group theory

# Contents

## Idea

The Baer sum is the natural addition operation on abelian group extensions.

For $G$ a group and $A$ an abelian group, the extensions of $G$ by $A$ are classified by the degree-2 group cohomology

${H}_{\mathrm{Grp}}^{2}\left(G,A\right)={H}^{2}\left(BG,A\right)=H\left(BG,{B}^{2}A\right)\phantom{\rule{thinmathspace}{0ex}}.$H^2_{Grp}(G,A) = H^2(\mathbf{B}G, A) = H(\mathbf{B}G, \mathbf{B}^2 A) \,.

On cocycles $BG\to {B}^{2}A$ there is a canonical addition operation coming from the additive structure of $A$, and the Baer sum is the corresponding operation on the extensions that these cocycles classify.

## Definition

Below are discussed several different equivalent ways to define the Baer sum

### On concrete cocycles

A cocycle in degree-2 group cohomology ${H}_{\mathrm{Grp}}^{2}\left(G,A\right)$ is a function

$c:G×G\to A$c : G \times G \to A

satisfying the cocycle property.

###### Definition

Given two coycles ${c}_{1},{c}_{2}:G×G\to A$ their sum is the composite

$\left({c}_{1}+{c}_{2}\right):G×G\stackrel{{\Delta }_{G×G}}{\to }\left(G×G\right)×\left(G×G\right)\stackrel{\left({c}_{1},{c}_{2}\right)}{\to }A×A\stackrel{+}{\to }A$(c_1 + c_2) : G \times G \stackrel{\Delta_{G \times G}}{\to} (G \times G) \times (G \times G) \stackrel{(c_1,c_2)}{\to} A \times A \stackrel{+}{\to} A

of

• the diagonal on $G×G$;

• the direct product $\left(f,g\right)$;

• the group operation $+:A×A\to A$.

Hence for all ${g}_{1},{g}_{2}\in G$ this sum is the function that sends

$\left({c}_{1}+{c}_{2}\right):\left({g}_{1},{g}_{2}\right)↦{c}_{1}\left({g}_{1},{g}_{2}\right)+{c}_{2}\left({g}_{1},{g}_{2}\right)$(c_1 + c_2) : (g_1, g_2) \mapsto c_1(g_1,g_2) + c_2(g_1, g_2)

### On abstract cocycles

As discussed at group cohomology, a cocycle $c:G×G\to A$ is equivalently a morphism of 2-groupoids from the delooping groupoid $BG$ of $G$ to the double-delooping 2-groupoid ${B}^{2}A$ of $A$:

${c}_{1},{c}_{2}:BG\to {B}^{2}A\phantom{\rule{thinmathspace}{0ex}}.$c_1,c_2 : \mathbf{B}G \to \mathbf{B}^2 A \,.

Since $A$ is an abelian group, $matbf{B}^{2}A$ is naturally an abelian 3-group, equipped with a group operation $+:\left({B}^{2}A\right)×\left({B}^{A}\right)\to {B}^{2}A$.

With respect to this the sum operation is

${c}_{1}+{c}_{2}:BG\stackrel{{\Delta }_{BG}}{\to }BG×BG\stackrel{\left({c}_{1},{c}_{2}\right)}{\to }{B}^{2}A×{B}^{2}A\stackrel{+}{\to }{B}^{2}A$c_1 + c_2 : \mathbf{B}G \stackrel{\Delta_{\mathbf{B}G}}{\to} \mathbf{B}G \times \mathbf{B}G \stackrel{(c_1,c_2)}{\to} \mathbf{B}^2 A \times \mathbf{B}^2 A \stackrel{+}{\to} \mathbf{B}^2 A

### On short exact sequences

For $0\to A\to {\stackrel{^}{G}}_{i}\to G\to 0$ for $i=1,2$ two short exact sequences of abelian groups, their Baer sum is

${\stackrel{^}{G}}_{1}+{\stackrel{^}{G}}_{2}≔{+}_{*}{\Delta }^{*}{\stackrel{^}{G}}_{1}×{\stackrel{^}{G}}_{2}$\hat G_1 + \hat G_2 \coloneqq +_* \Delta^* \hat G_1 \times \hat G_2

The first step forms the pullback of the short exact sequence along rhe diagonal on $G$:

$\begin{array}{ccc}A\oplus A& \to & A\oplus A\\ ↓& & ↓\\ {\Delta }^{*}\left({\stackrel{^}{G}}_{1}\oplus {\stackrel{^}{G}}_{2}\right)& \to & {\stackrel{^}{G}}_{1}\oplus {\stackrel{^}{G}}_{2}\\ ↓& & ↓\\ G& \stackrel{{\Delta }_{G}}{\to }& G\oplus G\end{array}$\array{ A \oplus A &\to& A \oplus A \\ \downarrow && \downarrow \\ \Delta^* (\hat G_1 \oplus \hat G_2) &\to& \hat G_1 \oplus \hat G_2 \\ \downarrow && \downarrow \\ G &\stackrel{\Delta_G}{\to}& G\oplus G }

The second forms the pushout along the addition map on $A$:

$\begin{array}{ccc}A\oplus A& \stackrel{+}{\to }& A\\ ↓& & ↓\\ {\Delta }^{*}\left({\stackrel{^}{G}}_{1}\oplus {\stackrel{^}{G}}_{2}\right)& \to & {+}_{*}{\Delta }^{*}\left({\stackrel{^}{G}}_{1}\oplus {\stackrel{^}{G}}_{2}\right)\\ ↓& & ↓\\ G& \to & G\end{array}$\array{ A \oplus A &\stackrel{+}{\to}& A \\ \downarrow && \downarrow \\ \Delta^* (\hat G_1 \oplus \hat G_2) &\to& +_* \Delta^*(\hat G_1 \oplus \hat G_2) \\ \downarrow && \downarrow \\ G &\to& G }

## References

Lecture notes include for instance

• Patrick Morandi, Ext Groups and Ext Functors (pdf)

Revised on October 1, 2012 12:48:45 by Urs Schreiber (82.113.121.177)