f-Vectors of Triangulated Balls

TL;DR

This paper introduces methods to determine which vectors cannot be f-vectors of homology d-balls, disproves a conjecture for dimensions five and higher, and constructs triangulated balls with various f-vectors, covering all f-vectors in dimensions three and four.

Contribution

It provides new methods to identify invalid f-vectors, disproves a longstanding conjecture, and constructs all possible f-vectors for certain dimensions.

Findings

Disproved Billera and Lee's conjecture for dimensions five and higher.

Developed two methods to show a vector cannot be an f-vector of a homology d-ball.

Constructed all f-vectors for 3- and 4-dimensional balls, conjectured to extend to dimension five.

Abstract

We describe two methods for showing that a vector can not be the f-vector of a homology d-ball. As a consequence, we disprove a conjectured characterization of the f-vectors of balls of dimension five and higher due to Billera and Lee. We also provide a construction of triangulated balls with various f-vectors. We show that this construction obtains all possible f-vectors of three and four dimensional balls and we conjecture that this result also extends to dimension five.