key: cord-0209607-fbdiumbj
authors: Gu, Xing
title: On the Brown-Peterson cohomology of $BPU_n$ in lower dimensions and the Thom map
date: 2020-05-06
journal: nan
DOI: nan
sha: 4498c14056cac0652d549fc29bd5585f058232ad
doc_id: 209607
cord_uid: fbdiumbj

For an odd prime $p$, we determined the Brown-Peterson cohomology of $BPU_n$ in dimensions $-(2p-2)leq ileq 2p+2$, where $BPU_n$ is the classifying space of the projective unitary group $PU_n$. We construct a family of $p$-torsion classes $eta_{p,k}in BP^{2p^{k+1}+2}(BPU_n)$ for $p|n$ and $kgeq 0$ and identify their images under the Thom map with well understood cohomology classes in $H^*(BPU_n;mathbb{Z}_{(p)})$.

Let p be an odd prime number, and let BP be the corresponding Brown-Peterson spectrum. The Brown-Peterson cohomology BP * (BG) of the classifying space of a compact Lie group or a finite group G is the subject of various works such as Kameko and Yagita [8] , Kono and Yagita [11] , Leary and Yagita [12] , and Yan [21] .

One case that BP * (BG) is particularly interesting is when G is homotopy equivalent to a complex algebraic group via a group homomorphism. In this case the Chow ring of BG, CH * (BG) is defined by Totaro [16] , and one has the cycle class map (1.1) cl : CH * (BG) → H even (BG; Z)

which is a ring homomorphism from the Chow ring to the subring of H * (BG) of even dimensional classes. Although for complex algebraic varieties, Chow rings are in general much more complicated than ordinary cohomology, it is shown in many cases that CH * (BG) is simpler than H * (BG; Z). On the other hand, Totaro [16] shows that the cycle class map (1.1) factors as where MU denotes the complex cobordism theory, and the second map T is the Thom map. The first mapcl is called the refined cycle class map. Therefore, the BP theory, being a p-local approximation of the MU theory, plays the role of a bridge between the Chow ring and the ordinary cohomology of BG. Indeed, it is an interesting problem to find out for which G is the refined cycle class map cl : CH * (BG)c l − → MU * (BG) ⊗ MU * Z an isomorphism. For this to hold, it is necessary that BP * (BG) concentrates in even dimensions. This property is studied for various G by Kono and Yagita [11] .

Key words and phrases. the Brown-Peterson cohomology, the classifying spaces of the projective unitary groups.

The author would like to thank the Max Planck Institute for Mathematics for their hospitality and financial support.

In this paper we focus on BP * (BP U n ), where P U n is the nth projective unitary group. The algebraic invariants of BP U n are much less known compared to BG for most of the other compact Lie groups G. The Chow ring of BP U 3 is determined, up to one relation, by Vezzosi [18] . The additive structure of CH * (BP U 3 ) is independently determined by Kameko and Yagita [8] . Vistoli [19] improved Vezzosi's method and determined the additive structures of the Chow ring and ordinary cohomology with integral coefficients of BP U p for an odd prime p and much of the ring structures, and completed Vezzosi's study of the Chow ring of BP U 3 . The ordinary mod p cohomology and Brown-Peterson cohomology of BP U p are studied by Kameko and Yagita [8] , Kono and Yagita [11] , and Vavpetič and Viruel [17] . The mod 2 ordinary cohomology ring of BP U n for n ≡ 2 (mod 4) is determined by Kono and Mimura [10] and Toda [15] .

For a general positive integer n, the cohomology groups H k (BP U n ; Z) for k ≤ 3 are trivial and are briefly discussed in Section 3. The group H 4 (BP U n ; Z) is determined by Woodward [20] and H 5 (BP U n ; Z) by Antieau and Williams [2] . The ring structure of H * (BP U n ; Z) in dimensions less than or equal to 10 is determined by the author [5] . The author [6] also studies some p-torsion classes of CH * (BP U n ) for n with p-adic valuation 1, i.e., p|n but p 2 ∤ n. To the author's best knowledge, BP * (BP U n ) for n not a prime number has not been studied in any published work before.

