Localizable invariants of combinatorial manifolds and Euler characteristic

TL;DR

This paper proves that any local PL-invariant of closed combinatorial manifolds, depending solely on vertex link f-vectors, must be proportional to the Euler characteristic, highlighting its unique role among such invariants.

Contribution

The paper establishes a uniqueness result for local PL-invariants, showing they are essentially scalar multiples of the Euler characteristic when based on vertex link f-vectors.

Findings

Any local PL-invariant depending only on link f-vectors is proportional to Euler characteristic.

The result constrains possible local invariants in combinatorial topology.

Highlights the fundamental nature of Euler characteristic among local invariants.

Abstract

It is shown that if a real value PL-invariant of closed combinatorial manifolds admits a local formula that depends only on the f-vector of the link of each vertex, then the invariant must be a constant times the Euler characteristic.