[Paper Review] A combinatorial formula for the Pontrjagin classes

Israel M. Gelfand, Robert MacPherson|arXiv (Cornell University)|Apr 1, 1992

Advanced Operator Algebra Research2 references4 citations

TL;DR

This paper presents a combinatorial formula for the Pontrjagin classes of a triangulated manifold using oriented matroid theory and a modified Chern-Weil approach. The key contribution is a purely combinatorial expression for these characteristic classes, valid for piecewise linear manifolds, bypassing differential geometry and relying on discrete structures.

ABSTRACT

A combinatorial formula for the Pontrjagin classes of a triangulated manifold is given. The main ingredients are oriented matroid theory and a modified formulation of Chern-Weil theory.

Motivation & Objective

• To develop a purely combinatorial formula for the Pontrjagin classes of a triangulated manifold.
• To extend the classical Chern-Weil theory to a discrete setting using oriented matroid structures.
• To eliminate reliance on smooth structures or differential forms in computing characteristic classes.
• To provide a formula that is intrinsic to the triangulation and combinatorial data of the manifold.
• To establish a bridge between algebraic topology and combinatorics via characteristic classes.

Proposed method

• Utilizes oriented matroid theory to encode the combinatorial structure of the triangulated manifold.
• Adapts the Chern-Weil homomorphism to a discrete setting using curvature-like invariants defined on the 1-skeleton of the triangulation.
• Constructs characteristic classes via polynomial invariants of the combinatorial curvature form.
• Applies a modified version of the Chern-Weil construction that avoids differential forms and instead uses discrete connections.
• Defines the Pontrjagin classes as cohomology classes in the simplicial cochain complex of the manifold.
• Establishes the invariance of the formula under subdivision and PL isomorphisms, ensuring topological consistency.

Experimental results

Research questions

• RQ1 Can the Pontrjagin classes of a triangulated manifold be expressed purely combinatorially, without smooth structures?
• RQ2 How can the Chern-Weil theory be adapted to work in a discrete, combinatorial setting?
• RQ3 What role do oriented matroids play in encoding the curvature and characteristic classes of a PL manifold?
• RQ4 Is the resulting formula invariant under PL homeomorphisms and subdivisions?
• RQ5 Can the combinatorial formula reproduce the classical Pontrjagin classes when the manifold is smooth?

Key findings

• The paper constructs a well-defined, combinatorial formula for the Pontrjagin classes of a triangulated manifold using oriented matroid data.
• The formula is invariant under subdivision of the triangulation, confirming its topological naturality.
• The construction generalizes the classical Chern-Weil theory to the piecewise linear category via discrete curvature invariants.
• The resulting characteristic classes lie in the simplicial cohomology of the manifold and agree with the standard Pontrjagin classes when the manifold is smooth.
• The method provides a discrete, algebraic-topological alternative to differential-geometric methods for computing characteristic classes.
• The approach demonstrates that characteristic classes can be defined and computed using only the combinatorics of the 1-skeleton and orientation data.