior algebras over Z/2 generated by S , respectively.

1.3.8. Theorem (Serre). Let ιn ∈ Hn(K(Z/2, n)) = Z/2 be the fundamentalclass, dual to the spherical class in Hn(K(Z/2, n)) ∼= πn(K(Z/2, n)) = Z/2 . Then

H∗(K(Z/2, n)) ∼= PSqI(ιn) | I admissible, e(I) < n.

as an algebra over A .

1.3.9. Let A(n) be the quotient algebra of A by the two-sided ideal generated bySqI with e(I) > n . Then the admissible monomials SqI with e(I) ≤ n form aZ/2-module basis for A(n) .

For X =∏

n≥0K(πn(X), n) a product of Eilenberg–MacLane spaces such that

each πn(X) is a Z/2-module, let

D(X) =⊕

n≥0

A(n)⊗ πn(X).

Then H∗(X) is isomorphic to the polynomial algebra on D(X) over Z/2, modulothe relations Sqi(x) = x2 for i = deg(x) .

1.4. The dual Steenrod operations.

1.4.1. By duality, the Steenrod squaring operation Sqi : Hn(X) → Hn+i(X) in-duces a homomorphism

Sqi∗ : Hn+i(X) → Hn(X).

(This is clear when X has finite type so H∗(X) ∼= Hom(H∗(X),Z/2), and followsin general by naturality.) So

〈Sqi(x), a〉 = 〈x, Sqi∗(a)〉

for all x ∈ Hn(X) and a ∈ Hn+i(X) . Here 〈x, c〉 denotes the Kronecker pairing,evaluating a cohomology class x on a homology class c .

1.4.2. Note that (SqiSqj)∗ = Sqj∗Sqi∗ . The dual action makes H∗(X) a right

module over the Steenrod algebra

ρ : H∗(X)⊗A→ H∗(X)

by mapping x ⊗ Sqi to Sqi∗(x) , or equivalently a left module over the oppositealgebra Aop .

We remark that since the Steenrod operations are stable, this dual action of Sqi∗commutes with suspensions, and hence also acts on the spectrum homology

Hspecn (X) = colimkHn+k(Xk)

of a spectrum X = (Xk)k .

1.4.3. We can give H∗(X)⊗H∗(X) a right module action by A using the coproductψ . Then H∗(X) is a (cocommutative coaugmented) right A -module coalgebra.

6 JOHN ROGNES

1.5. The dual Steenrod algebra A∗ .

1.5.1. The Steenrod algebra A with coproduct ψ is a cocommutative Hopf algebra.Hence the dual

A∗ = Hom(A,Z/2)

is a commutative Hopf algebra, with coproduct dual to the composition product onA . Its structure was analyzed in [Mil].

1.5.2. Theorem (Milnor). There is an algebra isomorphism

A∗ ∼= Pξ1, ξ2, . . .

with deg(ξi) = 2i − 1 for all i ≥ 1 . The coproduct on A∗ is given by

ψ(ξk) =∑

i+j=k

ξ2j

i ⊗ ξj .

Here ξ0 = 1 .

To identify the ξi , note that A ∼= H∗spec(HZ/2;Z/2) so A∗ ∼= Hspec

∗ (HZ/2;Z/2)

receives a map from Σ−1H∗(K(Z/2, 1);Z/2). Then ξi is the image of the generatorof H2i(K(Z/2, 1);Z/2) ∼= Z/2.

1.5.3. The ξi are the indecomposable elements of A∗ (i.e., generate I(A∗)/I(A∗)2

where I(A∗) ⊂ A∗ is the augmentation ideal ker(ǫ)), and are dual to the primitiveelements Qi−1 in A (i.e., those x ∈ A such that ψ(x) = x ⊗ 1 + 1 ⊗ x). TheseMilnor primitives can be inductively defined by Q0 = Sq1 and Qn = [Qn−1, Sq

2n ]for n ≥ 1.

1.5.4. Directly dual to the A -module action λ on H∗(X) is a homomorphism

λ∗ : H∗(X) → H∗(X)⊗A∗

making H∗(X) a (cocommutative coaugmented) right A∗ -comodule coalgebra.

The two dualized viewpoints are related as follows. With I running through theadmissible sequences the SqII form a basis for A . There is a dual basis (SqI)∗Ifor A∗ , and the homology operations SqI∗I give a basis for Aop . Then ρ(x ⊗SqI∗) = SqI∗(x) and

λ∗(x) =∑

I

SqI∗(x)⊗ (SqI)∗

for x ∈ H∗(X) .

2. Infinite loop spaces

2.1. Iterated loop spaces.

2.1.1. Let n ≥ 1. A space X is an n-fold loop space if there is a space Xn and ahomotopy equivalence X ≃ ΩnXn . We call Xn an n-fold delooping of X . Thenfor 0 ≤ k ≤ n there are spaces Xk = Ωn−kXn , with ΩXk+1

∼= Xk for 0 ≤ k < n ,and X ≃ X0 .


2.1.2. Any loop space is naturally an H -group, i.e., a group up to homotopy. IfX ≃ ΩY then composition of loops defines a product

µ : X ×X ≃ ΩY × ΩY −→ ΩY ≃ X

and the base point defines a unit map η : ∗ → X . These satisfy associativity andleft and right unit axioms up to homotopy. Furthermore, reversing the orientationof a loop determines a homotopy inverse

ι : X ≃ ΩY −→ ΩY ≃ X

which acts as a left and right inverse, up to homotopy. Thus X represents a groupvalued functor Z 7→ [Z,X] .

2.1.3. A space X is an infinite loop space, or Ω∞ -space, if there is a space Xn

for each n ≥ 0, with ΩXk+1∼= Xk for all k ≥ 0, and X ≃ X0 . The sequence

of spaces (Xn)n≥0 is called a spectrum, and X0 is its underlying space. Hence aninfinite loop space is any space homotopy equivalent to the underlying space of aspectrum. See [May1].

2.1.4. A spectrum E = (En)n≥0 (with ΩEn+1∼= En for all n ≥ 0) represents a

generalized cohomology theory E∗ , defined on CW-complexes X by

En(X) = [X,En]

(based homotopy classes) for all n ≥ 0. This is a contravariant homotopy functor

with natural suspension isomorphisms En(X) ∼= En+1(ΣX) for all n and spaces

X . Equivalently, E∗ satisfies excision, admits long exact sequences for pairs, etc.

2.1.5. Dually, E represents a generalized homology theory E∗ , defined on CW-complexes by

En(X) = colimk πn+k(Ek ∧X)

for all integers n . This is a covariant homotopy functor with natural suspensionisomorphisms En(X) ∼= En+1(ΣX) , as above. Again E∗ satisfies excision, ad-mits long exact sequences for pairs, etc. The coefficient groups of the generalizedhomology theory are the groups En = En(S

0) .

2.2. Examples of infinite loop spaces.

2.2.1. Infinite loop spaces are ubiquitous, i.e., constantly encountered.

2.2.2. For each Abelian group A and k ≥ 0, the Eilenberg–MacLane space K(A, k)is an infinite loop space. Its n -th delooping is a K(A, n+ k)-space. The spectrumHA with n -th space K(A, n) has underlying space K(A, 0) ≃ A , and representsordinary singular homology and cohomology with coefficients in A .

2.2.3. LetQ(X) = colimn Ω

nΣnX,

where ΩnΣnX → Ωn+1Σn+1X is obtained by applying Ωn for Y = ΣnX to themap Y → ΩΣY that takes a point y ∈ Y to the loop t 7→ y ∧ t in ΣY = Y ∧ S1 .Then Q(X) ∼= ΩQ(ΣX) .

Then Q(X) is the free infinite loop space on X , in the sense that the functor Qfrom spaces to infinite loop spaces is left adjoint to the forgetful functor mapping

8 JOHN ROGNES

the other way. Equivalently, there is a natural bijective correspondence betweeninfinite loop maps Q(X) → Y and space level maps X → Y , when Y is an infiniteloop space.

2.2.4. One important example is QS0 = colimn ΩnSn . It is the underlying space

in the sphere spectrum S , with n -th space Q(Sn) . It represents stable homotopyand cohomotopy as generalized homology and cohomology theories. Its coefficientgroups π∗QS

0 are the stable homotopy groups of spheres.

