[Paper Review] Cohomology rings and formality properties of nilpotent groups

Anca Măcinic|arXiv (Cornell University)|Jan 31, 2008

Homotopy and Cohomology in Algebraic Topology12 references4 citations

TL;DR

This paper introduces k-formality for nilpotent groups using partial minimal models, linking it to resonance varieties and cohomology ring structure. It proves that k-formal finitely generated nilpotent groups have trivial resonance varieties up to degree k and cohomology generated in degree 1 up to degree k+1; for 2-step nilpotent groups, this is both necessary and sufficient. The paper computes these invariants for Heisenberg-type groups, showing they are (m−1)-formal but not m-formal when the symplectic rank is 2m.

ABSTRACT

We introduce partial formality and relate resonance with partial formality properties. For instance, we show that for finitely generated nilpotent groups that are k-formal, the resonance varieties are trivial up to degree k. We also show that the cohomology ring of a nilpotent k-formal group is generated in degree 1, up to degree k+1; this criterion is necessary and sufficient for 2-step nilpotent groups to be k-formal. We compute resonance varieties for Heisenberg-type groups and deduce the degree of partial formality for this class of groups.

Motivation & Objective

• To define and study k-formality for nilpotent groups using partial minimal models, extending the classical notion of formality.
• To relate k-formality to the structure of cohomology rings and resonance varieties, particularly in the context of finitely generated nilpotent groups.
• To determine the degree of partial formality for Heisenberg-type groups via explicit computation of resonance varieties and cohomology ring generators.
• To provide obstructions to formality using resonance and cohomological generation, especially for 2-step nilpotent groups.
• To address the Serre problem on projective groups by showing that certain Heisenberg-type groups cannot be fundamental groups of smooth projective varieties with trivial higher homotopy groups.

Proposed method

• Introduces k-formality via the k-minimal model, testing formality up to degree k+1 using finite data, rather than the full minimal model.
• Uses the theory of minimal models and differential graded algebras (DGA) over a field of characteristic zero to analyze rational homotopy types.
• Applies the Künneth formula and tensor product decomposition to compute resonance varieties of tensor products of algebras.
• Employs the resonance variety definition: 𝒮₁ⁱ(G) ⊆ H¹(G, 𝕜) consists of cohomology classes ξ for which multiplication by ξ has nontrivial cohomology in degree i.
• Analyzes the minimal model of Heisenberg-type groups as 𝒜 = ⋀(t₁,…,tₙ) ⊗ ⋀(z), with d(z) = ω ∈ ⋀²(t₁,…,tₙ), and ω in canonical form x₁y₁ + ⋯ + xₘyₘ.
• Uses the decomposition 𝒜 ≅ ℋₘ ⊗ ⋀(t₂ₘ₊₁,…,tₙ) to compute resonance via Proposition 5.7: 𝒮₁ⁱ(𝒜) = ⋃_{p+q=i} 𝒮₁ᵖ(ℋₘ) × 𝒮₁^{q}(⋀(t₂ₘ₊₁,…,tₙ)).

Experimental results

Research questions

• RQ1 What is the relationship between k-formality and the structure of the cohomology ring of a finitely generated nilpotent group?
• RQ2 How do resonance varieties constrain the formality properties of nilpotent groups?
• RQ3 For which k is a given Heisenberg-type group k-formal, and what is the precise degree of partial formality?
• RQ4 Can k-formality be characterized algebraically by the generation of cohomology in degree 1 up to degree k+1?
• RQ5 Which nilpotent groups can arise as fundamental groups of smooth projective complex varieties, and what obstructions exist?

Key findings

• For a finitely generated k-formal nilpotent group, the cohomology ring H^{≤k+1}(G) is generated by H¹(G), establishing a strong algebraic constraint.
• For 2-step nilpotent groups, k-formality is equivalent to the cohomology ring being generated in degree 1 up to degree k+1.
• The resonance variety 𝒮₁ⁱ(G) is trivial (i.e., {0}) for all i ≤ k if G is k-formal, providing a cohomological obstruction to formality.
• For a Heisenberg-type group with symplectic rank 2m, the resonance variety 𝒮₁ⁱ(G) is trivial for i ≤ m−1 and equals 𝕜²ᵐ for i = m.
• A Heisenberg-type group with symplectic rank 2m is (m−1)-formal but not m-formal, establishing the exact degree of partial formality.
• Such groups cannot be fundamental groups of smooth projective complex varieties M with π_{≤m}(M̃) = 0, due to the conflict between m-formality of M and non-m-formality of the group.