Nonabelian H1

Non-abelian \(H^1\) is the pointed set classifying torsors under a non-commutative group. It is the first cohomology that survives when the coefficient group is non-abelian, but only as a pointed set — group structure is lost.

Definition

Let \(\Gamma\) be a topological group and \(A\) a topological \(\Gamma\)-group (a group on which \(\Gamma\) acts continuously by group automorphisms, not necessarily abelian).

A 1-cocycle is a continuous map \(c : \Gamma \to A\) satisfying \[ c_{\sigma \tau} \;=\; c_\sigma \cdot \sigma(c_\tau). \]

Two cocycles \(c, c'\) are cohomologous if there exists \(a \in A\) with \[ c'_\sigma \;=\; a^{-1} c_\sigma \sigma(a) \quad \text{for all } \sigma. \]

The set of equivalence classes is the first cohomology pointed set: \[ H^1(\Gamma, A) \;:=\; Z^1(\Gamma, A) / \sim, \] with the trivial cocycle \(c \equiv 1\) as basepoint.

Why "pointed set" and not "group"

When \(A\) is non-abelian, the natural multiplication of cocycles \((c \cdot c')_\sigma = c_\sigma c'_\sigma\) does not generally satisfy the cocycle condition. So \(H^1\) is only a set with distinguished basepoint, not a group.

For \(A\) abelian: \(H^1(\Gamma, A)\) recovers the usual group structure.

Long exact sequence (partial)

For a short exact sequence of \(\Gamma\)-groups \(1 \to A \to B \to C \to 1\):

\[ 1 \to A^\Gamma \to B^\Gamma \to C^\Gamma \to H^1(\Gamma, A) \to H^1(\Gamma, B) \to H^1(\Gamma, C) \]

is exact as pointed sets. The sequence does NOT extend to higher cohomology in general.

If \(A\) is central in \(B\), the sequence extends one more step: \[ \ldots \to H^1(\Gamma, C) \to H^2(\Gamma, A), \] because \(A\) is then abelian and the obstruction map to "lifting" a \(C\)-cocycle to a \(B\)-cocycle is well-defined.

Twisting and "different basepoints"

For \(c \in Z^1(\Gamma, A)\) and a \(\Gamma\)-\(A\)-set \(X\), the twist \(X^c\) has the same underlying set as \(X\) but with new \(\Gamma\)-action \(\sigma \cdot_c x = c_\sigma \sigma(x)\). Twisting yields:

A bijection of pointed sets (after re-basing) \(H^1(\Gamma, A) \cong H^1(\Gamma, A^c)\) where \(A^c\) is the inner-twisted group;
A change of basepoint: the basepoint of \(H^1(\Gamma, A^c)\) corresponds to \([c]\) in \(H^1(\Gamma, A)\).

This is the standard tool for comparing different forms.

Examples

\(A = \mathrm{PGL}_n\)

\(H^1(k, \mathrm{PGL}_n) = \mathrm{Br}(k)[n]\): central simple \(k\)-algebras of degree \(n^2\) modulo Brauer equivalence.

\(A = \mathrm{O}(q)\)

\(H^1(k, \mathrm{O}(q))\) = quadratic forms of the same rank and discriminant as \(q\).

\(A = G_{\mathrm{ad}}\) for \(G\) reductive

\(H^1(k, G_{\mathrm{ad}})\) classifies inner forms of \(G\) over \(k\). The exact sequence \(1 \to Z(G) \to G \to G_{\mathrm{ad}} \to 1\) gives: \[ G(k) \to G_{\mathrm{ad}}(k) \to H^1(k, Z(G)) \to H^1(k, G) \to H^1(k, G_{\mathrm{ad}}) \to H^2(k, Z(G)). \]

\(A = S_n\) (a finite group)

\(H^1(\Gamma, S_n)\) classifies degree-\(n\) étale algebras over \(k\) — more concretely, \(n\)-element sets with a \(\Gamma\)-action up to isomorphism.

Vanishing theorems

Hilbert 90. \(H^1(k, \mathrm{GL}_n) = 0\) for any field \(k\).
Lang's theorem. \(H^1(\mathrm{Gal}(\bar{\mathbb{F}_q}/\mathbb{F}_q), G) = 0\) for any connected algebraic group \(G\) over \(\mathbb{F}_q\).
Steinberg. For \(G\) simply connected semisimple over a field of cohomological dimension \(\le 1\): \(H^1(k, G) = 0\).
Bruhat-Tits, Kneser. For \(G\) simply connected semisimple over a non-archimedean local field \(F\): \(H^1(F, G) = 0\). (Fails for real local fields.)

Springer-style obstruction theory

The non-abelian \(H^1\) admits an analogue of long-exact sequence with obstructions in higher dimensional gerbe-cohomology. The modern formulation uses 2-groups (gr-stacks) and 2-cocycles to capture the obstruction more cleanly.

This is the technical underpinning of Kaletha's "rigid inner forms" and the Galois-gerbe formalism for L-packets.

Role in Langlands

Forms of reductive groups: classified by \(H^1(k, \mathrm{Aut}(G_0))\).
Inner twists: classes in \(H^1(k, G_{\mathrm{ad}})\).
Endoscopic data: include sections of certain non-abelian \(H^1\) sequences.
Stable conjugacy: two semisimple elements are stably conjugate iff the cocycle linking them is trivial in \(H^1(k, Z(G^*))\) for the centraliser \(G^*\) — a non-abelian comparison.