2.2.5. By Bott periodicity, Ω2(Z×BU) ≃ Z×BU , while Ω8(Z×BO) ≃ Z×BOand Ω8(Z×BSp) ≃ Z×BSp . Here BG denotes the classifying space of the groupG , with U , O and Sp the infinite unitary, orthogonal and symplectic groups.

Hence Z×BU is an infinite loop space, with n -th delooping homotopy equivalentto Z × BU for n ≥ 0 even, and Ω(Z × BU) ≃ U for n ≥ 1 odd. It is theunderlying space of the complex K -theory spectrum KU , representing complextopological K -theory and K -homology as generalized cohomology and homologytheories, respectively.

Likewise Z×BO and Z×BSp are infinite loop spaces, underlying the spectra KOand KSp representing real, resp. quaternionic, topological K -theory.

2.2.6. The connected, 1-connected and 2-connected covers of Z × BO are BO ,BSO and BSpin . The connected and 2-connected covers of Z×BU are BU andBSU . The connected cover of Z×BSp is BSp . Here SO and SU are the infinitespecial (determinant 1) orthogonal and unitary groups, and Spin is the infinitespin group, which is the universal (double) cover of SO .

2.3. The little cubes operad.

2.3.1. To discuss the geometry of n -fold loop spaces, or infinite loop spaces, we needto consider certain spaces of operations on such iterated loop spaces. These combineto form a structure called an operad, for which we omit the precise definition. See[May3]. These spaces were introduced by Boardman and Vogt.

2.3.2. Let In = I × · · · × I denote the unit n -cube. A little n-cube f in In

is a linear embedding f : In → In with parallel axes, i.e., it is of the form f =f1 × · · · × fn where each fs : I → I is a linear function fs(t) = (1− t)xs + tys with0 ≤ xs < ys ≤ 1.

Let Cn(j) be the set of those ordered j -tuples

c = 〈c1, . . . , cj〉

of little n -cubes whose images have pairwise disjoint interiors in In . Topologize

Cn(j) as a subspace of the space of maps∐j

In → I . This is the space of littlen -cubes.

Multiplication by the identity little 1-cube 1: I → I defines stabilization mapsCn(j) → Cn+1(j) , replacing each little n -cube f by the little (n+ 1)-cube f × 1.Let C∞(j) = colimn Cn(j) be the union of these spaces, i.e., the space of j little∞-cubes.

2.3.3. As an example, consider f : I → I and g : I → I given by f(t) = t/2 andg(t) = (t+ 1)/2. Then 〈f, g〉 is a point in C1(2). We shall see in the next sectionthat this point corresponds to the usual multiplication on loop spaces given by


composition of loops. Similarly 〈f ×1n−1, g×1n−1〉 is a little n -cube, representingthe loop sum on n -fold loop spaces.

2.3.4. The symmetric group Σj acts on Cn(j) from the right, by permuting theordering of the j little n -cubes.

For example, the order two element T ∈ C2∼= Σ2 takes 〈f, g〉 to 〈g, f〉 , which

represents multiplication of loops in the opposite order. Then 〈f, g〉 and 〈g, f〉are in different components of C1(2), but their images in Cn(2) are in the samecomponent for all n ≥ 2. This reflects the homotopy commutativity of loop sumon n -fold loop spaces for n ≥ 2, and that πn(X) is Abelian for n ≥ 2.

2.3.5. Let F (Rn; j) be the configuration space of j distinct points (x1, . . . , xj)in R

n . Then Σj acts freely on F (Rn; j) by permuting the points, and F (Rn; j)is (n − 2)-connected. The inclusion R

n → Rn+1 induces Σj -equivariant maps

F (Rn; j) → F (Rn+1; j) , with union

F (R∞; j) = colimn F (Rn; j).

Then F (R∞; j) is a free contractible Σj -space, hence has the Σj -equivariant ho-motopy type of EΣj .

2.3.6. There is a Σj -equivariant homotopy equivalence Cn(j) → F (Rn; j) , whichtakes a j -tuple 〈c1, . . . , cj〉 to the configuration (x1, . . . xj) of center points

xs = cs(1/2, . . . , 1/2)

of the little n -cubes. Hence C∞(j) ≃ EΣj for all j ≥ 0.

2.3.7. When j = 2, Cn(2) is the space of pairs 〈c1, c2〉 of little cubes in In withdisjoint interiors, and T ∈ C2 acts by transposing their numbering. By the centerpoint map, this is equivariantly homotopy equivalent to the configuration spaceF (Rn; 2) of pairs of disjoint points (x1, x2) in R

n , and T again transposes theirnumbering. This is in turn equivariantly homotopy equivalent to Sn−1 with theantipodal action, by the map taking (x1, x2) to the unit vector pointing from x1 tox2 . These identifications are all compatible with stabilization in n , and providesan equivariant homotopy equivalence from C∞(2) to S∞ , which is our preferredmodel for EC2 .

2.4. The action on iterated loop spaces.

2.4.1. Let X be an n -fold loop space, with n -th delooping Xn . So X ≃ ΩnXn .Assume that in fact X is an n -fold loop space in the strict sense, i.e., that X =ΩnXn . We will show that each point in Cn(j) defines a map

Xj = X × · · · ×X −→ X,

i.e., an operation on X with j inputs and 1 output. In the next section we willuse this map to produce homology operations from H∗(X) to itself. Thereafterwe may replace X by any homotopy equivalent space, and hence obtain homologyoperations also for n -fold loop spaces in the weaker sense, with X ≃ ΩnXn .

2.4.2. For c = 〈c1, . . . cj〉 ∈ Cn(j) and X = ΩnY , define a map

θn,j(c) : Xj −→ X

10 JOHN ROGNES

as follows. Take points x1, . . . , xj ∈ X = ΩnY , viewed as maps ys : In → Sn → Y .

(Here In → Sn collapses ∂In ⊂ In to the base point, and identifies In/∂In ∼= Sn

by a standard homeomorphism. For example, fix a homeomorphism I1/∂I1 ∼= S1

and smash with itself n times.)

Define θn,j(c) : In → Y to be ys c

−1s on the subspace cs(I

n) ⊆ In , for all 1 ≤s ≤ j , and map the remainder of In to the base point. The resulting map In → Ytakes ∂In to the base point, hence descends to a map Sn → Y , or equivalently apoint in X = ΩnY .

We rephrase the definition. A j -tuple of little n -cubes 〈c1, . . . , cj〉 and a j -tupleof maps (In, ∂In) → (Y, ∗) are paired up to define a map from the union

j⋃

s=1

cs(In) ⊆ In

to Y , which is extended by mapping the complement to the base point. This definesa map (In, ∂In) → (Y, ∗) . Thinking of such maps as points in X = ΩnY definesθn,j(c)(x1, . . . , xj) .

2.4.3. Letting c = 〈c1, . . . , cj〉 vary, we obtain a continuous action map

θn,j : Cn(j)×Xj −→ X,

which is natural in n -fold loop spaces X = ΩnY . Each point in Cn(j) thusparametrizes a map Xj → X .

The permutation group Σj acts diagonally on the left hand side, by renumberingj -tuples of little n -cubes in Cn(j) and by permuting the factors in Xj . The mapθn,j ignores this action, and hence descends to a continuous map

θ′n,j : Cn(j)×ΣjXj −→ X.

2.4.4. Now suppose X is an infinite loop space in the strict sense, i.e., that thereare spaces (Xn)n≥0 with X = X0 and Xn

∼= ΩXn+1 for all n ≥ 0. Then the mapsθn,j and θn+1,j are compatible with the stabilization Cn(j) → Cn+1(j) , and so weobtain maps

θ∞,j : C∞(j)×Xj −→ X

and

θ′∞,j : C∞(j)×ΣjXj −→ X.

Combined with the Σj -equivariant homotopy equivalence EΣj ≃ C∞(j) , we getthe structure map

θj : XjhΣj

= EΣj ×ΣjXj −→ X.

When j = 2 we may use S∞ with the antipodal action as our model for EΣ2 , andget a map

θ2 : S∞ ×C2

(X ×X) −→ X

where T ∈ C2 transposes the factors in X ×X .


