Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the p-primary subgroups of the cohomology of the classifying spaces of PUn

Zhilei Zhang, Linan Zhong

Abstract

Let $PU_n$ denote the projective unitary group of rank $n$, and let $BPU_n$ be its classifying space. We extend our previous results to a description of $H^s(BPU_n;\mathbb{Z})_{(p)}$ for $s<2p+9$ by showing that $p$-primary subgroups of $H^s(BPU_n;\mathbb{Z})$ is $\mathbb{Z}/p$ for $s=2p+5$ and are trivial for $s = 2p+7$ and $s = 2p+8$, where $p$ is an odd prime.

On the p-primary subgroups of the cohomology of the classifying spaces of PUn

Abstract

Let denote the projective unitary group of rank , and let be its classifying space. We extend our previous results to a description of for by showing that -primary subgroups of is for and are trivial for and , where is an odd prime.
Paper Structure (6 sections, 12 theorems, 48 equations, 1 figure)

This paper contains 6 sections, 12 theorems, 48 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The spectral sequences
  3. The Serre spectral sequence $^U\! E$
  4. The auxiliary spectral sequences $^T\! E$ and $^K\! E$
  5. Proof of lemmas
  6. The proof of Theorem \ref{['thm: p>2, main thm, 2p+5,7,8 ']}

Key Result

Proposition 1.2

Suppose $x\in H^*(BPU_n)$ is a torsion class. Then there exists some $i\geq 0$ such that $n^ix = 0$.

Figures (1)

  • Figure 1: Some nontrivial differentials in the spectral sequence

Theorems & Definitions (21)

  • Proposition 1.2: gu_zzz, Proposition 1.1
  • Theorem 1
  • Remark 1.1
  • Proposition 2.1
  • Remark 2.1
  • Proposition 2.2: gu2019cohomology, Corollary 2.16
  • Remark 2.2
  • Proposition 2.3: gu2019cohomology, Proposition 3.2
  • Remark 2.3
  • Proposition 2.4: gu_zzz, Lemma 3.1
  • ...and 11 more