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Central Extensions and Cohomology

Rohit Joshi, Steven Spallone

Abstract

Let G be a group which is topologically a CW-complex, BG a classifying space for G, and A a discrete abelian group. To a central extension of G by A, one can associate a cohomology class in $H^2(BG,A)$. We show this association is injective, and bijective in many cases. A homomorphism to G lifts to the extension iff the pullback of the associated cohomology class vanishes.

Central Extensions and Cohomology

Abstract

Let G be a group which is topologically a CW-complex, BG a classifying space for G, and A a discrete abelian group. To a central extension of G by A, one can associate a cohomology class in . We show this association is injective, and bijective in many cases. A homomorphism to G lifts to the extension iff the pullback of the associated cohomology class vanishes.
Paper Structure (43 sections, 46 theorems, 109 equations)

This paper contains 43 sections, 46 theorems, 109 equations.

Table of Contents

  1. Introduction
  2. Notation and Preliminaries
  3. Groups
  4. Topological Groups
  5. Topology on Products
  6. Central Extensions
  7. Functorial Properties
  8. Baer Sums
  9. Relation to Lifting Problems
  10. An Exact Sequence
  11. The Weil group
  12. Extensions of Semidirect Products
  13. $G$-bundles
  14. Definition
  15. Baer Sum of $A$-Bundles
  16. ...and 28 more sections

Key Result

Theorem 1.1

The map $\alpha_G$ is an injection. It is an isomorphism when $G$ is connected. When $A=\mathbb Z/p\mathbb Z$ and $G$ is the semidirect product of a discrete group and a connected group, then $\alpha_G$ is an isomorphism.

Theorems & Definitions (98)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 3.1
  • Lemma 3.2
  • ...and 88 more