Resumen de 2-LC triangulated manifolds are exponentially many

Bruno Benedetti, Marta Pavelka

• We introduce “t-LC triangulated manifolds” as those triangulations obtainable from atree of d-simplices by recursively identifying two boundary (d − 1)-faces whose intersectionhas dimension at least d − t − 1. The t-LC notion interpolates between the class of LCmanifolds introduced by Durhuus-Jonsson (corresponding to the case t = 1), and the class ofall manifolds (case t = d). Benedetti–Ziegler proved that there are at most 2d2 N triangulated1-LC d-manifolds with N facets. Here we prove that there are at most 2 d32 N triangulated2-LC d-manifolds with N facets. This extends an intuition by Mogami for d = 3 to alldimensions.We also introduce “t-constructible complexes”, interpolating between constructiblecomplexes (the case t = 1) and all complexes (case t = d). We show that all t-constructiblepseudomanifolds are t-LC, and that all t-constructible complexes have (homotopical) depthlarger than d − t. This extends the famous result by Hochster that constructible complexesare (homotopy) Cohen–Macaulay.Details, proofs, and more can be found in our preprint https://arxiv.org/pdf/2106.12136.pdf.