Before stating the main conclusions of this paper, we fix some notations. For a spectrum A, we denote by A * its homotopy groups, or the group of coefficients of the homology theory A, considered as a graded abelian group. Denote by A * the group of coefficients of the cohomology theory associated to A. Then A * and A * are isomorphic, but the gradings are opposite to each other. For instance, we have

Theorem 1.1. Assume p|n. There is a homomorphism of Z (p) -algebras

which is an isomorphism in dimensions −(2p − 2) ≤ i ≤ 2p + 2. Here we have dim η p,0 = 2p + 2, dim v 1 = −(2p − 2) and R * a graded Z (p) -algebra such that (a) the Thom map T : BP * (BP U n ) → H * (BP U n ; Z (p) ) takes η p,0 to y p,0 , and (b) R is free as a graded Z (p) -module, and (c) we have an isomorphism

with dim e i = 2i, and (d) the following homomorphism induced from (1.3)

is an isomorphism in dimensions 0 ≤ i ≤ 2p + 2. In particular, in dimensions −(2p − 2) ≤ i ≤ 2p + 2, BP * (BP U n ) concentrates in even dimensions.

For the next conclusion, we note that there are p-torsion classes y p,k ∈ H 2p k+1 +2 (BP U n ; Z (p) ) which are studied in [5] and discussed in more details in Section 3. 

such that T (η p,k ) = y p,k , where T is the Thom map.

The class η p,0 is already given in Theorem 1.1.

Remark 1.2.1. Localizing at p the homotopy fiber sequence BZ n → BSU n → BP U n , we obtain a p-local homotopy equivalence BSU n ≃ (p) − −− → BP U n in the case p ∤ n. Therefore this case is not very interesting.

Remark 1.2.2. In [6] , the author constructs p-torsion classes ρ p,k , k ≥ 0 in the Chow ring of BP U n when the p-adic valuation of n is 1, satisfying cl(ρ p,k ) = y p,k . This suggests that the the classes ρ p,k should exist for n with p-adic valuation greater than 1 and that the refined cycle class map ( [16] ) should take ρ p,0 to η p,0 . This paper is organized as follows. In Section 2 we present some preliminaries on the the cohomology theories P(n) defined by Johnson and Wilson [7] . In Section 3 we give some results on the ordinary cohomology of BP U n , most of which are proved in [5] and [6] , and presented in slightly different forms. In Section 4 and Section 5, we prove Theorem 1.1 and Theorem 1.2, respectively. This paper is written during a visit to the Max Planck Institute for Mathematics (MPIM), and also in the middle of the COVID-19 pandemic. The author would like to thank MPIM for their supports in various ways during this difficult time.

In [7] , Johnson and Wilson consider variations of the BP cohomology theory P (n), n ≥ 0 defined inductively in terms of the following cofiber sequences of spectra

where P(0) = BP and v 0 = p. In particular, we have the cofiber sequence

We immediately deduce

In particular, in the case n = 0 and n = 1, we have

We consider the Atiyah-Hirzebruch spectral sequence of the cohomology theories P(1) and BP. Brown and Peterson [3] studied the Postnikov tower of the spectrum BP, of which the 0th level is the Eilenberg-Mac Lane spectrum HZ (p) . To obtain the 1st level, we consider the map

is the wedge sum of the spectra Σ 2p i −1 HZ (p) . By convention, P k denotes the kth Steenrod reduced power operation and δ denotes the connecting homomorphism associated to the short exact sequence Z (p)

The 1st level of the Postnikov tower is the homotopy fiber of this map. Therefore we have the following lemma that gives the first nontrivial differential of the Atiyah-Hirzebruch spectral sequence of BP in the universal case:

Lemma 2.1. The first nontrivial differential in the Atiyah-Hirzebruch spectral se-

The Postnikov tower of P(1) is constructed in a similar way, with the 0th level HZ p and the 1st level the homotopy fiber of the map

Hence we have the following Lemma 2.2. The first nontrivial differential in the Atiyah-Hirzebruch spectral se-

Finally we take notes on a comparison between the cohomology operations for BP theory and the mod p ordinary cohomology theory due to Kane [9] . Let E = (e 1 , e 2 , · · · ) be a sequence of non-negative integers with only finitely many nonzero terms, and let P E be the operation defined by Milnor [13] . Then we have an BP cohomology operation r E ∈ BP * BP and the following diagram (display (1.1) of [9] ):

where c : A * → A * is the canonical anti-automorphism of the mod p Steenrod algebra A , and the vertical arrows labelled by T are the reduced Thom isomorphisms, i.e., the composition of the Thom map and the mod p reduction.

