Group cohomology extends homological algebra to study groups through their actions on modules. It uses cochain complexes and 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
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consists of a sequence of abelian groups or modules Cn and homomorphisms (coboundary operators) dn:Cn→Cn+1
Coboundary operators satisfy the condition dn+1∘dn=0 for all n, meaning the composition of any two consecutive coboundary operators is the zero map
Elements of Cn are called n-cochains
are elements x∈Cn such that dn(x)=0, forming the kernel of dn
are elements x∈Cn such that x=dn−1(y) for some y∈Cn−1, forming the image of dn−1
Computing Cohomology Groups
Cohomology groups Hn(C∙) are defined as the quotient of the kernel of dn (cocycles) by the image of dn−1 (coboundaries)
Hn(C∙)=ker(dn)/im(dn−1)
Cohomology groups measure the "obstruction" to a cochain being a coboundary, similar to how homology groups measure the "holes" in a chain complex
is a specific type of resolution used to compute the cohomology of a group with coefficients in a
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∙ is a collection of group homomorphisms G→Aut(Cn) for each n, compatible with the coboundary operators
Compatibility means that for any g∈G and x∈Cn, g⋅dn(x)=dn(g⋅x)
Group action induces a group action on the cohomology groups Hn(C∙)
is a bilinear operation on cochains, ⌣:Cp×Cq→Cp+q, that is compatible with the coboundary operators and induces a graded-commutative product on cohomology
Cohomological Operations
is a map from the cohomology of a group G to the cohomology of a subgroup H, induced by the inclusion H↪G
is a map from the cohomology of a quotient group G/N to the cohomology of G, induced by the quotient map G↠G/N
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" Erp,q, where each page is obtained from the previous one by taking cohomology
relates the cohomology of a group extension 1→N→G→Q→1 to the cohomology of N and Q
E2p,q=Hp(Q;Hq(N;A))⇒Hp+q(G;A), where A is a G-module and the action of Q on Hq(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 drp,q:Erp,q→Erp+r,q−r+1 that square to zero, and the cohomology of dr gives the next page Er+1