Modular coinvariants and the mod p homology of QSk

In this paper, we use the modular coinvariants theory to establish a complete set of relations of the mod p homology of {QSk }k≥0, for p odd, as a ring object in the category of coalgebras, so called a coalgebraic ring or a Hopf ring. Beside, we also describe the action of the mod p DyerLashof algebra as well as one of the mod p Steenrod algebra on the coalgebraic ring

Modular coinvariants and the mod p homology of QSk

Phan Hoàng Chơn

Abstract

In this paper, we use the modular coinvariants theory to establish a complete set of relations

of the mod p homology of {QS k}k≥0, for p odd, as a ring object in the category of coalgebras,

so called a coalgebraic ring or a Hopf ring Beside, we also describe the action of the mod p Dyer-Lashof algebra as well as one of the mod p Steenrod algebra on the coalgebraic ring.

1 Introduction

Let G∗(−) be an unreduced multiplicative cohomology theory Then, G∗(−) can be

represented unstably by the infinite loop spaces G n of its associated Ω-spectrum (i.e G k (X) ∼=

[X, G k ] naturally and ΩG k+1 ' G k , where we denote by [X, Y ] the homotopy classes of unbased maps from X to Y ) The collection of these spaces G∗= {G k}k∈Z is considered as a gradedring space with the loop sum

m : G k × G k → G k

and the composition product

µ : G k × G ` → G k+` Therefore, the homology of {G k}k∈Z (beside the usual addition and coproduct) has two

operations, which are denoted by ? and ◦, respectively, induced by m and µ These operations make the homology of {G k}k∈Z a ring object in the category of coalgebras, which is called a

Hopf ring or coalgebraic ring (see Ravenel-Wilson [26], and Hunton-Turner [9]) The Hopf ring

structure actually becomes an important tool to study the homology of Ω-spectrum as well

as the unreduced generalized multiplicative cohomology theory, and it is of interest in study

of algebraic topologists For example, the Hopf ring for complex cobordism M U is studied

by Ravenel-Wilson [26], the Hopf ring for Morava K-theory is studied by Wilson [28] and for connective Morava K-theory by Kramer[19], Boardman-Kramer-Wilson [2] Recently, the Hopf ring structure for BP and KO, KU are respectively investigated by Kashiwabara [11],

Kashiwabara-Strickland-Turner [16] and Mortion-Strickland [23].

Let QS k = lim

−→ΩnΣn S k be the infinite loop space of the sphere S k Then {QS k}k≥0 is an

Ω-spectrum, called the sphere spectrum, therefore, the mod p homology of {QS k}k≥0also has

a Hopf ring structure Moreover, it is well known that all spectra are module spectra over the

sphere spectrum, so the mod p homology of any infinite loop space becomes an H∗QS0-module

or {H∗QS k}k≥0-module object in the category of coalgebras, which is called a coalgebraic

module As is well-known, (see Kashiwabara [14]) the mod p homology of an infinite loop

space has an A-H∗QS0-coalgebraic module structure Beside, from the result of May [3], the

mod p homology of an infinite loop space also has a so-called A-R-allowable Hopf algebra, i.e.,

it is a Hopf algebra on which both the Steenrod and the Dyer-Lashof algebra act satisfying

some compatibility conditions Thus, understanding the coalgebraic ring structure of H∗QS0

2000 Mathematics Subject Classification 55P47, 55S12 (Primary), 55S10, 20C20 (Secondary).

This work is partial supported by a NAFOSTED gant.

plays important role in the study of homology of infinite loop spaces as well as in the study

of the category of A-H∗QS0-coalgebraic modules and one of A-R-allowable Hopf algebras, and

relationship between them

By the results of Araki-Kudo [1], Dyer-Lashof [5] and May [3], the mod p homology of

{QS k}k≥0 is generated as a Hopf ring by Q i [1], i ≥ 0, σ (for p = 2) and by Q i [1], i ≥ 0,