Since r E is a map of BP-modules, which in particular commutes with the multiplication BP ×p − − → BP, it reduces to an operation r ′ E ∈ P(1) * P(1), and yields a commutative diagram

3. On the ordinary cohomology of BP U n In this section we consider the ordinary cohomology of BP U n . For the most part of this section, we reformulate some results in [5] . By definition we have a short exact sequence of Lie groups 1 → Z n → SU n → P U n → 1 which gives a universal cover of P U n and shows π 1 (P U n ) ∼ = Z n . It follows from the Hurewicz theorem and the universal coefficient theorem that we have

Consider the short exact sequence of Lie groups

which defines the Lie group P U n . Here S 1 is the complex unit circle, and U n is the nth unitary group. Taking classifying spaces, we obtain a homotopy fiber sequence

Notice that BS 1 is of the homotopy type of the Eilenberg-Mac Lane space K(Z, 2). Delooping BS 1 , we obtain another homotopy fiber sequence

Here the map

where e is the p-adic valuation of n), which we call the canonical Brauer class of BP U n .

The homotopy fiber sequence (3.1) induces a Serre spectral sequence ( U E * , * * , U d * , * * ) converging to H * (BP U n ; Z (p) ): 3) ; H t (BU n ; Z (p) )) ⇒ H s+t (BP U n ; Z (p) ).

Then an element in U E s,t 2 is the sum of elements of the form ϑ ⊗ ξ where ϑ ∈ H * (BP U n ; Z (p) ) and ξ ∈ H * (K(Z, 3); Z (p) ). The cohomology of BU n is a polynomial ring in the universal Chern classes:

As in [5] , we consider the linear operator ∇ : H * (BU n ; Z (p) ) → H * −2 (BU n ; Z (p) ) determined by ∇(c i ) = (n − i + 1)c i−1 and the Leibniz formula ∇(ab) = a∇(b) + ∇(a)b. The differential U d 3 is completely determined in [5] : Lemma 3.1 (Corollary 3.10, [5] ). Let ϑ = ϑ[c 1 , · · · , c n ] be a class in H * (BP U n ; Z), and let ξ ∈ H * (K(Z, 3); Z). Let x 1 ∈ H 3 (K(Z, 3); Z) be the canonical Brauer class.

Let T Un and T P Un be the respective standard maximal tori of U n and P U n . Then we have

where each t i is of dimension 2. It follows from an elementary calculation that the canonical projection T Un → T P Un identifies H * (BT P Un ; Z) as the subring of H * (BT Un ; Z) generated by t i − t j for all i = j. Let W be the Weyl group of T P Un in P U n . Then it is well known that the pullback

is an isomorphism, where H ( BT Un ; Z) W is the invariant subring of the Weyl group action. Furthermore, an elementary calculation yields the following Remark 3.3. In general H * (BT Un ; Z) W is not necessarily a polynomial ring. As shown by Vezzosi [18] , there is a class y 12 ∈ H 12 (BT U3 ; Z) W which is not a nontrivial product, but 3y 12 is.

The cohomology of K(Z, 3), in principle, is determined in [4] , where the homology of K(π, n) for any finitely generated abelian group π and any n > 0 is described in full. In [5] , the author uses an alternative description of the cohomology of K(Z, 3), which is consistent with the notations in this paper. The following lemma is well known and can be easily deduced from Section 2 of [5] .

Let x 1 ∈ H 3 (K(Z, 3); Z (p) ) be the canonical Brauer class. Then the group H 2p+2 (K(Z, 3); Z (p) ) ∼ = Z p is generated by a class y p,0 = δP 1 (x 1 ), where δ is the connecting homomorphism and P 1 is the first Steenrod reduced power operation. The mod p reduction of y p,0 , denoted byȳ p,0 , is equal to Q 1 (x 1 ), where Q 1 is one of the Milnor's operations defined in [13] .

Proposition 3.5. In dimension 0 ≤ i ≤ 2p + 2 we have (H * (BT P Un ; Z) W ⊗ Z (p) [x 1 , y p,0 ]/(nx 1 , py p,0 )) ≤2p+2 ∼ = H ≤2p+2 (BP U n ; Z (p) ).

