Milnor operations and classifying spaces

We give an example of a nonzero odd degree element of the classifying space of a connected Lie group such that all higher Milnor operations vanish on it. It is a counterexample for a conjecture of Kono and Yagita.


Introduction
For each prime number p, there are the mod p and Brown-Peterson cohomology.For a compact connected Lie group G, the mod p cohomology of the classifying space BG has no nonzero odd degree element if the integral cohomology of G has no p-torsion.So does the Brown-Peterson cohomology.On the one hand, if the integral homology of G has p-torsion, the mod p cohomology of BG has a nonzero odd degree element.On the other hand, for the Brown-Peterson cohomology, Kono and Yagita conjectured the following: Conjecture 1.1 (Kono and Yagita, (1) in Conjecture 4 in [KY93]).There is no nonzero odd degree element in the Brown-Peterson cohomology of the classifying space of a compact Lie group.
Conjecture 1.1 is interesting in conjunction with Totaro's conjecture on the cycle map from the Chow ring of the classifying space of a complex linear algebraic group G to its Brown-Peterson cohomology.In [Tot97], Totaro showed that the cycle map from the Chow ring of a complex smooth algebraic variety to its ordinary cohomology factors through the Brown-Peterson cohomology after localized at p.In [Tot99], he defined the Chow ring CH * (BG) of a linear algebraic group G and conjectured the following.
Conjecture 1.2 (Totaro, p.250 in [Tot99]).For a complex linear algebraic group G, if there is no nonzero odd degree element in the Brown-Peterson cohomology BP * (BG), the cycle map is an isomorphism.
With Conjectures 1.1 and 1.2, we expect a close connection between the Chow ring in algebraic geometry and the Brown-Peterson cohomology in algebraic topology.In [KY93], Kono and Yagita confirmed Conjecture 1.1 for some compact connected Lie groups with p-torsion by computing the Atiyah-Hirzebruch spectral sequences.However, the non-triviality of Milnor operations on odd degree elements yields non-trivial differentials sending odd degree elements to non-zero elements, so odd degree elements do not survive to the E ∞ -term.With their computational results on the Brown-Peterson cohomology of classifying spaces, Kono and Yagita conjectured the following: Conjecture 1.3 (Kono and Yagita, Conjecture 5 in [KY93]).For each nonzero odd degree element x of the mod p cohomology of the classifying space of a compact connected Lie group, there exists an integer i such that for m ≥ i, Conjecture 1.3 is interesting in the cohomology theory of classifying spaces of non-simply connected Lie groups.In [VV05], Vavpetič and Viruel showed that if p is an odd prime, Conjecture 1.3 holds for the projective unitary group P U (p).Moreover, the cohomology of classifying spaces of non-simply connected Lie groups has recently enjoyed renewed interest.Many mathematicians have studied it in various contexts.Antieau, Gu and Williams ([AW14], [Gu19], [Gu20], [GZZZ22]) studied it for the topological period-index problem.Antieau, the author and Tripathy ([Ant16], [Kam15], [Kam17], [Tri16]) studied it for integral Hodge conjecture modulo torsion.Furthermore, the Atiyah-Hirzebruch spectral sequence is used in theoretical physics to study anomalies, cf.García-Etxebarria and Montero [GEM19].
In this paper, we give a counterexample for Conjecture 1.3 in the case p = 2. Our result is as follows: Let H be the quaternions.Let Sp(1) ⊂ H be the symplectic group consisting of unit quaternions.Let G be the quotient of the 3-fold product Sp(1) 3 of the symplectic groups Sp(1) by the subgroup Γ 2 generated by (−1, −1, 1) and (−1, 1, −1).
Theorem 1.4.In the mod 2 cohomology of the classifying space of the compact connected Lie group G above, there exists a nonzero element x 13 of degree 13 such that Q m x 13 = 0 for m ≥ 1.This paper is organized as follows.In Section 2, we describe the action of Milnor operations on the mod 2 cohomology of BSO(3).In Section 3, we prove Theorem 1.4 as Proposition 3.5.
The author would like to thank the anonymous referee for suggestions to improve the exposition of this paper.

