[Paper Review] Small rational model of subspace complement

Sergey Yuzvinsky|ArXiv.org|Jun 26, 1998

Advanced Topics in Algebra15 references4 citations

TL;DR

This paper constructs a small rational model for the rational cohomology ring of complex subspace complements using a differential graded subalgebra quasi-isomorphic to the De Concini-Procesi wonderful model. It provides explicit presentations of the cohomology ring for two classes of arrangements, offering a computationally efficient framework for studying their topology via combinatorial data.

ABSTRACT

In the paper we compute the ring strucure on the rational cohomology of a complex subspace complement. For that we construct a small differential graded subalgebra of the De Concini-Procesi wonderful model that is quasi isomorphic to this model. For two classes of arrangements explicit presentation of the ring is given.

Motivation & Objective

• To compute the rational cohomology ring structure of complex subspace complements.
• To develop a small, computationally efficient differential graded algebra (DGA) model quasi-isomorphic to the De Concini-Procesi wonderful model.
• To provide explicit presentations of the cohomology ring for two specific classes of hyperplane arrangements.
• To bridge combinatorial data of arrangements with topological invariants via rational homotopy theory.

Proposed method

• Construct a differential graded subalgebra within the De Concini-Procesi wonderful model that captures the rational cohomology.
• Use quasi-isomorphism to ensure the small DGA computes the same cohomology as the full wonderful model.
• Leverage combinatorial properties of hyperplane arrangements to define generators and relations in the DGA.
• Apply techniques from rational homotopy theory to simplify the model while preserving cohomological structure.
• Utilize the structure of the wonderful model to isolate a minimal subcomplex that supports the rational cohomology.
• Derive explicit presentations of the cohomology ring for two classes of arrangements using the constructed DGA.

Experimental results

Research questions

• RQ1 How can one construct a small rational model for the cohomology of a complex subspace complement that is both computationally efficient and quasi-isomorphic to the De Concini-Procesi wonderful model?
• RQ2 What is the explicit presentation of the rational cohomology ring for two specific classes of hyperplane arrangements?
• RQ3 How do the combinatorial invariants of an arrangement relate to the algebraic structure of its rational cohomology ring?
• RQ4 Can a differential graded algebra be constructed within the wonderful model that captures the full rational cohomology with minimal complexity?
• RQ5 What are the necessary and sufficient conditions on the arrangement for the cohomology ring to admit a presentation via generators and relations derived from the DGA?

Key findings

• A small differential graded subalgebra is constructed that is quasi-isomorphic to the De Concini-Procesi wonderful model, enabling efficient computation of rational cohomology.
• The rational cohomology ring of the subspace complement is fully determined by the combinatorics of the arrangement via the DGA construction.
• Explicit presentations of the cohomology ring are provided for two classes of arrangements, including generators and relations.
• The method reduces the complexity of computing rational cohomology by replacing the full wonderful model with a smaller, combinatorially controlled DGA.
• The quasi-isomorphism ensures that the cohomology ring structure is preserved, validating the model's topological accuracy.
• The results demonstrate that rational cohomology of subspace complements can be effectively computed using algebraic models derived from arrangement combinatorics.