Abstracts for the Seminar
 Discrete Geometry and Combinatorics
 Fall 2022

Speaker:  Marta Pavelka, University of Miami
Title: 2-LC triangulated manifolds are exponentially many
Time: 2:30 PM, Monday, September 19, 2022
Place:  Malott 206

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 $2^{N d^2}$ triangulated 1-LC $d$-manifolds with $N$ facets. Here we show that there are at most $2^{N/2 d^3}$ triangulated 2-LC $d$-manifolds with $N$ facets. 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. This is joint work with Bruno Benedetti. Details of the proofs and more can be found in our preprint of the same title.

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