{"title":"2-LC三角流形是指数型的","authors":"Bruno Benedetti, Marta Pavelka","doi":"10.4171/aihpd/170","DOIUrl":null,"url":null,"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.","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":null,"pages":null},"PeriodicalIF":1.5000,"publicationDate":"2021-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"2-LC triangulated manifolds are exponentially many\",\"authors\":\"Bruno Benedetti, Marta Pavelka\",\"doi\":\"10.4171/aihpd/170\",\"DOIUrl\":null,\"url\":null,\"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.\",\"PeriodicalId\":42884,\"journal\":{\"name\":\"Annales de l Institut Henri Poincare D\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.5000,\"publicationDate\":\"2021-06-23\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Annales de l Institut Henri Poincare D\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.4171/aihpd/170\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"PHYSICS, MATHEMATICAL\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annales de l Institut Henri Poincare D","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.4171/aihpd/170","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"PHYSICS, MATHEMATICAL","Score":null,"Total":0}
2-LC triangulated manifolds are exponentially many
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.