Determine the precise nature of the Hochschild-Serre spectral sequence's infinite page in relation to group cohomology over rational numbers | Step-by-Step Solution

Hello! This is a problem in algebraic topology and group theory. Let's break it down together!

What We're Solving:

We need to understand what happens to the Hochschild-Serre spectral sequence when it converges and how this relates to group cohomology when working over the rational numbers.

The Approach:

Spectral sequences are a systematic way to compute complicated cohomology groups by breaking them down into simpler pieces. The Hochschild-Serre spectral sequence specifically helps us understand the cohomology of a group extension.

Step-by-Step Solution:

Step 1: Set up the context

• Consider a short exact sequence of groups: 1 → N → G → H → 1
• We want to relate H(G, ℚ) to H(N, ℚ) and H*(H, ℚ)
• The Hochschild-Serre spectral sequence has E₂^{p,q} = H^p(H, H^q(N, ℚ))

Step 2: Understand what "infinite page" means

• The spectral sequence has differentials d_r: E_r^{p,q} → E_r^{p+r,q-r+1}
• Eventually, for large enough r, these differentials become zero
• The "infinite page" E_∞^{p,q} is what remains after all possible differentials

Step 3: Connection to group cohomology

• The spectral sequence converges to H*(G, ℚ)
• This means H^n(G, ℚ) has a filtration whose associated graded pieces are ⊕_{p+q=n} E_∞^{p,q}

Step 4: Special properties over ℚ

• Over ℚ, many technical difficulties disappear
• Rational cohomology is often more manageable than integral cohomology
• Torsion elements vanish, making the analysis cleaner

Step 5: The precise relationship

• We get: E_∞^{p,q} ≅ Gr^p H^{p+q}(G, ℚ)
• Where Gr^p represents the p-th graded piece of the filtration on H^{p+q}(G, ℚ)

The Answer:

The infinite page E_∞^{p,q} of the Hochschild-Serre spectral sequence provides the associated graded pieces of a natural filtration on H*(G, ℚ). Specifically:

• E_∞^{p,q} ≅ F^p H^{p+q}(G, ℚ)/F^{p+1} H^{p+q}(G, ℚ)
• The filtration F^p is induced by the group extension structure
• Over ℚ, this filtration often splits, giving us: H^n(G, ℚ) ≅ ⊕_{p+q=n} E_∞^{p,q}

Memory Tip:

Think of the spectral sequence as a "computation machine" that starts with easily understood pieces (cohomology of N and H separately) and systematically combines them to recover the cohomology of G. The infinite page shows you the "final simplified form" after all the internal cancellations and relationships have been accounted for.

Spectral sequences don't just give you the answer - they give you the answer with extra structure (the filtration) that often provides geometric or algebraic insight into why that answer has the form it does!

Keep exploring these beautiful connections between algebra and topology - you're working with some of the most elegant tools in modern mathematics!