9.2 Group cohomology

Group cohomology extends homological algebra to study groups through their actions on modules. It uses cochain complexes and cohomology groups to measure obstructions and analyze group structures.

This approach connects to broader cohomology theories by providing tools for understanding group extensions, representations, and algebraic invariants. It also introduces key concepts like spectral sequences for computing complex cohomological information.

Cochain Complexes and Cohomology Groups

Defining Cochain Complexes

• Cochain complex consists of a sequence of abelian groups or modules $C^n$ and homomorphisms (coboundary operators) $d^n: C^n \to C^{n+1}$
• Coboundary operators satisfy the condition $d^{n+1} \circ d^n = 0$ for all $n$, meaning the composition of any two consecutive coboundary operators is the zero map
• Elements of $C^n$ are called $n$-cochains
• Cocycles are elements $x \in C^n$ such that $d^n(x) = 0$, forming the kernel of $d^n$
• Coboundaries are elements $x \in C^n$ such that $x = d^{n-1}(y)$ for some $y \in C^{n-1}$, forming the image of $d^{n-1}$

Computing Cohomology Groups

• Cohomology groups $H^n(C^\bullet)$ are defined as the quotient of the kernel of $d^n$ (cocycles) by the image of $d^{n-1}$ (coboundaries)
• $H^n(C^\bullet) = \ker(d^n) / \operatorname{im}(d^{n-1})$
• Cohomology groups measure the "obstruction" to a cochain being a coboundary, similar to how homology groups measure the "holes" in a chain complex
• Bar resolution is a specific type of resolution used to compute the cohomology of a group with coefficients in a module
• Normalized cochains are a subcomplex of the cochains in the bar resolution, obtained by considering only those cochains that vanish on degenerate elements

Group Actions and Cohomological Operations

Group Actions on Cochain Complexes

• Group action on a cochain complex $C^\bullet$ is a collection of group homomorphisms $G \to \operatorname{Aut}(C^n)$ for each $n$, compatible with the coboundary operators
• Compatibility means that for any $g \in G$ and $x \in C^n$, $g \cdot d^n(x) = d^n(g \cdot x)$
• Group action induces a group action on the cohomology groups $H^n(C^\bullet)$
• Cup product is a bilinear operation on cochains, $\smile: C^p \times C^q \to C^{p+q}$, that is compatible with the coboundary operators and induces a graded-commutative product on cohomology

Cohomological Operations

• Restriction is a map from the cohomology of a group $G$ to the cohomology of a subgroup $H$, induced by the inclusion $H \hookrightarrow G$
• Inflation is a map from the cohomology of a quotient group $G/N$ to the cohomology of $G$, induced by the quotient map $G \twoheadrightarrow G/N$
• Transfer map is a homomorphism from the cohomology of a subgroup $H$ to the cohomology of the ambient group $G$, induced by averaging over cosets of $H$ in $G$
• These operations allow for the comparison of cohomology groups of related groups and the study of the behavior of cohomology under group homomorphisms

Spectral Sequences

The Lyndon-Hochschild-Serre Spectral Sequence

• Spectral sequence is a tool for computing cohomology groups by successively approximating them with "pages" $E_r^{p,q}$, where each page is obtained from the previous one by taking cohomology
• Lyndon-Hochschild-Serre spectral sequence relates the cohomology of a group extension $1 \to N \to G \to Q \to 1$ to the cohomology of $N$ and $Q$
• $E_2^{p,q} = H^p(Q; H^q(N; A)) \Rightarrow H^{p+q}(G; A)$, where $A$ is a $G$-module and the action of $Q$ on $H^q(N; A)$ is induced by the action of $G$ on $A$ and the conjugation action of $G$ on $N$
• Spectral sequence converges to the cohomology of $G$ with coefficients in $A$, providing a way to compute it from the cohomology of the subgroup $N$ and quotient $Q$
• Differentials on each page are homomorphisms $d_r^{p,q}: E_r^{p,q} \to E_r^{p+r,q-r+1}$ that square to zero, and the cohomology of $d_r$ gives the next page $E_{r+1}$