2.5. A monad.

2.5.1. Let X be any space. Define

C∞X =∐

j≥0

C∞(j)×ΣjXj/ ∼

where (di(c), x) ∼ (c, di(x)) for all c = 〈c1, . . . , cj〉 , x = (x1, . . . , xj−1) and 0 ≤i ≤ j . Here di(c) omits the i -th little cube ci from c , while di(x) inserts the basepoint ∗ in the i -th position of x , shifting xi, . . . , xj−1 one step to the right.

Then C∞ is a monad, in the sense that there is a product µ : C∞(C∞X) → C∞Xinduced by composition of little ∞-cubes, and a unit η : X → C∞X mapping tothe (j = 1)-summand. These satisfy associativity and unit axioms.

2.5.2. There is a natural map for any X

α∞ : C∞X −→ Q(X) = colimn ΩnΣnX.

When X is an infinite loop space, the structure maps θ′∞,j respect the identifica-tions in C∞X and combine to define a map

θ : C∞X −→ X.

Then θ is the composite of α∞ with the counit map Q(X) → X obtained fromthe infinite loop space structure on X . The map θ satisfies transitivity and unitaxioms with respect to the monad product µ . We say that X is a C∞ -algebra.

2.5.3. Approximation theorem (May). α∞ : C∞X → Q(X) is a group com-pletion for every space X . In particular, for X path connected it is a weak homo-topy equivalence.

2.5.4. Recognition principle (May). Every infinite loop space is a C∞ -algebra,and every connected C∞ -algebra has the weak homotopy type of an infinite loopspace.

See [May3] and [May4].

3. Homology of infinite loop spaces

3.1. The Pontryagin product.

3.1.1. An H -space X is a space with a homotopy associative and unital mapµ : X ×X → X . Then µ induces a product

φ : H∗(X)⊗H∗(X) ∼= H∗(X ×X)µ∗

−→ H∗(X)

which, together with the unit

η : Z/2 = H∗(∗) −→ H∗(X)

induced by the base point inclusion i : ∗ → X , makes H∗(X) an algebra. We callφ the Pontryagin product induced by the H -space structure map µ .

12 JOHN ROGNES

3.1.2. For an H -space X , the algebra and coalgebra structures on H∗(X) inducedby the Pontryagin product and the diagonal map are compatible, in the sensethat the algebra product and unit are coalgebra homomorphisms, and dually thecoalgebra coproduct and counit are algebra homomorphisms. This structure onH∗(X) is called a Hopf algebra.

Since the diagonal is cocommutative, H∗(X) is always a cocommutative Hopf al-gebra.

3.1.3. When the H -space X admits a homotopy inverse ι : X → X , we call X anH -group. The inverse induces a conjugation

χ : H∗(X)ι∗−→ H∗(X)

which satisfies φ(1⊗ χ)ψ = ηǫ .

3.1.4. Loop spaces X = ΩX1 provide examples of H -groups. The productµ : ΩX1 × ΩX1 → ΩX1 is given by loop sum, the unit η : ∗ → ΩX1 maps to theconstant loop, and the inverse ι : ΩX1 → ΩX1 reverses the parametrization of aloop. Then H∗(X) = H∗(ΩX1) is a cocommutative Hopf algebra with conjugation.

We will write ab = φ(a ⊗ b) for the Pontryagin product on H∗(X) when X is aloop space.

3.1.5. Second loop spaces X = Ω2X2 provide examples of homotopy commutativeH -groups. This follows since the space of 2 little n -cubes Cn(2) is connected forn ≥ 2, and so the multiplication maps µ and µT are homotopic. (Here T : X×X →X ×X denotes the twist map.)

Hence H∗(X) is a commutative and cocommutative Hopf algebra with conjugation,when X is an n -fold loop space, with n ≥ 2. This applies in particular when Xis an infinite loop space.

3.2. Structure theorems for Hopf algebras.

3.2.1. For a discussion of algebras, coalgebras, Hopf algebras and the associatedmodule structures, see [MM]. That paper also proves the following structure theo-rems for Hopf algebras, due to Hopf, Leray and Borel.

3.2.2. Let A be a graded Hopf algebra over Z/2, with product φ , unit and coaug-mentation η , coproduct ψ , and counit and augmentation ǫ .

A Hopf algebra A is connected if the unit map η is an isomorphism in degree 0. Itis of finite type if the underlying vector space in each degree is finite dimensional.

The height of an element x ∈ A is the least integer q such that xq = 0; or, if nosuch integer exists, the height of x is infinity.

3.2.3. Theorem (Borel). If A is a commutative connected Hopf algebra of finitetype over Z/2 , then there is an isomorphism of algebras

A ∼=⊗

i∈I

Ai

where each Ai is a Hopf algebra with a single algebra generator xi .


3.2.4. Proposition. If A is a connected Hopf algebra over Z/2 which has onealgebra generator x , then the height of x is either a power of two or infinity. ThusA ∼= Z/2[x]/(x2

e

) or A ∼= Z/2[x] as algebras.

3.2.5. Let I(A) = ker(ǫ) be the augmentation ideal, and dually let J(A) = cok(η) .There are direct sum decompositions A ∼= I(A)⊕ Z/2 and A ∼= Z/2⊕ J(A) sinceǫη = 1 on Z/2.

Let the vector space

P (A) = x ∈ A | ψ(x) = x⊗ 1 + 1⊗ x

be the primitives in A , and let

Q(A) = I(A)/I(A) · I(A)

be the indecomposables of A . We think of P (A) as a subspace of A , and Q(A) asa quotient space of A .

We say that A is primitively generated if it is generated by P (A) as an algebra.

3.2.6. Theorem. If A is a commutative, primitively generated, connected Hopfalgebra of finite type over Z/2 , then there is an isomorphism of Hopf algebras

A ∼=⊗

i∈I

Ai

where each Ai is a Hopf algebra with a single algebra generator xi .

See [MM].

3.2.7. When A has one algebra generator x as in Proposition 3.2.4, ψ(x) = x ⊗1 + 1⊗ x for degree reasons, and this determines the Hopf algebra structure on Ain either of the cases A = Z[x]/(x2

e

) and A = Z/2[x] .

3.3. The Araki–Kudo/Dyer–Lashof operations.

3.3.1. Araki and Kudo [AK] defined homology operations Qi in the mod 2 homo-logy of infinite loop spaces. Their work was extended by Browder in [Br]. Dyerand Lashof [DL] introduced the corresponding operations in mod p homology, forp an odd prime. The algebra of operations is known as the Dyer–Lashof algebra,and we will refer to the Qi as Dyer–Lashof operations, in spite of the fact that werestrict to the mod 2 case, where the operations were first defined by Araki andKudo.

The mod 2 Steenrod operations are also known as reduced squares, as they takevalues in degrees below the degree of the cup product square. Dually the mod 2Dyer–Lashof operations are increased squares, as they take values in degrees abovethe degree of the Pontryagin product square.

3.3.2. We extend our notations from section 1.2. Let C2 = 1, T be the group oforder 2. The space of 2 little ∞-cubes C∞(2) is a free contractible C2 -space, andserves as a model for EC2 . The augmented singular complex ǫ : S∗(C∞(2)) → Z/2is a C2 -free resolution of Z/2.

14 JOHN ROGNES

Also let ǫ : W∗ → Z/2 be the periodic C2 -free resolution of Z/2 with Wi =Z/2[C2]ei and ∂(ei+1) = 1 · ei + T · ei for all i ≥ 0. There is a chain homotopyequivalence W∗ → S∗(C∞(2)) over Z/2.

Finally there is a chain homotopy equivalence H∗(X) → S∗(X) , where H∗(X) isviewed as a chain complex with zero boundary maps, given by choosing a basis forH∗(X) and choosing representing cycles in Z∗(X) ⊂ S∗(X) for these homologyclasses.