βQ i [1], i ≥ 1, σ (for p odd), where Q i is the ith homology operation (which is called the Lashof operation), [1] ∈ H∗QS0 is the image of the non-base point generator of H0S0 under

Dyer-the homomorphism H0S0→ H0QS0induced by the inclusion S0,→ QS0and σ is the image of the basis element of H1S1under the homomorphism H1S1→ H1QS1induced by the inclusion

S1,→ QS1 This actually corresponds the fact that the Quillen’s approximation map of finite

groups by elementary abelian subgroups is a monomorphism [25] However, a long time, no

one undertook to sudy the relations until the importance of the coalgebraic ring structure of

H∗QS k is clearly made again from works of Hunton-Turner [9] and Kashiwabara [13] (which

develop the homological algebra for the category of modules over a Hopf ring) These works are

maybe the main motivation for study in [27] and [6], which give a description of a complete set

of relations as a Hopf ring of H∗QS k for p = 2 Later, it was discovered in [12] that the nice

description of the complete set of relations comes from the fact that the Quillen’s map for the

symmetric groups is actually an isomorphism at the prime 2 (see [7]) Also according to [7],

the map is no longer an isomorphism for odd primes, therefore, it is difficult to generalize the

results in [27] and [6] for odd primes However, in the Brown-Peterson cohomology theory, the Quillen’s map of the symmetric groups is also an isomorphism [8] This fact allows to generalize

the results in [27] and [6] for the Bockstein-nil homology of H∗QS k [15] Thus, the describing

of a complete set of relations as a Hopf ring for {H∗QS k}k≥0 is not only important but alsodifficult

In this work, we discover that the isomorphism between the dual of R[n] and the image of the

restriction map from the cohomology of the symmetric group Σp n to the elementary abelian

p-group of rank n, V n, is the main key to establish the nice description of the complete set

of relations as above discussion, where R[n] denote the subspace of the Dyer-Lashof algebra

spanned by all monomials of length n Using this idea and modifying the framework in [27]

allows us to obtain a nice description of the complete set of relations as a Hopf ring of

{H∗QS k}k≥0 for p odd In more detail, we construct a new basis for B[n]∗, which is thedual of the image of the restriction map from the cohomology of the symmetric group Σp n to

the cohomology of V n [24] Using the basis and combining with the fact that the induced in

homology of the Kahn-Priddy transfer, tr∗(n) , is multiplicative and GL n-invariant to investigate,

we obtain an analogous description of a complete set of relation of {H∗QS k}k≥0as a coalgebraicring for odd primes This fact again confirms the closely correspondence between the Hopf ring

structure of {H∗QS k}k≥0and the Quillen’s map of the symmetric groups The results in [27],

[6] as well as in [15] can be deduce from our results by letting p = 2 or killing the action of

the Bockstein operation for p odd It should be noted that much of our work rests on previous

results with a suitable modifying For example, relations (4.1)-(4.3) (see Proposition 4.7) can be

followed from the multiplicativity and the GL2-invariant of tr∗(2)as the case of p = 2 However, the relation (4.4) is here difference In deed, for p = 2 or for the Bocktein-nil homology, the

general case of the relation can be simple implied from the case of the length 1 and otherrelations, but here it is impossible because of the action of Bocktein operation

In this paper, additive base of (H∗BV n)GL n , (H∗BV n)GL n as well as the cokernel of

the restriction map H∗BΣ p n // (H∗BV n)GL n are also established Beside, using the similar

method of Turner [27], we give descriptions of the action of the mod p Steenrod algebra A and

the action of the mod p Dyer-Lashof algebra R on the Hopf ring as relative results.

