From Hodge theory to combinatorics and geometry

Introduction

We propose an approach to the Dodziuk-Singer conjecture, a central conjecture in geometric topology, specifically concerning the great challenge that understanding aspherical manifolds can pose. This approach is based on newly developed tools from combinatorial commutative algebra and combinatorial Hodge Theory. We will discuss several intermediate problems along the way.

Main Idea

The main idea is based on a connection to commutative algebra via the partition complex. This is an interpretation of local cohomology that allows for a translation between data contained in the L2 cohomology of a manifold and Lefschetz properties of toric varieties associated with them.

Connections to Other Approaches

Additionally, we outline connections to other approaches to the Dodziuk-Singer conjecture as well as special cases, such as:

1. The Hopf conjecture
2. The Charney-Davis conjecture

We propose ideas to connect the different aspects of these viewpoints into one cohesive framework.

Problems and Focus Areas

Finally, we discuss problems related to the methods proposed, particularly focusing on unrealized and unexploited relations between:

• Combinatorics
• Hodge theory
• Geometry

We discuss in particular deformations of polyhedra and metrics, as well as expansion and connectivity.