A homotopical revisit to group extensions (v2.0)
I. Central extensions of groups \(1 \to A \to G \to Q \to 1\) are classified by (homotopy classes of) maps \(\Bund Q \to \Bund ^2 A.\)
It's a basic result in group theory/group cohomology that the second cohomology of \(Q\) with coefficients in a trivial module \(A\) classifies such central extensions. More generally, if \(A\) carries some \(Q\)-action, the cohomology with twisted coefficients \(\mathrm{H}^2(Q; A)\) classifies extensions where the conjugation action of \(Q\) on \(A\) coincides with the given one. The zero class in \(\mathrm{H}^2(Q; A)\) corresponds to the semi-direct product in either case.
The usual algebraic proof often consists of directly manipulating 2-cocycles to write down the multiplication rule, a boring and tedious process that is hardly illuminating. In this blogpost. I hope to shed some light onto the result by putting on the lenses of homotopy theory. We shall focus on the case of central extensions. Here we have the following Lemma 3.4.2 in May-Ponto's More Concise book. With the point-set mess cleaned up, it says:
- The fiber sequence of \(f\) extends one step to the right: \[\Bund ^n A \to X \xrightarrow{f} Y \xrightarrow{k} \Bund ^{n+1} A.\]
- The fundamental group \(\pi_1(Y)\) acts trivially on \(\pi_n(\fib f) = A\).
In this case, The cohomology class \(k \in \mathrm{H}^{n+1} (Y; A)\) represented by \(k : Y \to \Bund^{n+1} A\) is the only obstruction to the lifting of maps \(T \to Y\) along \(f\). These are famously known under the names of \(k\)-invariants and characteristic classes. This result is of great utility in the theory of Postnikov-Whitehead towers.
If we apply the lemma to the map \(\Bund G \to \Bund Q\) with fiber \(\Bund A\), the action of \(Q\) on \(A\) referred to above is by conjugation (see later), so it's exactly the central extensions that are classified by \(\mathrm{Map}_\Ani (\Bund Q, \Bund ^2A)\). This is a \(2\)-truncated space with \(\pi_n = \mathrm{H}^{2-n}(Q; A)\).
The proof presented in May-Ponto uses the Serre spectral sequence. In this blogpost, I'll try to explain the statement within the context of (higher) group extensions.
II. The threefold way of group extensions
Let's put on out homotopy hat and apply the machinery to (discrete) groups. We want to describe arbitrary extensions of groups \(1 \to N \to G \to Q \to 1\) .II.1 Group extension? That's just a bundle.
Famously from classical topology, \(F\)-bundles with structure group \(G\) are classified by (unpointed) maps into \(\Bund G\), and they're the same as principal \(G\)-bundles.
A direct inspection of the Puppe sequence shows that arbitrary extensions of groups \(1 \to N \to G \to Q \to 1\) are precisely the \(\Bund N\)-bundles over \(\Bund Q\). Therefore they are classified by \(\Bund \Aut_\Ani (\Bund N)\).
II.2 The very important point: Semi-direct products
The careless reader (like me) might have remembered that \(Q \to \Aut _\Grp(N)\) classifies only the split extensions (semi-direct products)! In other words, since \(\Aut _\Grp(N)\) fixes the unit of \(N\), there is a canonical section of these \(\Bund N\)-bundles, splitting the extension... What's making it different here?
The answer is that I've mixed up \(\Aut_\Ani (\Bund N)\) with \(\Aut_\AniB (\Bund N)\).- \(\Aut_\AniB (\Bund N)\) classifies pointed bundles, so you get split extensions. At same time it's equivalent to \(\Bund \Aut_\Grp (N)\) by delooping.
- \(\Aut_\Ani (\Bund N)\) receives a map from the latter through forgetting the (datum witnessing the preservation of) basepoint.