Milnor operations
In this section, we recall Milnor operations and the mod 2 cohomology of the classifying space BSO(3).Milnor operations Q m are defined by They have the following properties: These formulae are essential in our proofs Propositions 2.2 and 3.5.The mod 2 cohomology of BSO(3) is a polynomial ring generated by two elements w 2 , w 3 of degree 2, 3, respectively.The action of Steenrod squares on these elements is well-known as the Wu formula.In particular, we have By the Wu formula and by the definition and elementary properties of Milnor operations stated above, it is easy to obtain This section aims to prove the following lemma on the action of Milnor operations on the mod 2 cohomology of BSO(3).
Lemma 2.1.For m ≥ 2, there exists a polynomial g m in w 2 2 and w 2 3 such that we have To prove Lemma 2.1, we recall the relation between Dickson invariants and Milnor operations as Proposition 2.2.The connection between Dickson invariants and Milnor operations is an exciting subject in algebraic topology.Thus, we refer the reader to the classical work of Adams and Wilkerson ([AW80], [Wil83]) for more detail on the background of this section.However, to make this paper self-contained as far as possible, we give detailed proof for Lemma 2.1 without mentioning Dickson invariants and the above background.
Let (Z/2) 2 = Z/2×Z/2 be the elementary abelian 2-subgroup of SO(3) generated by diagonal matrices We denote by ι : (Z/2) 2 → SO(3) the inclusion map.The induced homomorphism is injective, and its image is the subring generated by the following elements.
Then, we have D m x = 0.
Proof.Here, in the proof of Proposition 2.2, by the mod 2 cohomology ring, we mean the mod 2 cohomology ring of B(Z/2) 2 unless otherwise stated explicitly.
Recall that for x, y ∈ H * (X; Z/2).For i = 1 and 2, we have Thus, for elements x, y in the mod 2 cohomology ring, we have Therefore, since the mod 2 cohomology ring is generated by s 1 , s 2 , the fact that D m s i = 0 for i = 1, 2 implies that D m x = 0 for each element x in the mod 2 cohomology ring.Now, for m ≥ 2, we describe the action of the Milnor operation Q m in terms of certain polynomials f m,0 , f m.1 in w 2 2 and w 2 3 and Milnor operations Q 0 , Q 1 .Since the induced homomorphism is injective, by Proposition 2.2, for each x in the mod 2 cohomology of BSO(3), we have We may write it in the following form.
Let us define a matrix A m whose coefficients are polynomials in w 2 2 , w 2 3 as follows: Furthermore, let us define polynomials f m,0 , f m,1 by Then, for x in the mod 2 cohomology of BSO(3), we have Proof of Lemma 2.1.We have the following congruence.
Hence, we have f m,0 ≡ 0 mod (w 2 3 ).Therefore, there exists a polynomial g m in w 2 2 and w 2 3 such that f m,0 = g m w 2 3 .
Recall the fact that 3 .Example 2.3.For m = 2, 3, 4, elements Q m x and polynomials g m in Lemma 2.1 are as follows:

The nonzero odd degree element
In this section, we prove Theorem 1.4 as Proposition 3.5.
We begin with recalling the definition of the connected Lie group G in Section 1 and set up notations.Let us consider the 3-fold product of symplectic groups Sp(1) ⊂ H consisting of unit quaternions.Let Γ 3 = {(±1, ±1, ±1)} be the center of Sp(1) 3 .Let Γ 2 be its subgroup generated by (−1, 1, −1), (1, −1, −1) and Let Z/2 = {(±1, 1, 1)} ⊂ Γ 3 .Then, Z/2 and Γ 2 generate Γ 3 .Moreover, we have Therefore, we have the following fiber sequence: We denote by {E p,q r , d r : E p,q r → E p+r,q−r+1 r } the Leray-Serre spectral sequence associated with this fiber sequence.Let us denote its E r -term by We compute the mod 2 cohomology of BG using the above Leray-Serre spectral sequence.Although it is easy, we quickly review it.See [Kam19] for more detail.We describe the E 2 -term and compute the first non-trivial differential d 2 .Let be the map induced by the projection to the i th factor for i = 1, 2, 3. We denote by w ′ i , w ′′ i , w ′′′ i the cohomology classes Bπ * 1 (w i ), Bπ * 2 (w i ), Bπ * 3 (w i ), respectively.Let u 1 be the generator of the mod 2 cohomology H 1 (BZ/2; Z/2) ∼ = Z/2 of the fibre BZ/2.The E 2 -term is given by To compute the differential d 2 , we consider the Leray-Serre spectral sequence { Ēp,q r , dr : Ēp,q r → Ēp+r,q−r+1 r } associated with the fiber sequence BZ/2 → BSp(1) → BSO(3).
Recall that its E 2 -term is given as follows: and its first nontrivial differential d2 is given by d2 (u 1 ) = w 2 .
Then we have the following commutative diagram, Furthermore, we have 2 ) = w 2 .Now, we are ready to compute the differential d 2 .Suppose that the first nontrivial differential d 2 is given as follows: we obtain is given as follows: Let us recall the relation between the transgression and Steenrod squares.For r ≥ 2, the transgression d r : E 0,r−1 r → E r,0 r commutes with Steenrod squares Sq i .In other words, if d r (x) = y then we may have an element Sq i x ∈ E 0,r−1+i s for r ≤ s, an element Sq i y ∈ E r+i,0 r+i and there hold that d s (Sq i x) = 0 for r ≤ s < r + i and that d r+i (Sq i x) = Sq i y.
Let It is clear that the ring homomorphism φ is injective.Thus, M is isomorphic to the subring φ(M ) of Z/2[w ′ 2 , w ′′ 2 , t 1 ].Therefore, both M and φ(M ) are integral domains.
This work was supported by JSPS KAKENHI Grant Number JP17K05263.
is the bottom line of the E 9 -term of the spectral sequence such that The ring homomorphism φ induces the weight-preserving ring homomorphismφ : M → Z/2[w ′ 2 , w ′′ 2 , t 1 ].
To compute higher terms and differentials, let us consider the ring homomorphism