# Simplicial Complexes with Large Homological Systoles

Tuesday, April 29, 2014 - 9:00am - 9:50am

Keller 3-180

Matthew Kahle (The Ohio State University)

(This is work in progress with Dominic Dotterrer and Larry Guth.)

In a graph, the girth is the length of the smallest cycle. How large

the girth can be for a graph on n vertices and m edges is a very well

studied problem in combinatorics. More generally, in a d-dimensional

simplicial complex, we define the d-systole to be the smallest

nonempty collection of closed d-dimensional faces whose union has no

boundary, and we measure the size of a systole in terms of volume,

i.e. the number of faces. It is natural to ask what is the largest

possible d-systole for a simplicial complex on n vertices with m

top-dimensional faces.

We show the existence of simplicial complexes with large systoles

using random simplicial complexes with modifications, and we also

require some new results on estimating the number of triangulated

surfaces on a given number of vertices. On the other hand, we show

that the systoles can not be much larger than this, so these results

are essentially optimal. In the higher-dimensional setting, there are

surprising contrasts with the classical graph theoretic picture, and

in particular the systoles can be quite large.

In a graph, the girth is the length of the smallest cycle. How large

the girth can be for a graph on n vertices and m edges is a very well

studied problem in combinatorics. More generally, in a d-dimensional

simplicial complex, we define the d-systole to be the smallest

nonempty collection of closed d-dimensional faces whose union has no

boundary, and we measure the size of a systole in terms of volume,

i.e. the number of faces. It is natural to ask what is the largest

possible d-systole for a simplicial complex on n vertices with m

top-dimensional faces.

We show the existence of simplicial complexes with large systoles

using random simplicial complexes with modifications, and we also

require some new results on estimating the number of triangulated

surfaces on a given number of vertices. On the other hand, we show

that the systoles can not be much larger than this, so these results

are essentially optimal. In the higher-dimensional setting, there are

surprising contrasts with the classical graph theoretic picture, and

in particular the systoles can be quite large.

MSC Code:

05E45

Keywords: