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Counterexamples to a conjecture of Adams

Feifei Fan

TL;DR

The paper refutes Adams' conjecture on detection of $H^*(BPU(n); _p)$ by elementary abelian $p$-subgroups for odd primes when $p^2ig| n$, by constructing explicit cohomology classes $oldsymbol{eta}_1,oldsymbol{eta}_2ig extrm{ in }H^3(BPU(n); _p)$ with $oldsymbol{eta}_1+oldsymbol{eta}_2$ undetectable under the restriction map to all maximal elementary abelian $p$-subgroups. The proof combines a detailed Serre spectral sequence analysis (via GuGu21) with Vistoli’s computation for $BPU(p)$, invariant theory for Weyl groups acting on elementary abelian subgroups, and Milnor operations to produce nilpotent elements. The paper further derives consequences for Brown–Peterson cohomology and related Chow rings, showing potential counterexamples to long-standing conjectures in that area. Overall, it deepens understanding of detection phenomena in $p$-local cohomology of classifying spaces of projective unitary groups and provides constructive tools for exploring nilpotence in generalized cohomology theories.

Abstract

For any odd prime $p$ and any integer $n>0$ with $p^2|n$, we show that the mod $p$ cohomology ring of the classifying space of the projective unitary group $PU(n)$ is not completely detected by elementary abelian $p$-subgroups, providing counterexamples to a conjecture due to J. F. Adams. We also give an application involving Milnor operations and Brown-Peterson cohomology.

Counterexamples to a conjecture of Adams

TL;DR

The paper refutes Adams' conjecture on detection of by elementary abelian -subgroups for odd primes when , by constructing explicit cohomology classes with undetectable under the restriction map to all maximal elementary abelian -subgroups. The proof combines a detailed Serre spectral sequence analysis (via GuGu21) with Vistoli’s computation for , invariant theory for Weyl groups acting on elementary abelian subgroups, and Milnor operations to produce nilpotent elements. The paper further derives consequences for Brown–Peterson cohomology and related Chow rings, showing potential counterexamples to long-standing conjectures in that area. Overall, it deepens understanding of detection phenomena in -local cohomology of classifying spaces of projective unitary groups and provides constructive tools for exploring nilpotence in generalized cohomology theories.

Abstract

For any odd prime and any integer with , we show that the mod cohomology ring of the classifying space of the projective unitary group is not completely detected by elementary abelian -subgroups, providing counterexamples to a conjecture due to J. F. Adams. We also give an application involving Milnor operations and Brown-Peterson cohomology.
Paper Structure (10 sections, 30 theorems, 77 equations)

This paper contains 10 sections, 30 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. A spectral sequence converging to $H^*(BPU(n))$
  3. Cohomology ring of $K(\mathbb {Z},3)$
  4. On $H^*(BPU(p))$ and $E_*^{*,*}(BPU(p))$
  5. On $E_*^{*,*}(BPU(n))$ with $p^2|n$
  6. Elementary abelian $p$-subgroups of $PU(n)$
  7. The group $PU(p)$
  8. The group $PU(n)$
  9. Proof of $\Phi(\alpha_1+\alpha_2)=0$
  10. Milnor operations and nilpotent elements in $H^*(BPU(p^km);\mathbb {F}_p)$

Key Result

Theorem 1.2

Let $p$ be an odd prime, and assume that $n>0$ is an integer such that $p^2|n$. Then the restriction map is not injective. Precisely, for a nonzero element $\chi\in H^3(BPU(n);\mathbb {F}_p)\cong \mathbb {F}_p$, let Then $\alpha_1$ and $\alpha_2$ are linearly independent in $H^*(BPU(n);\mathbb {F}_p)$ and $\Phi(\alpha_1+\alpha_2)=0$.

Theorems & Definitions (49)

  • Conjecture 1.1: Adams, see VV05
  • Theorem 1.2
  • Remark 1.3
  • Proposition 2.1: Gu21
  • Theorem 2.2: CG21
  • Proposition 3.1: Gu21b
  • Proposition 3.2: See Tam99 or Gu21
  • Proposition 3.3
  • Theorem 4.1: Vistoli Vis07
  • Proposition 4.2
  • ...and 39 more