As the middle map is a group morphism, this sequence extends to the right into \[\Aut_\Ani (\Bund N) \to \Bund N \to \Bund \Aut_\AniB (\Bund N) \to \Bund \Aut_\Ani (\Bund N),\] where the conjugation action of \(N\) on itself inducese the map \(\Bund N \to \Bund \Aut _\Grp(N) \simeq \Bund \Aut _{\AniB}(\Bund N)\). We see that \[\pi_0\Aut_\Ani (\Bund N) = \mathrm{Out}(N), \pi_1\Aut_\Ani (\Bund N) = Z(N).\]
If we further assume that \(N = \Bund^n A, n \geq 0\) is (a delooping of) an abelian group, then the (pointed) map \(\Bund^{n+1} A \to \Bund \Aut_\AniB (\Bund^n A)\)
comes equipped with a nulhomotopy, witnessing a splitting \[\Aut_\Ani (\Bund^n A) \simeq \Aut_\AniB (\Bund^n A) \times \Bund^{n+1} A \simeq \Aut_\Grp(A) \times \Bund^{n+1}A.\]
I should warn you myself that this splitting is only one of spaces, not of groups.
Exercise The map \(\Aut_\Ani (\Bund N) \to \Bund N\) is in general NOT a morphism of groups.
As an irrelevant side note, the extension \(1 \to G/Z(G) \to \Aut (G) \to \mathrm{Out}(G) \to 1\) doesn't always split for (non-abelian) groups.
II.3 (FIXED and REFURBISHED like BRAND NEW!) Another attempt at central extensions
Take now the group \(A\) to be discrete abelian, then the space \(\Bund A\) is canonically a commutative group, as well as all its higher deloopings \(\Bund^n A\). Its action on itself by translation provides a group morphism \[\Bund^n A \xrightarrow{\lambda} \Aut_\Ani (\Bund^n A).\] Moreover, the \(\Bund ^n A\)-bundle classified by the delooping \[\Bund^{n+1} A\xrightarrow{\Bund\lambda} \Bund\Aut_\Ani (\Bund^n A)\] is exactly the path fibration \(\mathbf{E}\Bund^n A \to \Bund^{n+1}A\), a.k.a the action (\(\infty\)-)groupoid of the group \(\Bund^nA\). This is a classical construction of delooping, known as the bar construction.
We now really make use of \(A\) being DISCRETE ABELIAN.
I claim that the morphism \(\Bund\lambda\) exhibits \(\Bund^{n+1}\) as
an \((n+1)\)-connective cover of \(\Bund \Aut_\Ani (\Bund^n A)\).
To see this, observe the following pullback square.
\[\begin{CD}
\bullet @>>> \mathrm{pt}\\
@VVV @VVV\\
\mathbf{E}\Bund^n A @>>> \Bund\Aut_\AniB (\Bund^n A)\\
@VVV @VVV\\
\Bund^{n+1} A @>>> \Bund\Aut_\Ani (\Bund^n A)
\end{CD}\]
The top left corner \(\bullet\) is the fiber of \(\Bund \lambda\). Seeing as \(\mathbf{E}\Bund^nA\) is contractible,
it's also equivalent to \(\Loop \Bund \Aut _\AniB(\Bund^n N)= \Aut _\AniB(\Bund^n N)\).
Because \(\Bund^n A\) is Eilenberg-Mac Lane, this is just the discrete group \(\Aut _\Grp(N)\).
This shows that the map \(\Bund\lambda\) is a map with discrete fiber mapping out of a
simply connected space into a connected space. This means that it is a universal cover, and hence the fiber
of the \(1\)-truncation map
\[\Bund\Aut_\Ani (\Bund^n A) \to \tau_{\leq1} \Bund\Aut_\Ani (\Bund^n A).\]
The latter can be identified as a \(\Bund\Aut _\Grp(N)\), and it classifies the action of fundamental groups on the fibers. (EXERCISE!)