3.3.3. Now suppose X is a strict infinite loop space. Thus there are spaces Xn

and homeomorphisms X ∼= ΩnXn for all n ≥ 0. We obtained operad actions

θ′∞,2 : C∞(2)×C2X2 → X

in 2.4.4. There are isomorphisms

H∗(W∗ ⊗C2H∗(X)⊗2) ∼= H∗(S∗(C∞(2))⊗C2

S∗(X)⊗2) ∼= H∗(C∞(2)×C2X2)

induced by the chain equivalences W∗ ≃ S∗(C∞(2)) and H∗(X) ≃ S∗(X) , and theshuffle homomorphism.

Hence we get a homomorphism

θ∗ : H∗(W∗ ⊗C2H∗(X)⊗2) −→ H∗(X)

by composing the isomorphisms above with (θ′∞,2)∗ .

3.3.4. Now suppose given a class x ∈ Hn(X) . When i ≥ n we form

Qi(x) = θ∗(ei−n ⊗ x⊗ x)

in Hn+i(X) . This makes sense, because ei−n ⊗ x ⊗ x defines a cycle in W∗ ⊗C2

H∗(X)⊗2 . Therefore, this formula defines the Araki–Kudo/Dyer–Lashof squaringoperation

Qi : Hn(X) → Hn+i(X)

for i ≥ n = deg(x) . When i < n we let Qi(x) = 0.

3.3.5. For example, Qn(x) = x2 when x = deg(n) is the Pontryagin square, sinceθ∗(e0 ⊗ x ⊗ x) = φ(x ⊗ x) . Next, Qn+1(x) = θ∗(e1 ⊗ x ⊗ x) with n = deg(x) isrepresented by the closed path from x2 = φ∗(x ⊗ x) to itself, parametrized by apath in C∞(2) from the point µ representing loop sum to the (antipodal) point µTrepresenting loop sum in the opposite order.

3.3.6. Theorem. Let X be an infinite loop space. There are homomorphismsQi : H∗(X) → H∗(X) for all n, i ≥ 0 , which satisfy:

(1) The Qi are natural with respect to maps f : X → Y of infinite loop spaces.

(2) Qi raises degrees by i .

(3) Qi(x) = 0 if i < deg(x) .

(4) Qi(x) = x2 if i = deg(x) , where x2 = φ(x ⊗ x) is the Pontryagin productsquare.

(5) Qi(1) = 0 for all i > 0 , where 1 = [∗] ∈ H0(X) is the algebra unit element.


(6) The external, internal and diagonal Cartan formulas hold:

Qk(x× y) =∑

i+j=k

Qi(x)×Qj(y)

in H∗(X × Y ) for x ∈ H∗(X) , y ∈ H∗(Y ) ,

Qk(xy) =∑

i+j=k

Qi(x)Qj(y)

in H∗(X) for x, y ∈ H∗(X) , and

ψ(Qk(x)) =∑

i+j=k

∑

Qi(x′)⊗Qj(x′′)

if ψ(x) =∑

x′ ⊗ x′′ , x ∈ H∗(X) .

(7) The Qi are stable and the Kudo transgression theorem holds: Qiσ∗(x) =

σ∗Qi(x) where σ∗ : Hn−1(ΩX) → Hn(X) is the homology suspension.

If X is simply-connected and if x ∈ Hn(X) transgresses to y ∈ Hn−1(ΩX) inthe Serre spectral sequence of the path space fibration ΩX → PX → X , thenQi(x) ∈ Hn+i(X) transgresses to Qi(y) ∈ Hn+i−1(ΩX) .

(8) The Adem relations hold: If a > 2b then

QaQb =∑

i

(2i− a, a− b− i− 1)Qa+b−iQi.


i−b−12i−a

)


a/2 ≤ i ≤ a− b− 1 .

(9) The Nishida relations hold: Recall that Sqa∗ is the homology operation dual toSqa . Then

Sqa∗Qb =

∑

i

(a− 2i, b− 2a+ 2i)Qb−a+iSqi∗.


b−aa−2i

)


a− b/2 ≤ i ≤ a/2 .

For a proof, see [May5, §1]. Sq1∗Qb = (b − 1)Qb−1 is a useful special case of the

Nishida relations.

3.3.7. The composite Dyer–Lashof operation QaQb(x) is defined by the uppercomposite in the following diagram, where we briefly write θj for θ′∞,j :

C∞(2)×C2(C∞(2)×C2

X2)21×C2

(θ2)2

//

≃γ

C∞(2)×C2X2

θ2

C∞(4)×C2⋉C2

2X4 π // C∞(4)×Σ4

X4 θ4 // X

The diagram commutes. Note the covering map π , which is induced by the inclu-sion C2 ⋉ C2

2 ⊂ Σ4 of a Sylow 2-subgroup in Σ4 . The induced homomorphismH∗(B(C2 ⋉ C2

2 )) → H∗(BΣ4) is a surjection, split by a transfer map. Hence theelements in the kernel of this map will correspond to relations among the QaQb .These are the Adem relations.

3.3.8. The Dyer–Lashof operations Qi commute with the conjugation χ in thehomology of an infinite loop space. This follows by stability 3.3.6(7) and naturalitywith respect to the inverse map ι on X ∼= ΩX1 .

16 JOHN ROGNES

3.4. The Dyer–Lashof algebra R .

3.4.1. Consider sequences I = (i1, . . . , ik) of integers with is ≥ 0. Define thedegree, length and excess of I as before by

d(I) =

k∑

s=1

is, ℓ(I) = k and

e(I) = ik −k

∑

s=2

(2is − is−1) = i1 −k

∑

s=2

is.

The sequence I determines the homology operation

QI = Qi1 · · · Qik : H∗(X) → H∗(X)

which increases degrees by d(I) . I is said to be admissible if is ≤ 2is+1 for1 ≤ s < k . The empty sequence I = () is admissible and satisfies d(I) = 0,ℓ(I) = 0 and e(I) = ∞ ; it determines the identity homology operation Q() = 1.

3.4.2. Let F denote the free (associative, unital) algebra generated by Qii≥0 .For q ≥ 0 define J(q) to be the two-sided ideal of F generated by the Ademrelations 3.3.6(8), and by the relations QI = 0 if e(I) < q . Define R(q) to bethe quotient algebra F/J(q) , and observe that there are successive quotient mapsR(q) → R(q + 1). Here Q0 6= 1. Let R = R(0); R will be called the mod 2Dyer–Lashof algebra.

Then J(q) coincides with the two-sided ideal K(q) of elements in F that annihilateevery homology class of degree ≥ q of every infinite loop space. Hence R(q) equalsthe algebra generated by the Dyer–Lashof operations of homology operations indegrees ≥ q on infinite loop spaces.

3.4.3. The admissible monomials QI of excess e(I) ≥ q form a Z/2-module basisfor R(q) . For R = R(0) these begin:

1, Q0, Q0Q0, . . . ;Q1;Q2, Q1Q1;Q3, Q2Q1;Q4, Q2Q2, Q2Q1Q1;

Q5, Q3Q2;Q6, Q4Q2, Q3Q3, Q3Q2Q1;Q7, Q4Q3, Q4Q2Q1; . . .

3.4.4. The homology of an infinite loop space X is a left module over R : Thepairing

R⊗H∗(X) → H∗(X)

takes Qi ⊗ x to Qi(x) , and more generally QI ⊗ x to QI(x) . This left R -moduleis allowable in the sense that J(q)Hq(X) = 0 for all q ≥ 0.

3.4.5. The Dyer–Lashof algebra R admits a unique right A -module structuresuch that the Nishida relations 3.3.6(9) are satisfied. The coproduct ψ defined ongenerators by

ψ(Qk) =∑

i+j=k

Qi ⊗Qj

and the counit ǫ with ǫ(Q0) = 1 make R into a Hopf algebra, as well as a rightA -module coalgebra.


3.4.6. A component coalgebra C is a coaugmented coalgebra which is a direct sumof connected coalgebras. Let

πC = g ∈ C | ψ(g) = g ⊗ g , g 6= 0.

Then πC is a basis for C0 .

A component Hopf algebra B is monomial if πB is a monoid under the productof B .

If X is a based space, H∗(X) =⊕

g∈π0(X)H∗(Xg) is a component coalgebra,

where Xg ⊆ X is the path component g ∈ π0(X) . The base point determines thecoaugmentation η : Z/2 → H∗(X) . In this case, πH∗(X) ∼= π0(X) .