Proof. It follows from Lemma 3.4 that in the spectral sequence U E * , * * , the only possibly nontrivial differentials from U E 0,t * for t ≤ 2p + 2 are U d 0,2t 3 . Therefore, by Lemma 3.1 we have U E 0,t ∞ ∼ = Ker ∇, and by Proposition 3.2, we have U E 0,t ∞ ∼ = H t (BT P Un ; Z) W for t ≤ 2p + 2. The class 1 ⊗ x 1 being an n-torsion permanent cocycle and 1 ⊗ y p,0 being a permanent cocycle follow from Theorem 1.1 and Theorem 1.2 of [5] . Therefore the groups U E s,t ∞ for s + t ≤ 2p + 2 is determined, and we conclude.

In the cohomology ring H * (K(Z, 3); Z (p) ) we have p-torsion classes y p,k = δP p k P p k−1 · · · P 1 (x 1 ), k ≥ 0 of dimension 2p k+1 + 2, where x 1 ∈ H 3 (BP U n ; Z) is the canonical Brauer class. In [6] the author shows that these classes have nontrivial images under χ : H * (K(Z, 3); Z (p) ) → H * (BP U n ; Z (p) ). Theorem 3.6 ((1) of Theorem 1.1, [6] ). In H 2p k+1 +2 (BP U n ; Z (p) ), we have ptorsion classes y p,k = 0 for all odd prime divisors p of n and k ≥ 0.

As pointed out in Section 1, the canonical map BSU n → BP U n is a rational homotopy equivalence. This means that in principal we do not need to worry about the non-torsion classes in H * (BP U n ; Z (p) ) or BP * (BP U n ). We formulate this argument as the following Proposition 3.7. We have

where dim e i = 2i, and

, e 3 , · · · , e n , v 1 , v 2 , · · · ], where BP Q denotes the localization of BP with respect to Q.

In this section we prove Theorem 1.1 by studying an Atiyah-Hirzeburch spectral sequence, and the Thom map for the Eilenberg-Mac Lane space K(Z, 3). Indeed, the Thom map for Eilenberg-Mac Lane spaces are studied in Tamanoi [14] , and we make use of one of his main conclusions. In what follows, we denote by A * the mod p Steenrod algebra, and we use the notations for the stable cohomology operations in Milnor [13] . is an A * -invariant polynomial subalgebra with infinitely many generators:

where τ n+2 ∈ H n+2 (K(Z, n + 2); Z p ) is the fundamental class.

Consider the Atiyah-Hirzebruch spectral sequence (E(1) * , * * , d * , * * ) converging to P(1) * (BP U n ):

Proof. Consider the Atiyah-Hirzebruch spectral sequence converging to P(1)

We use overhead bars to indicate mod p reductions of classes in H * (BP U n ; Z (p) ). The only nontrivial entry E(1) s,t 2 with s + t = 3 is E(1) 3,0 2 , which is generated by 1 ⊗x 1 . It follows from Lemma 2.2 that in the spectral sequence E(1) * , * * , we have d 2p−1 (1 ⊗x 1 ) = v 1 ⊗ Q 1 (x 1 ) = v 1 ⊗ȳ p,0 = 0, and we conclude. Proof. Let E * , * * be the Atiyah-Hirzebruch spectral sequence for BP * (BP U n ):

Consider the cofiber sequence (2.2)

which induces a long exact sequence of BP * -modules

By Proposition 3.5, we have

where e is the p-adic valuation of n, i.e., we have n = p e n ′ where p ∤ n ′ . Therefore, BP 3 (BP U n ) is a subring of Z/p e . Let i = 3 in the long exact sequence (4.2). Then we have a monomorphism

By Lemma 4.2, we have P(1) 3 (BP U n ) = 0. Hence we have BP 3 (BP U n ) = 0.

Lemma 4.4. The class 1 ⊗ y p,0 ∈ E 2p+2,0 2 is a permanent cocycle.

Thus we denote by η p,0 the class represented by 1⊗y p,0 . The Thom map therefore takes η p,0 to y p,0 .