From this discussion, we have obtained the fiber sequence we're looking for \[\Aut_\Grp (A) \to \Bund^{n+1} A \to \Bund\Aut_\Ani (\Bund ^n A) \to \Bund\Aut_\Grp (A).\] and a non-computational proof of the Lemma 3.4.2 presented at the start.
III. Another well-known example
The theory of classifying objects work in a more general context. Most notably for (sheaf) topoi. (I'm not too confident on the details, but this seems to be what's going on in principle. PLEASE DO CORRECT ME, I VERY WELL MAY BE WRONG)
In particular, we have a central extension of group sheaves on affine schemes (with your favorite topology): \[ 1 \to \mathbb{G}_m \to \mathrm{GL}_n \to \mathrm{PGL}_n \to 1, \] These should correspond to fiber sequences \[\Bund \mathbb{G}_m \to \Bund \mathrm{GL}_n \to \Bund \mathrm{PGL}_n \to \Bund ^2\mathbb{G}_m\] giving corresponding long exact sequences on cohomology (=global sections).
A. Further links and miscellany
- nLab page on \(\infty\)-group extension
- David Jaz Myers: Higher Schreier Theory
- Georg Lehner: Group completion via the action \(\infty\)-category. Contains juicy stuff about groups
- May-Ponto's "More Concise" book. Grabbed from Ranicki's website, TOC added by me.
i. Very much classical geometry
The space \(\Bund G\) really classifies principal \(G\)-bundles. With the datum of a \(G\)-action on \(F\), these induce \(F\)-bundles as follows: \[\begin{array}{rcl} (B \xrightarrow{f} \Bund G) &\simeq &(P = f^\ast \mathbf{E}G \to B : \text{principal bundle})\\ &\simeq &(P \times_G F \to B : \text{\(F\)-bundle with \(G\)-structure})\end{array}\]
ii. The homotopy action of \(\pi_1(Y)\) on \(F := \fib (X \to Y)\) (May-Ponto 1.4-1.5)
In the sections May-Ponto 1.4-1.5, the authors defined for connected pointed spaces \(X\) and \(Y\) an action of the fundamental group \(\pi_1(Y)\) on \(\pi_0\mathrm{Map}_\AniB (X,Y)\). Upon inspection of the geometric definition, this should arise from an \(S^1\)-comodule structure on \(X\) by "extruding a neighborhood around the basepoint". This is where the conjugation action comes from. There's quite some potential confusion chasing the basepoint around.
There's another action on the fiber given heuristically by path concatenation. With the following description of the fiber \[F = X \times_Y 1 = \big\{(x : X, \alpha : y_0 = f(x)\big\},\] we can describe this action as \[\begin{align*} F \times \Loop Y &= X \times_Y 1 \times 1 \times_Y 1 \\ &= X \times_Y 1 \times_Y 1 \\ &\xrightarrow{y_0} X \times_Y Y \times_Y 1 \\ &= X \times_Y 1 \\ &= F,\\ \left(\left(x : X, \alpha : y_0 = f\left(x\right)\right), \beta : y_0 = y_0\right) &\mapsto \left(x : X, \alpha \circ \beta : y_0 = f\left(x\right)\right).\end{align*}\] One can also view this as given by the straightening \(Y \to \Ani, y \mapsto X \times_Y \{y\}\), plus the equivalence \(Y \simeq \Bund \Loop Y\) determined by a basepoint. At first glance, this looks much like a right translation rather than conjugation. Moreover, this map has no reason to preserve basepoints. (One should think about how these two actions relate to each other.)
Future objectives: Rework the appendix on \(\pi_1\)-actions.
Update 2.0 (26.dec.2025): Moved to Katex
Update 1.3 (15.aug.2025): Reworked section III completely with a (nice and) non-sketchy proof.
Update 1.2e (14.aug.2025): Fatal error found and hotfixed.
Update 1.2 (13.aug.2025): Rewording and grammatical fixes, reorganized MathJax code.
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