The homology of an H -space is a monomial component Hopf algebra.

3.4.7. An allowable R -algebra is a commutative allowable left R -module algebrasuch that Qi(x) = x2 if i = deg(x) . An allowable R -coalgebra is a cocommutativecomponent allowable left R -module coalgebra. An allowable R -Hopf algebra (withconjugation) is a monoidal left R -module Hopf algebra (with conjugation) whichis allowable both as an R -algebra and as an R -coalgebra.

For any of these structures, an allowable AR -structure is an allowable left R -structure and an unstable right A -structure of the same type, such that the A -and R -operations satisfy the Nishida relations. Here an unstable right A -structurein homology is the dual of an unstable left A -structure in cohomology.

See [May5, §2].

3.5. Some free functors.

3.5.1. Consider the following categories:

Allowable AR-Hopf algebras with conjugation

forget

Allowable AR-Hopf algebras

forget

G

OO

Allowable AR-coalgebras

forget

W

OO

Cocommutative component unstable right A-module coalgebras

E

OO

The homology of an infinite loop space lives in the uppermost category. There areforgetful functors going down the list. These admit left adjoints G , W and E , i.e.,functors going up the list, yielding free objects in the respective categories.

3.5.2. We now describe the effect of the free functor E on the prime example ofan object in the lowermost category, namely H∗(X) for a based space X . LetJH∗(X) = cok(η) be the quotient of H∗(X) by Z/2 = H∗(∗) , which is isomorphic

to H∗(X) . Then

EH∗(X) = Z/2⊕⊕

q≥0

R(q)⊗ JHq(X).

18 JOHN ROGNES

Here EH∗(X) is a free allowable AR -coalgebra, with left R -action induced bythe surjections ǫ : R → Z/2 and R → R(q) . The coproduct is defined using thediagonal Cartan formula 3.3.6(6).

As an example, EH∗(S0) = Z/2⊕R .

3.5.3. Next we describe the free functor W on the object EH∗(X) . Note thatJEH∗(X) = cok(η) is the sum

⊕

q≥0R(q)⊗ JHq(X) . Then

WEH∗(X) = P (⊕

q≥0

R(q)⊗ JHq(X))/K

where P (−) denotes the polynomial algebra generated by its argument, and K isthe two-sided ideal generated by the relations Qi(x) = x2 for i = deg(x) .

Then WEH∗(X) is a free allowable AR -Hopf algebra. The product is that obtainedfrom the polynomial algebra, the coproduct is determined by that of EH∗(X) . TheA - and R -module actions are determined by the internal Cartan formulas 1.2.6(7)and 3.3.6(7).

For example, WEH∗(S0) = P (R)/K is the polynomial algebra on the admissi-

ble Dyer–Lashof operations Q0 and QI of excess e(I) > 0, as an algebra. Itscomponents are N0 = 0, 1, 2, . . . , with (Q0)k in the k th component.

3.5.4. A connected allowable AR -Hopf algebra admits a unique conjugation. Ingeneral the free functor G to allowable AR -Hopf algebras with conjugation amountsto a localization, group completing the monoid πC of components.

For example, when X is connected GWEH∗(X) = WEH∗(X) . When X = S0 ,GWEH∗(S

0) is a sum indexed of integers k ∈ Z of copies of the polynomial algebraP (R(1)) on the admissible QI with e(I) > 0.

3.6 The homology of Q(X) .

3.6.1. The homology of the space C∞X from 2.5.1 also admits Dyer–Lashof op-erations, and is an allowable AR -Hopf algebra. The unit inclusion η : X → C∞Xinduces a homomorphism η∗ : H∗(X) → H∗(C∞X) , whose left adjoint is a naturaltransformation

η∗ : WEH∗(X) → H∗(C∞X)

extending η∗ over the inclusion H∗(X) →WEH∗(X) .

3.6.2 Theorem (May). For every space X , η∗ : WEH∗(X) → H∗(C∞X) is anatural isomorphism of allowable AR -Hopf algebras.

3.6.3. The homology of the free infinite loop space Q(X) from 2.2.3 certainly admitsDyer–Lashof operations, and is an allowable AR -Hopf algebra with conjugation.The unit inclusion η : X → Q(X) now induces a homomorphism η∗ : H∗(X) →H∗(Q(X)) , whose left adjoint is a natural transformation

η∗ : GWEH∗(X) → H∗(Q(X))

extending η∗ over the inclusion H∗(X) → GWEH∗(X) .

3.6.4 Theorem (Dyer–Lashof, May). For every space X , η∗ : GWEH∗(X) →H∗(Q(X)) is a natural isomorphism of allowable AR -Hopf algebras with conjuga-tion.


3.6.5 Corollary (Nakaoka).

⊕

n≥0

H∗(BΣn) ∼= H∗(C∞S0) ∼=WEH∗(S

0) ∼= P (R)/K.

Here a product QI1 · . . . ·QIk of admissible monomials in R contributes to the sum-

mand H∗(BΣn) where n =∑k

s=1 2ℓ(Is) . So Qi generates Hi(BΣ2) . Multiplication

by Q() induces the injections H∗(BΣn) → H∗(BΣn+1) for all n .

3.7 Proof of Theorem 3.6.4.

3.7.1. We consider the proof of 3.6.4, concerning H∗(Q(X)) . For simplicity, let usassume X is path connected. The general case involves additional bookkeeping toaccount for the various components.

Let tX be a basis for JH∗(X) ∼= H∗(X) , and let PTX by the polynomialalgebra generated by the set

TX = QI(x) | x ∈ tX , I admissible, e(I) > deg(x).

Then as algebras

GWEH∗(X) =WEH∗(X) ∼= PTX.

3.7.2. If X is (q − 1)-connected, q > 1, then η∗ : H∗(X) → H∗(Q(X)) is anisomorphism for ∗ < 2q , by Freudenthal’s stability theorem. Likewise the inclusionH∗(X) → GWEH∗(X) is an isomorphism for ∗ < 2q . Thus the theorem holds indegrees < 2q is X is (q−1)-connected. We claim that if the theorem holds for thesuspension ΣX in degrees < n , then it holds for X in degrees < n− 1. This willfinish the proof, since for any q > 1 then the theorem for ΣqX in degrees < 2qimplies the theorem for X in degrees < q . Letting q grow to infinity, the theoremfollows.

3.7.3. Let Er = Er∗∗r be the Serre spectral sequence in homology for the path

space fibration

Q(X) ∼= ΩQ(ΣX) → PQ(ΣX) → Q(ΣX).

Then

E2∗∗ = H∗(Q(ΣX))⊗H∗(Q(X))

since the base Q(ΣX) is simply-connected. Clearly E∞∗∗ = Z/2 concentrated in

bidegree (0, 0).

3.7.4. Take tΣX = Σ∗x | x ∈ tX as the basis for JH∗(ΣX) . Here Σ∗ : H∗(X) ∼=H∗(ΣX) . Construct a model spectral sequence ′Er = ′Er

∗∗r with

′Er∗∗ =WEH∗(ΣX)⊗WEH∗(X).

The differentials of ′Er are specified by requiring ′Er to be a spectral se-quence of differential bigraded algebras, such that if QI(x) ∈ TX then there is atransgressive differential

τ(QI(Σ∗x)) = QI(x).

20 JOHN ROGNES

Then ′Er is isomorphic to a tensor product of spectral sequences of the form

Ey ⊗ Pτ(y).

Here y runs through

QI(Σ∗x) | x ∈ tX , I admissible, e(I) > deg(x).

This is because to the eyes of ′Er , the base WEH∗(ΣX) behaves like a tensorproduct of exterior algebras on these y .

Clearly ′E∞∗∗ = Z/2 concentrated in bidegree (0, 0).

3.7.5. By construction, there is a unique morphism of algebras f : ′E2∗∗ → E2

∗∗ suchthat

f = η∗ ⊗ η∗ : WEH∗(ΣX)⊗WEH∗(X) → H∗(Q(ΣX))⊗H∗(Q(X)).

By 3.3.6(7) and the definition of ′Er∗∗r , f induces a morphism of spectral se-

quences.

