2-LC triangulated manifolds are exponentially many

TL;DR

This paper introduces and analyzes $t$-LC triangulated manifolds, establishing bounds on their number and exploring their topological properties, thereby extending known results in combinatorial topology.

Contribution

It defines $t$-LC manifolds and complexes, proves exponential bounds on their counts, and links $t$-constructibility with topological depth, generalizing prior concepts.

Findings

Number of 2-LC $d$-manifolds with $N$ facets is at most $2^{rac{d^3}{2}N}$.

All $t$-constructible pseudomanifolds are $t$-LC.

$t$-constructible complexes have homotopical depth > $d-t$.

Abstract

We introduce "$t$-LC triangulated manifolds" as those triangulations obtainable from a tree of $d$-simplices by recursively identifying two boundary $(d-1)$-faces whose intersection has dimension at least $d-t-1$. The $t$-LC notion interpolates between the class of LC manifolds introduced by Durhuus--Jonsson (corresponding to the case $t=1$), and the class of all manifolds (case $t=d$). Benedetti--Ziegler proved that there are at most $2^{d^2 \, N}$ triangulated $1$-LC $d$-manifolds with $N$ facets. Here we prove that there are at most $2^{\frac{d^3}{2}N}$ triangulated $2$-LC $d$-manifolds with $N$ facets. This extends to all dimensions an intuition by Mogami for $d=3$. We also introduce "$t$-constructible complexes", interpolating between constructible complexes (the case $t=1$) and all complexes (case $t=d$). We show that all $t$-constructible pseudomanifolds are $t$-LC, and that all $t$-constructible complexes have (homotopical) depth larger than $d-t$. This extends the famous result by Hochster that constructible complexes are (homotopy) Cohen--Macaulay.