The paper is divided into five sections The first two sections are preliminaries In Section

2, we review some main points of the Dickson-Mùi algebra, the image of the restriction from

the cohomology of the symmetric group Σp n to the cohomology of the elementary abelian group of rank n, V n , as well as the mod p Dyer-Lashof algebra In Section 3, we construct new additive base for (H∗BV n)GL n , (H∗BV n)GL n, the cokenel as well as the dual of the image of

p-the restriction map H∗BΣ p n // (H∗BV n)GL n By the results of May [3], the new basis of the

dual of the image of the restriction map is considered as an additive basis of the subspace R[n]

of the mod p Dyer-Lashof algebra The Hopf ring for {H∗QS k}k≥0 as well as the actions of

the Steenrod algebra and the Dyer-Lashof algebra on {H∗QS k}k≥0are respectively presented

in two final sections

Fp [x1, · · · , x n ] is the polynomial algebra generated by x i’s

Let GL n denote the general linear group GL n = GL(V n ) The group GL n acts on V n and

then on H∗BV n according to the following standard action

The algebra of all invariants of H∗BV n under the actions of GL n is computed by Dickson [4]

and Mùi [24] We briefly summarize their results For any n-tuple of non-negative integers

(r1, , r n ), put [r1, · · · , r n ] := det(x p i rj), and define

L n,i := [0, · · · , ˆi, , n]; L n := L n,n; q n,i := L n,i /L n , for any 1 ≤ i ≤ n In particular, q n,n = 1 and by convention, set q n,i = 0 for i < 0 The degree

For 0 ≤ i1< · · · < i k ≤ n − 1, we define

M n;i1, ,i k := [k; 0, · · · , ˆi1, · · · , ˆi k , · · · , n − 1],

R n;i1,··· ,i k := M n;i1, ,i k L p−2 n The degree of M n;i1,··· ,i k is k + 2((1 + · · · + p n−1 ) − (p i1+ · · · + p i k)) and then the degree of

R n;i1,··· ,i k is k + 2(p − 1)(1 + · · · + p n−1 ) − 2(p i1+ · · · + p i k)

We put P n:= Fp [x1, · · · , x n ] The subspace of all invariants of H∗BV n under the action of

GL n is given by the following theorem

Theorem 2.1 (Dickson [4], Mùi [24]).

(i) The subspace of all invariants of P n under the action of GL n is given by

D[n] := P GL n

n = Fp [q n,0 , · · · , q n,n−1 ].

(ii) As a D[n]-module, (H∗BV n)GL n is free and has a basis consisting of 1 and all elements

of {R n;i1,··· ,i k : 1 ≤ k ≤ n, 0 ≤ i1< · · · < i k ≤ n − 1} In other words,

Mùi shows that, [24], the algebraB[n] is the image of the restriction from the cohomology

of the symmetric group Σp n to the cohomology of the elementary abelian p-group of rank n,

V n

In [3], May shows thatL

n≥1 B[n] is isomorphic to the dual of the Dyer-Lashof algebra.

2.2 The Dyer-Lashof algebra

Let us recall the construction of the Dyer-Lashof algebra Let F be the free algebra generated

by {f i |i ≥ 0} and {βf i |i > 0} over F p , with |f i | = 2i(p − 1) and |βf i | = 2i(p − 1) − 1 Then

F becomes a coalgebra equipped with coproduct ψ : F → F ⊗ F given by

ψf i =Xf i−j ⊗ f j; ψβf i=Xβf i−j ⊗ f j+Xf i−j ⊗ βf j

Elements of F are of the form

f I,ε = β 1f i1· · · β  n f i n , where (I, ε) = (1, i1, · · · ,  n , i n ) with  j ∈ {0, 1} and i j ≥  j for 1 ≤ j ≤ n The degree of f I,ε

is equal to 2(p − 1)(i1+ · · · + i n ) − (1+ · · · +  n ) Let l(f I,ε ) = n denote the length of (I, ε) or

f I,ε and let the excess of (I, ε) or f I,ε be denoted and defined by exc(f I,ε ) = 2i1− 1− |f I0,ε0|,

where (I0, ε0) = (2, i2, · · · ,  n , i n) In other words,

The excess is defined ∞ if (I, ε) = ∅ and we omit  j if it is 0 The element f I,ε is as

having non-negative excess if f I t ,ε t is non-negative excess for all 1 ≤ t ≤ n, where (I t , ε t) =

( t , i t , · · · ,  n , i n)

The algebra F is a Hopf algebra with unit η : F p → F and augmentation  : F → F psending

f0 to 1 and others to zero

Let T = F /I exc , where I exc is the two-sided ideal of F generated by all elements of negative

excess Then T inherits the structure of a Hopf algebra Denote the image of f I,ε by e I,ε The

degree, length, excess described above passes to T

Let I Adem be the two-sided ideal of T generated by elements

These elements are called Adem relations The quotient algebra R = T /I Adem is called the

Dyer-Lashof algebra We denote the image of e I,ε by Q I,ε , then Q i and βQ i satisfy the Ademrelations:

and non-negative excess, which is called the admissible basis

Next, we recall the structure of the dual of the Dyer-Lashof algebra For p = 2, the structure

is studied by Madsen [20] He shows that R[n]∗ is isomorphic to the Dickson algebra For p

odd, May [3] shows that R[n]∗is isomorphic to a proper subalgebra of the Dickson-Mùi algebra

(see also Kechagias [18]).

For convenience we shall write I instead of (I, ε).

Let I n,i , J n;i , K n;s,i be admissible sequences of non-negative excess and length n as follows

I n,i = (p i−1 (p n−i − 1), · · · , p n−i − 1, p n−i−1 , · · · , 1);

J n;i = (p i−1 (p n−i − 1), · · · , p n−i − 1, (1, p n−i−1 ), · · · , 1);

K n;s,i = (p i−1 (p n−i − 1) − p s−1 , · · · , p i−s (p n−i − 1) − 1),

(1, p i−s−1 (p n−i − 1)), p i−s−2 (p n−i − 1), · · · , p(p n−i − 1), (1, p n−i − 1), p n−i−1 , · · · , 1).

Then the excess of Q I n,i is 0 if 0 < i ≤ n − 1 and 2 if i = 0; and

exc(Q J n;i ) = 1, 0 ≤ i ≤ n − 1;

exc(Q K n;s,i ) = 0, 0 ≤ s < i ≤ n − 1.

Let ξ n,i = (Q I n,i)∗, 0 ≤ i ≤ n − 1, τ n;i = (Q J n;i)∗, 0 ≤ i ≤ n − 1, and σ n;s,i = (Q K n;s,i)∗, 0 ≤

s < i ≤ n − 1, with respect to the admissible basis of R[n].

The following theorem gives the structure of the dual of the Dyer-Lashof algebra

Theorem 2.2 (May [3], see also Kechagias [18]). As an algebra, R[n]∗ is isomorphic to

the free associative commutative algebra over Fp generated by the set {ξ n,i , τ n;i , σ n;s,i : 0 ≤ i ≤

n − 1, 0 ≤ s < i}, subject to relations:

(i) τ2

n,i = 0, 0 ≤ i ≤ n − 1;

(ii) τ n;s τ n;i = σ n;s,i ξ n,0 , 0 ≤ s < i ≤ n − 1;

(iii) τ n;s τ n;i τ n;j = τ n;s σ n;i,j ξ n,0 , 0 ≤ s < i < j ≤ n − 1;

(iv) τ n;s τ n;i τ n;j τ n;k = σ n;s,i σ n;j,k ξ n,02 , 0 ≤ s < i < j < k ≤ n − 1.

The relationship between the dual of the Dyer-Lashof algebra and the modular invariants isgiven by the following theorem

Theorem 2.3 (Kechagias [17], [18]). As algebras over the Steenrod algebra, R[n]∗ is

isomorphic to B[n] via the isomorphism Φ given by Φ(ξ n,i ) = −q n,i , Φ(τ n;i ) = R n;i , 0 ≤ i ≤

n − 1 and Φ(σ n;s,i ) = R n;s,i , 0 ≤ s < i ≤ n − 1.

3 Additive base of modular (co)invariants

In this section, we construct a new basis for B[n]∗, which is a useful tool for the Section 4

Since R[n] ∼=B[n]∗, the basis can be considered is a basis of R[n] Beside, some additive base of the Dickson-Mùi invariants (H∗BV n)GL n , the Dickson-Mùi coinvariants (H∗BV n)GL n as well

as the cokernel of the restriction map of the symmetric group H∗BΣ p n // (H∗BV n)GL n areestablished

We order the set of tuples I = (1, i1, · · · ,  n , i n) by the ordering defined inductively as follows

It should be noted that, when  k = ω k = 0 for all k, the above ordering coincides with the

lexicographic ordering from the left

Then we obtain the following lemmas.

Lemma 3.1 For i s≥ 0, we have

The proof is complete

For any string of integers I = (1, i1, ,  n , i n ), with i1∈ Z, i s ≥ 0, 2 ≤ s ≤ n, and  s∈

{0, 1}, we put b(I) =P

s  s and m(I) = max{ s : 1 ≤ s ≤ n}.

Lemma 3.2 For I = (1, i1, ,  n , i n ), with i1∈ Z, i s ≥ 0, 2 ≤ s ≤ n,  s ∈ {0, 1}, and i1−

m(I) + b(I) ≥ 0, we have

L (p−1)(i1+i2 ) 1

Since i s ≥ 0, 2 ≤ s ≤ n and i1− m(I) + b(I) ≥ 0, applying the proof of Lemma 3.1, we get

n is the least monomial occurring non-trivially

in M n;s Indeed, it is sufficient to compare the order of n following monomials.

By directly checking, we have the assertion

Combining these facts, we have the assertion of the lemma

Proposition 3.3 For any n ≥ 1, as an F p -vector space, (H∗BV n)GL n has a basis

consisting of all elements q I = R 1

n;0 q i1

n,0 · · · R  n

n;n−1 q i n

n,n−1 for i1∈ Z, i s ≥ 0, 2 ≤ s ≤ n,  s ∈ {0, 1} and i1− m(I) + b(I) ≥ 0.

Proof From Theorem 2.1, {q I = R 1

Moreover, from Lemma 3.2, this set is linear independent

Proposition 3.4 For any n ≥ 1, the set of elements q I = R  n;01 q i1

n,0 · · · R  n

n;n−1 q i n

n,n−1 for

i1∈ Z, i s ≥ 0, 2 ≤ s ≤ n,  s ∈ {0, 1} and 2i1+ b(I) ≥ 0, provides an additive basis for B[n].

Proof From Proposition 3.3, we see that the set in the proposition is the subset of a basis

of (H∗BV n)GL n, therefore, it is linear independent

Moreover, since, for 0 ≤ s < t ≤ n − 1,

R n;s,t = R n;s R n;t q n,0−1,

every elements inB[n] can be written as a linear combination of elements of the set.

Corollary 3.5 For any n ≥ 1, as an F p-vector space, the cokernal of the restriction map

H∗(BΣ p n) // H∗(BV n)GL n has a basis consisting of all elements that are the images under

the quotient map of all elements of the form R 1

n;0 q i1

n,0 · · · R  n

n;n−1 q i n

n,n−1 for i1∈ Z, i s ≥ 0, 2 ≤

s ≤ n,  s ∈ {0, 1} and m(I) − b(I) ≤ i1< −b(I)/2.

For k ≥ 0, the subspace of B[n] generated by {R 1

Let u i ∈ H1BV n be the dual of e i and let v i ∈ H2BV n be the dual of x i Then the homology

of V n , H∗BV n , is the tensor product of the exterior algebra generated by u i’s and the divided

power algebra generated by v i ’s We denote by v i [t] the t-th divided power of v i Since R[n] is

isomorphic toB[n]∗, R[n] is considered the quotient algebra of (H∗BV n)GL n The followingtheorem provides an additive basis forB[n]∗ and then for R[n].