Proof. Recall that the mod p reduction of y p,0 isȳ p,0 = Q 1 (x 1 ). It follows from Theorem 4.1 thatȳ p,0 , as a class in H * (K(Z, 3); Z p ), is in the image of the reduced Thom map

where the first arrow is the Thom map and the second one is the mod p reduction. Moreover, the second arrow is an isomorphism, from which it follows that y p,0 ∈ H 2p+2 (K(Z, 3); Z (p) ) is in the image of T . Therefore, the class y p,0 ∈ H * (BP U n ; Z (p) ) is in the image of the reduced Thom map, since we have the following commutative diagram:

where the vertical arrows are the Thom maps, and the horizontal ones are induced by the canonical Brauer class χ defined in (3.2) . Therefore the class y p,0 ∈ H 2p+2 (BP U n ; Z (p) ) is in the image of T and we conclude.

Next we consider the classes

It follows from Proposition 3.7 that there are nonnegative integers λ i , i ≥ 2 and µ, such that p λi e i and p µ v 1 are permanent cocycles. Therefore, the groups E s,t ∞ for 0 ≤ s ≤ 2p + 2 and −(2p − 2) ≤ t ≤ 0 are determined up to nonzero multiples of non-torsion classes, from which Theorem 1.1 follows.

In this section we prove Theorem 1.2 by constructing the p-torsion classes η p,k . One might be tempted to consider copying the construction of η p,0 . Indeed, by Tamanoi's Theorem 4.1, one can show that the classes

is in the image of the Thom map. However, one may run into trouble trying to produce an anolog of the isomorphism in (4.3) in higher dimensions. Hence we take a different approach.

For k ≥ 0, we have the classes

Letȳ p,k denote mod p reduction of y p,k . Then the classes x p,k andȳ p,k may be obtained inductively on k as follows:

Proposition 5.1. For k ≥ 1, we have

x p,k = P p k (x p,k−1 ) andȳ p,k = P p k (ȳ p,k−1 ).

Proof. Recall that for positive integers a, b such that a ≤ pb, we have the Adem relation ([1] )

For k > 0, let a = p k and b = p k−1 . Then the only choice of i to offer something nontrivial on the right hand side of (5.3) is i = p k−1 , and (5.3) becomes

Then it follows by induction that we have P p k βP p k−1 · · · P p P 1 = βP p k · · · P p P 1 .

Compare the above with the definitions (5.1) and (5.2), and we conclude.

We abuse notations and let x p,k , y p,k andȳ p,k denote the image of themselves under χ * : H * (K(Z, 3); L) → H * (BP U n ; L) for L = Z (p) or Z p .

Lemma 5.2. The class x p,0 ∈ H * (BP U n ; Z p ) is in the image of the Thom map T : P(1) * (BP U n ) → H * (BP U n ; Z p ).

We denote by ξ p,0 ∈ P(1) 2p+1 (BP U n ) a class satisfying T (ξ p,0 ) = x p,0 .

Proof. It follows from the long exact sequence of BP * -modules (4.2)

that the p-torsion class η p,0 is in the image of δ : P(1) 2p+1 (BP U n ) → BP 2p+2 (BP U n ).

Let ξ p,0 ∈ P(1) 2p+1 (BP U n ) be a class satisfying δ(ξ p,0 ) = η p,0 . The commutative diagram

H 2p+1 (BP U n ; Z p ) H 2p+2 (BP U n ; Z (p) ). yields T (ξ p,0 ) = x p,0 , and we conclude.

We proceed to construct a collection of classes ξ p,k for k ≥ 0 by induction on k. For k = 0, the class η p,0 is given in Theorem 1.1. For the inductive argument, recall Kane's commutative diagram (2.6): The canonical anti-automorphism satisfies c 2 = id, and restricts to an anti-automorphism on the subalgebra of A * generated by P E for various sequences E. Therefore, for k ≥ 0, we have sequences E i such that c(P p k ) = i P Ei , or equivalently P p k = i c(P Ei ).

By (5.5) we have R k ∈ P(1) * P(1) such that the following diagram commutes: P(1) * (X) P(1) * (X) H * (X; Z p ) H * (X; Z p ).

By induction on k and Proposition 5.1, if x p,k is in the image of the Thom map T , then so is x p,k+1 = P p k (x p,k ). The induction also gives classes ξ p,k = R k (ξ p,k−1 ) such that T (ξ p,k ) = x p,k . The desired classes η p,k are thus given by η p,k = δ(ξ p,k ).

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