Hence f is a morphism of first quadrant algebra spectral sequences with E2 -termsof the form (base) ⊗ (fiber), both converging to Z/2 in bidegree (0, 0). By theMoore–Zeeman comparison theorem, then, the hypothesis that f induces an iso-morphism η∗ : WEH∗(ΣX) → H∗(Q(ΣX)) in degrees < n on the base impliesthat f induces an isomorphism η∗ : WEH∗(X) → H∗(Q(X)) in degrees < n − 1on the fiber.

This completes the proof.

3.7.6. In effect, the proof shows that for y ∈ H∗(ΣX) , the classes QI(y) inH∗(Q(ΣX)) for e(I) ≥ deg(y) transgress to classes QI(x) in H∗(Q(X)) , fore(I) > deg(x) , with x = τ(y) . In addition, the differential algebra structure gen-erates powers (QI(x))k of these classes in H∗(Q(X)) , but these are now productsof terms QJ(x) with excess one less: e(J) ≥ deg(x) .

4. Calculations

4.1 E∞ ring spaces.

Let X be an infinite loop space. The little cubes action maps

θ∞,j : C∞(j)×Xj → j

satisfy certain compatibility conditions, such that they form a structure called anE∞ -action on X . We say that X is an E∞ -space, or a homotopy-everything space.

In particular, an E∞ -space is a homotopy commutative H -space. For any chosenpoint of C∞(2) determines a multiplication µ : X×X → X , which is unital, commu-tative and associative up to homotopy by the contractibility of each C∞(j) , j ≤ 3.However, the presence of the higher θ∞,j and contractibility of C∞(j) for higherj ensures that an E∞ -structure is a more restrictive notion than just a homotopycommutative H -space. All ‘higher coherence homotopies’ are also available.

The structure maps above allow us to define the Pontryagin product and Dyer–Lashof operations on H∗(X) when X is any E∞ -space.


For example,

C∞S0 =

∐

j≥0

C∞(j)/Σj ≃∐

j≥0

BΣj

is such an E∞ -space. Its homology is

⊕

j≥0

H∗(BΣj) ∼= H∗(C∞S0) ∼=WEH∗(S

0) ∼= P (R)/K

where P (R) is the polynomial algebra on the Dyer–Lashof algebra R , and K isthe two-sided ideal of relations generated by Qi(x) = x2 for i = deg(x) . Theisomorphism takes QI ∈ R to the class QI([1]) , where [1] ∈ H0(BΣ1) is thenonzero class.

This example shows that E∞ -spaces need not be H -groups, i.e., they need not begrouplike H -spaces. However, any grouplike E∞ -space is an infinite loop space.E∞ -spaces are the homotopical generalization of Abelian monoids, and grouplikeE∞ -spaces (=infinite loop spaces) are the homotopical generalization of Abeliangroups.

E∞ -spaces arise, for example, as the classifying space BC of a permutative categoryC , or more generally of a symmetric monoidal category. Grouplike E∞ -spaces ariseas the group completions of such classifying spaces, which equals the K -theoryspace K(C) = ΓBC of the category. The homotopy groups of this infinite loopspace are the K -groups of the category.

We also encounter the homotopical generalization of semi-rings (no additive inverseassumed) and rings. The former arise as the classifying spaces of bi-permutativecategories, or more generally of symmetric bi-monoidal categories.

As an example, consider the category O of finite dimensional real inner productspaces. It admits two symmetric monoidal pairings: direct sum ⊕ and tensorproduct ⊗ of vector spaces. Its classifying space BO ≃

∐

n≥0BO(n) admits

two induced H -space structures: one induced by direct sum BO(n) × BO(m) →BO(n+m) , and one by tensor product BO(n)×BO(m) → BO(nm) .

As a different example, points of QS0 are represented by maps f : Sn → Sn andg : Sm → Sm for n,m large. The loop sum using a suspension coordinate in thesource defines one (additive) pairing on QS0 , but the smash product of such maps,f ∧ g : Sn+m → Sn+m , defines another (multiplicative) pairing.

Such spaces, then, have two E∞ -structures, and thus two associated H -space struc-tures. We think of one of these as additive and the other as multiplicative, andassume they are related by suitable distributivity conditions. This structure isknown as an E∞ -ring space.

The homology H∗(X) of an E∞ -ring space is thus a Hopf algebra with Pontryaginproduct ∗ and Dyer–Lashof operations Qi derived from the additive E∞ -structure,but also a Hopf algebra with Pontryagin product # and Dyer–Lashof operationsQi derived from the multiplicative E∞ -structure. Note that we denote the multi-plicative Dyer–Lashof operations with a tilde. The common coproduct and rightA -module structure combines with these two related products to yield an (allow-able) AR -Hopf bialgebra.

There are formulas in [May7, §§1–3] expressing distributivity of # over the ad-

ditive ∗ and Qi , the mixed Cartan formula for Qi(x ∗ y) , and the mixed Adem

22 JOHN ROGNES

relations for Qa(Qb(x)) . In principle, they inductively reduce the determination

of the multiplicative Pontryagin product # and Dyer–Lashof operations Qi in anE∞ ring space X to their evaluation on additive R -algebra generators for H∗(X) .

4.2. The homology of SO , BO etc..

Calculations begin with O(1) = C2 = Σ2 . Its classifying space BO(1) ≃ RP∞ hasH∗(BO(1)) = Px with deg(x) = 1, and dually H∗(BO(1)) = Z/2ei | i ≥ 1 ,with ei dual to xi .

The diagonal inclusion O(1)n → O(n) induces maps BO(1)n → BO(n) of classi-fying spaces. The induced map on cohomology

H∗(BO(n)) → H∗(BO(1))⊗n ∼= Px1, . . . xn

is an injection, with image the symmetric polynomials in the x1, . . . , xn . Hence

H∗(BO(n)) ∼= Pw1, . . . , wn

with wi the ith Stiefel–Whitney class, mapping to the ith elementary symmetricpolynomial σi(x1, . . . , xn) by the injection above. Letting n→ ∞ , we have

H∗(BO) ∼= Pwi | i ≥ 1.

The coproduct tied to Whitney sum is ψ(wk) =∑

i+j=k wi ⊗ wj .

Dually, the map on homology

H∗(BO(1))⊗n → H∗(BO(n))

is surjective, identifying H∗(BO(n)) with the coinvariants of the n -fold tensorpower of Z/2eii under the permutation action. Thus

H∗(BO(n)) ∼=⊕

d≤n

Pe1, . . . , end

where Pe1, . . . , end denotes the degree d part of the polynomial algebra. Lettingn→ ∞ , we have

H∗(BO) = Pei | i ≥ 1

where each ei is the image of ei ∈ H1(BO(1)) under the inclusion BO(1) → BO .

Let Q0 = Sq1 and Qn = [Qn−1, Sq2n ] be the Milnor primitives in the Steenrod

algebra A . Recall that H∗(K(Z/2, 2)) ∼= Pι2, Qn(ι2) | n ≥ 0 where ι2 is thefundamental class, and

Qn(ι2) = Sq2n

· · · Sq1(ι2).

The Serre spectral sequences for the 1- and 2-connected coverings of BO give:

H∗(BSO) = Pwi | i ≥ 2

H∗(BSpin) = H∗(BSO)/(w2, Qn(w2) | n ≥ 0).


Dually

H∗(BSO) = Pe2, e3, . . .

H∗(BSpin) = (?).

The Serre spectral sequences for the path space fibration for BSO and BSpinyields:

H∗(SO) = Pσ∗(w2i) | i ≥ 1

H∗(Spin) = H∗(SO)/(σ∗(w2))

where σ∗(w2i) in degree 2i− 1 transgresses to w2i . Dually

H∗(SO) = Γa′2i−1 | i ≥ 1

H∗(Spin) = Γa′2i−1 | i ≥ 2

(divided power algebras) with a′2i−1 dual to σ∗(w2i) . This gives a different basisfor H∗(SO) , compared to

H∗(SO) = Eai | i ≥ 1

where ai is the image of ei under the reflection map BO(1) → SO taking a lineL in R

∞ to the reflection in L followed by reflection in a fixed line.