Theorem 3.6 For k ≥ 0, the set of all elements

Hence, we have the assertion of the theorem

It should be noted that, when k = 0, the basis mentioned in Theorem 3.6 is not the dual

basis of the one in Proposition 3.4

Using the proof is similar to the proof of Theorem 3.6, we have the following proposition

Proposition 3.7 For n ≥ 1, the set of all elements

all monomial of length n Then, we have the following proposition.

Proposition 3.8 As algebras over the Steenrod algebra, R k [n]∗∼=Bk [n] via the

isomorphism give in Theorem 2.3

Proof For a string of integers e = (e1, · · · , e j ) such that 1 ≤ e1< · · · < e j ≤ n, we put

L n;e =



K n;e1,e2+ · · · + K n;e j−1 ,e j , if j is even,

K n;e1,e2+ · · · + K n;e j−2 ,e j−1 + J n;e j , if j is odd, and L n;e is the string of all zeros if e is empty Here we mean (1, i1, · · · ,  n , i n) +

(01, j1, · · · , 0n , j n ) to be the string (ω1, t1, · · · , ω n , t n ) with t s = i s + j s and ω s =  s+

0s(mod 2)

In [3, p.38], May shows that for any string I of non-negative excess, it can be uniquely

expressed in the form

3.7 in [3, p.29], we obtain that the set of all monomials

ξ i1

n,0 · · · ξ i n

n,n−1 (σ n;e1,e2· · · σ n;e j−2 ,e j−1)1τ 2

n;e j , 2i1+ 2≥ k provides an additive basis of R k [n]∗

Using relation (ii) in Theorem 2.2, above monomials can be written in the form (up to asign)

By the definition of Bk [n] and Theorem 2.3 we have the assertion of the proposition.

4 The Hopf ring structure of H∗QS k

In this section, we use results of the modular (co)invariants in above sections to describe a

complete set of relations for {H∗QS k}k≥0 as a Hopf ring

Let [1] ∈ H∗QS0 be the image of non-base point generator of H0S0 under the map induced

by the canonical inclusion S0,→ QS0 and let σ ∈ H∗QS1 be the image of the generator of

H1S1under the homomorphism induced by the inclusion S1,→ QS1 Note that the element σ

is usually known as the homology suspension element because σ ◦ x is the homology suspension

of x From the results of Dyer-Lashof [5] and May [3], we have

Theorem 4.1 (Dyer-Lashof [5], May [3]). The mod p homology of {QS k}k≥0is given by

H∗QS0= P [Q I [1] : I admissible, exc(I) + 1> 0] ⊗ F p [Z],

H∗QS k = P [Q I (σ ◦k ) : I admissible, exc(I) + 1> k], k > 0.

Some basic properties are given in the following theorem

Theorem 4.2 (May [3], [22]). For b, f ∈ H∗QS k,

The induced transfer tr∗(1): H∗(BV1)+→ H∗QS0 sends u  v [i(p−1)−] to β  Q i[1] and others tozero.

Let ψ : Σ m× Σn→ Σmn be the permutation product of symmetric groups; and let I n : V n→

in H∗BV n (with respect to the decomposition BV n ' BV r × BV n−r) to the circle product in

H∗QS0 In other words, we have

Since tr (n) = i ◦ BI n and GL n is the “Weyl group” of the inclusion V n ⊂ Σp n, we have an

important feature of the map tr∗(n) is that they factor through the coinvariant of the generallinear group In other words, the diagram

Moreover, we have the following proposition

Proposition 4.3 The transfer tr∗(n)factors through (B[n])∗ In other words, the diagram

Proof Since the induced transfer on cohomology tr∗(n) = (BI n)∗◦ i∗, the image of tr∗(n) is

contained in the image of the restriction (BI )∗: H∗BΣ → H∗BV Moreover, from Mùi