# Mathematics Colloquia and Seminars

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.