Some torsion classes in the Chow ring and cohomology of $BPGL_n$

Abstract

In the integral cohomology ring of the classifying space of the projective linear group P GLn (over C), we find a collection of p-torsions y p,k of degree 2(p k+1 + 1) for any odd prime divisor p of n, and k ≥ 0.If in addition, p 2 ∤ n, there are p-torsion classes ρ p,k of degree p k+1 + 1 in the Chow ring of the classifying stack of P GLn, such that the cycle class map takes ρ p,k to y p,k .We present an application of the above classes regarding Chern subrings.

On the Brown-Peterson cohomology of $BPU_n$ in lower dimensions and the Thom map

“…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…”

“…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.…”

“…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.…”

“…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 .…”

“…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.…”