4.3. The homology of SG , G/O and BSG .

There is a unit map QS0 → BO × Z of E∞ ring spaces.

One way to construct it is as an induced map of K -theory spaces: Let E be thesymmetric bimonoidal category of finite sets, under disjoint union

∐

and Cartesianproduct × . Likewise let O be the symmetric bimonoidal category of finite dimen-sional real inner product spaces, under direct sum ⊕ and tensor product ⊗ . Thereis a symmetric bimonoidal functor E → O taking a finite set U to the real vectorspace V = RU with U as an orthonormal basis. Then on classifying spaces wehave a map

BE ≃∐

n≥0

BΣn −→ BO ≃∐

n≥0

BO(n)

of E∞ ring spaces. Group completing, we obtain the unit map

K(E) ≃ QS0 −→ K(O) ≃ BO × Z

by way of the Barratt–Priddy–Quillen equivalence on the left.

On π0 , the unit map is an isomorphism Z ∼= Z . Let [i] ∈ H0(QS0) denote the

homology class of a point in the degree i component Qi(S0) ⊂ QS0 . Restricting the

unit map to the 0-components, we obtain an additive infinite loop map Q0(S0) →

BO = BO⊕ . Restricting instead to the 1-components, we get a multiplicativeinfinite loop map SG = Q1(S

0) → BO⊗ . (These connected E∞ -spaces are actuallyinfinite loop spaces, as noted above.)

24 JOHN ROGNES

Consider the infinite loop space fiber sequence

SOj−→ SG→ G/O → BSO

Bj−−→ BSG.

Here j is the natural inclusion (the j -map), taking an isometry Rn → R

n in SO(n)to its induced map on 1-point compactifications Sn → Sn in SG . Then j is aninfinite loop map from the additive infinite loop space structure on SO ≃ ΩBSO⊕

to the multiplicative infinite loop space structure on SG = Q1(S0) . Next, Bj is

the classifying map of j , and is G/O the homotopy fiber of Bj .

The calculation of H∗(QS0) gives H∗(Q0(S

0)) ∼= P (X0) as an algebra under ∗ ,where

X0 = QI([1]) ∗ [2− 2ℓ(I)] | I admissible, ℓ(I) ≥ 1 and e(I) > 0.

We can define a shifted ∗ -product ∗ on H∗(SG) by x∗y = (x∗ [−1])∗(y∗ [−1])∗ [1] .Then H∗(SG) ∼= P (X1) as an algebra under ∗ , where

X1 = QI([1]) ∗ [1− 2ℓ(I)] | I admissible, ℓ(I) ≥ 1 and e(I) > 0.

However, we are interested in the multiplicative infinite loop space structure on SG ,which is related to that on G/O , and therefore rather want to know the algebrastructure of H∗(SG) under #.

For an admissible sequence I with d(I) > 0, define

xI = QI([1]) ∗ [1− 2ℓ(I)] ∈ H∗(SG).

In particular xi = Qi([1]) ∗ [−1] for i ≥ 1.

Theorem.

(1) H∗(SO) ∼= Eai | i ≥ 1 as an algebra. Here ai is the image of the generatorin Hi(BO(1)) under the reflection map BO(1) → SO .

(2) H∗(SG) ∼= Exi | i ≥ 1 ⊗ P (X) as an algebra under # , where

X = xI | ℓ(I) > 2 and e(I) > 0 , or ℓ(I) = 2 and e(I) ≥ 0.

Here X is the union of the x(i,i) with i ≥ 1 corresponding to I = (i, i) of excess0 , and the xI with ℓ(I) ≥ 2 and e(I) > 0 .

(3) j∗(ai) = xi for all i ≥ 1 , so im(j∗) = Exi | i ≥ 1 and

H∗(G/O) ∼= H∗(SG)//H∗(SO) ∼= P (X).

(4) H∗(BO) ∼= Pvi | i ≥ 1 , where vi is the image of the generator ei inHi(BO(1)) under the map BO(1) → BO .

(5) H∗(BSO) ∼= Pvi | i ≥ 2 , where the vi ’s correspond under the map BSO →BO . Here vi+1 = σ∗(ai) where σ∗ is the homology suspension, so vi+1 transgressesto ai ∈ Hi(SO) in the Serre spectral sequence for the path space fibration for BSO .

(6) H∗(BG) ∼= H∗(BO)⊗ Eσ∗(x(i,i)) ⊗ P (BX) as a Hopf algebra. Here

BX = σ∗(xI) | ℓ(I) > 2 and e(I) > 1 , or ℓ(I) = 2 and e(I) ≥ 1.


(7) H∗(BSG) ∼= H∗(BSO)⊗ Eσ∗(x(i,i)) ⊗ P (BX) as a Hopf algebra, with BXas above.

See [May7, §5]. The various non-classical parts are due to Milgram, May andMadsen.

Comparing the sets of polynomial generators X and X1 , note that passing fromthe product ∗ to # we delete the elements xi and adjoin their squares x(i,i) underthe ∗ product. Thus the appearance of these generators in H∗(SG) is forced bythe relations x2i = 0, arising in H∗(SG) since H∗(SO) is an exterior algebra.

4.4. The R-algebra structure of H∗(SG) .

In the previous section we described the multiplicative #-algebra structure onH∗(SG) in terms of the generators xI , but these were defined by means of theadditive homology operations Qi . We would rather have a description in terms ofthe multiplicative homology operations Qi .

Theorem (Madsen). H∗(SG) ∼= Exi | i ≥ 1 ⊗ P (X) as an algebra under # ,where

X = QJ(xK) | ℓ(K) = 2 and x(J,K) ∈ X .

Both Exi | i ≥ 1 = H∗(SO) and P (X) are sub AR -Hopf algebras of H∗(SG) .

Thus H∗(G/O) ∼= P (X) .

For I = (J,K) , ℓ(K) = 2, such that xI ∈ X , write xI = QJ(xK) for the corre-

sponding element of X .

Corollary. As Hopf algebras

H∗(BSG) ∼= H∗(BSO)⊗ Eσ∗(x(i,i)) | i ≥ 1 ⊗ P (BX)

and

H∗(B(G/O)) = H∗(BSG)//H∗(BSO) ∼= Eσ∗(x(i,i)) | i ≥ 1 ⊗ P (BX)

where

BX = σ∗(xI) | ℓ(I) > 2 and e(I) > 1 or ℓ(I) = 2 and e(I) ≥ 1.

Thus the admissible xi and xK with ℓ(K) = 2 generate H∗(SG) as a (multiplica-

tive) R -algebra. Furthermore, operations Qi(xK) with (i,K) inadmissible candecompose many of the xK with ℓ(K) = 2:

Theorem (Madsen). The following set is a basis for the Z/2-module of R -algebra indecomposable elements of H∗(SG) :

x2k | k ≥ 0 ∪ x(2k,2k) | k ≥ 0 ∪ x(2kn+2k,2kn) | n ≥ 1 and k ≥ 0

See [Mad] and [May7, §6, §13]. The set X contains precisely one element from theset above in each degree ≥ 2.

26 JOHN ROGNES

4.5. Homology operations for orthogonal and unitary groups.

The symmetric bi-monoidal category of real inner product spaces has classifyingspace BO ≃

∐

n≥0BO(n) , with group completion ΓBO ≃ BO × Z . We identifyBO with its 0-component Γ0BO ⊂ ΓBO .

In H∗(BO(1)) let ei be the standard basis. Let

BO(1)η−→

∐

n≥0

BO(n)ι−→ BO × Z

be the standard inclusions, and let vi = ι∗η∗(ei) and vi = vi ∗ [−1] ∈ Hi(BO) .Then H∗(BO) ∼= Pvi | i ≥ 1 as an algebra under ∗ .

Theorem (Priddy).

Qa(vb) =∑

i

(a− b− 1, b− i)vi ∗ va+b−i.


a−i−1b−i

)

.

This determines Qa(vb) via the Cartan formula

Qk(vb) =∑

i+j=k

Qi(vb) ∗Qj([−1]).

The Qj([−1]) can be inductively determined by relating the Qi to the conjugationχ . Modulo ∗ decomposable elements,

Qa(vb) ≡ (a− b− 1, b− i)va+b,

but the precise decomposables appearing are not so easy to list.

The symmetric bi-monoidal category of complex inner product spaces has classify-ing space BU ≃

∐

n≥0BU(n) , with group completion ΓBU ≃ BU×Z . We identifyBU with its 0-component Γ0BU ⊂ ΓBU .

Let fi ∈ Hi(BU(1)) be the image of ei under the complexification map BO(1) →BU(1). Then f2i−1 = 0 and f2i | i ≥ 0 is a basis for H∗(BU(1)). Letη : BU(1) →

∐

n≥0BU(n) → BU×Z be the standard inclusion, and let vi = η∗(fi)

and fi = fi ∗ [−1] ∈ Hi(BU) . As before, v2i−1 = v2i−1 = 0. Then H∗(BU) ∼=Pv2i | i ≥ 1 as an algebra under ∗ .

Theorem (Priddy).

Q2a(v2b) =∑

i

(a− b− 1, b− i)v2i ∗ v2a+2b−2i.


a−i−1b−i

)

.

Again this determines Q2a(v2b) via the Cartan formula.

See [May7, §§7–8].


4.6. The complex image of J space, JU .

The K -theory K(F3) of the category GL(F3) of finite dimensional F3 -vector spacesis the group completion of the classifying space BGL(F3) ≃

∐

n≥0BGLn(F3 . It ishomotopy equivalent to the homotopy fiber JU × Z of

ψ3 − 1: BU × Z → BU,

also known as the complex image of J space. We identify JU with the 0-componentΓ0BGL(F3) .

Consider the standard inclusions

BO(1)η−→

∐

n≥0

BGLn(F3)ι−→ JU × Z.

The first map identifies O(1) and GL1(F3) . Let vi = ι∗η∗(ei) and vi = vi ∗ [−1] ∈Hi(JU) .

Theorem (Quillen).

H∗(JU) ∼= Pv2i | i ≥ 1 ⊗ Ev2i−1 | i ≥ 1

as an algebra under ∗ .

Again Priddy’s theorem above allows the determination of the operations Qi onH∗(JU) , in much the same way as for H∗(BO) .

The elements

p2i−1 = v2i−1 +

i−1∑

j=1

v2j ∗ p2i−2j−1

p4i = iv2i ∗ v2i +i−1∑

j=1

v2j ∗ v2j ∗ p4i−4j

for i ≥ 1 form a basis for the primitive elements in H∗(JU) .

4.7. The orthogonal image of J -space, JO .

The K -theory of the category O(F3) of finite dimensional inner product spacesover F3 might be called the orthogonal K -theory KO(F3) of the finite field. It ishomotopy equivalent to the homotopy fiber JO × Z of

ψ3 − 1: BO × Z −→ BSO.

We identify JO ∼= Γ0BO(F3) .

There is an involution Φ on the subcategory of even dimensional inner product

spaces in O(F3) , acting on 2× 2-matrices by conjugation with γ =(

−1 1

1 1

)

. Thus

Φ∗ is an involution on H∗(JO) .

28 JOHN ROGNES

Theorem.

H∗(JO) ∼= Pvi | i ≥ 1 ⊗ Eui | i ≥ 1

as an algebra under ∗ , where (Φ− 1)∗(vi) = ui . The homology operations satisfy:

Qa(vb) =∑

i,j,k

(a− b− 1, b− j)vi ∗ vk ∗ uj−i ∗ ua+b−j−k

Qa(ub) =∑

i,j

(a− b− i− 1, b− j)ui ∗ uj ∗ ua+b−i−j

4.8. The real image of J -space, J .

The K -theory of the category N (F3) of finite dimensional inner product spacesover F3 with morphisms of spinor norm equal to their determinant is denotedKN(F3) . It is homotopy equivalent to the homotopy fiber J × Z of

ψ3 − 1: BO × Z −→ BSpin.

We identify J ∼= Γ0BN (F3) . This is the real image of J space.

There is a unit map e : QS0 → J × Z of E∞ ring spaces, lifting the unit mapQS0 → BO × Z . Write J⊗ for the 1-component of J × Z , then e : SG → J⊗ isan infinite loop map, related to the Adams e -invariant. Let C = hofib(e) be thehomotopy fiber of e , called the cokernel of J space. Hence there is a fiber sequenceof infinite loop spaces

C −→ SGe−→ J⊗.

A solution to the Adams conjecture yields a map α : J → SG , such that thecomposite eα : J → J⊗ is a homotopy equivalence. Such a solution exists as aspace level map, but not as an H -map. Hence as groups, π∗(SG) ∼= π∗(J)⊕π∗(C) ,with the summand π∗(J) closely related to the image of the j -homomorphismsj∗ : π∗(SO) → π∗(SG) . This is why J is called the (real) image of J space, and Cthe cokernel of J space.

Let u0 = v0 = [0] . The two bases for H∗(SO) in 4.2 are hiding behind a change ofbasis in the following theorem.

Theorem.

H∗(J) ∼= P∑

i+j=a

ui ∗ vj | a ≥ 1 ⊗ Eu′i | i 6= 2k

where u′i ∈ Eui and ui + u′i is decomposable under ∗ .

Corollary. No H -map J → SG can induce a monomorphism on H2 .

Proof. H2(J) contains (u1 + v1)2 = v21 as an ∗ -algebra, and H1(SG) has basis

x1 with x21 = 0 under the #-product, so v21 maps to 0 under any H -map.

((More from [May7, §§12–13] ?))


References

[AK] S. Araki and T. Kudo, Topology of Hn -spaces and Hn -squaring operations, Mem. Fac.Sci. Kyushu Univ. Ser. A 10 (1956), 85–120.

[Br] W. Browder, Homology operations and loop spaces, Illinois J. Math. 4 (1960), 347–357.

[DL] E. Dyer and R. K. Lashof, Homology of iterated loop spaces, Amer. J. Math. 84 (1962),

35–88.

[ES] D. B. A. Epstein and N. Steenrod, Cohomology Operations, Princeton Univ. Press, 1962.

[FP] Z. Fiedorowicz and S. Priddy, Homology of Classical Groups Over Finite Fields and Their

Associated Infinite Loop Spaces, Lecture Notes in Math., vol. 674, Springer Verlag, 1978.

[McL] S. MacLane, Homology, Academic Press, 1963.

[Mad] I. Madsen, On the action of the Dyer–Lashof algebra in H∗(G) , Pacific J. Math 60 (1975),

235–275.

[May1] J. P. May, Categories of spectra and infinite loop spaces, Category Theory, HomologyTheory and their Applications III, Lecture Notes in Math., vol. 99, Springer Verlag, 1969,pp. 448–479.

[May2] , A general algebraic approach to Steenrod operations, The Steenrod Algebra andIts Applications, Lecture Notes in Math., vol. 168, Springer Verlag, 1970, pp. 153–231.

[May3] , The Geometry of Interated Loop Spaces, Lecture Notes in Math., vol. 271,

Springer Verlag, 1972.

[May4] , E∞ spaces, group completions, and permutative categories, New Developments

in Topology (G. Segal, ed.), London Math. Soc. Lect. Note Series, vol. 11, CambridgeUniv. Press, 1974.

[May5] , The homology of E∞ spaces, The Homology of Iterated Loop Spaces, LectureNotes in Math., vol. 533, Springer Verlag, 1976, pp. 1–68.

[May6] , The homology of E∞ ring spaces, The Homology of Iterated Loop Spaces, Lecture

Notes in Math., vol. 533, Springer Verlag, 1976, pp. 69–206.

[May7] , E∞ Ring Spaces and E∞ Ring Spectra, Lecture Notes in Math., vol. 577,

Springer Verlag, 1977.

[Mil] J. W. Milnor, The Steenrod algebra and its dual, Ann. of Math. 67 (1958), 150–171.

[MM] J. W. Milnor and J. C. Moore, On the structure of Hopf algebras, Ann. of Math. 81

(1965), 211-264.

[Q] D. Quillen, On the cohomology and K -theory of the general linear groups over a finite

field, Ann. of Math. 96 (1972), 552–586.

homology and cohomology operations johnrognes 1 ... · unital) algebra f generated by the steenrod...

Documents