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1. • CommentRowNumber1.
   • CommentAuthorTodd_Trimble
   • CommentTimeMay 24th 2010
   • (edited May 24th 2010)
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexI added a section on triangulable spaces and PL structures to [[simplicial complex]], but this is the type of thing which gets beyond my ken pretty quickly. My real motivation is to convince myself that a space is homeomorphic to the realization of a simplicial complex (in short, is triangulable) if and only if it is homeomorphic to the realization of a simplicial set -- perhaps this seems intuitively obvious, but it should be given a careful proof, and I want such a proof to have a home in the Lab. (Tim Porter said in a related discussion that there was a relevant article by Curtis in some early issue of Adv. Math., but I am not near a university library to investigate this.) I'll put down some preliminary discussion here. Let $P_{fin}(X)$ denote the poset of finite *nonempty* subsets of $X$. A simplicial complex consists of a set $V$ and a down-closed subset $\Sigma \subseteq P_{fin}(V)$ such that every singleton $\{v\}$ belongs to $\Sigma$. Thus $\Sigma$ is itself a poset, and we can take its nerve as a simplicial set. The first claim is that the realization of this nerve is homeomorphic to the realization of the simplicial complex. This I believe is or should be a basic result in the technique of subdivision. Hence realizations of simplicial sets subsume triangulable spaces. For the other (harder) direction, showing that realizations of simplicial sets are triangulable, I want a lemma: that the realization of a nerve of a poset is triangulable. Basically the idea is that we use the simplicial complex whose vertices are elements of the poset and whose simplices are subsets $\{x_1, x_2, \ldots, x_n\}$ for which we have a strictly increasing chain $x_1 \lt x_2 \lt \ldots \lt x_n$. Then, the next step would use the following construction: given a simplicial set $X$, construct the poset whose elements are *nondegenerate* simplices (elements) of $X$, ordered $x \lt y$ if $x$ is some face of $y$. The claim would be that the realization of $X$ is homeomorphic to the realization of the nerve of this poset. All of this could very well be completely standard, but it's hard for me to find an account of this in one place. Alternatively, my intuitions might be wrong here. Or, perhaps I'm going about it in a clumsy way.

   I added a section on triangulable spaces and PL structures to simplicial complex, but this is the type of thing which gets beyond my ken pretty quickly. My real motivation is to convince myself that a space is homeomorphic to the realization of a simplicial complex (in short, is triangulable) if and only if it is homeomorphic to the realization of a simplicial set – perhaps this seems intuitively obvious, but it should be given a careful proof, and I want such a proof to have a home in the Lab. (Tim Porter said in a related discussion that there was a relevant article by Curtis in some early issue of Adv. Math., but I am not near a university library to investigate this.)

   I’ll put down some preliminary discussion here. Let $P_{fin}(X)$ denote the poset of finite nonempty subsets of $X$. A simplicial complex consists of a set $V$ and a down-closed subset $\Sigma \subseteq P_{fin}(V)$ such that every singleton $\{v\}$ belongs to $\Sigma$. Thus $\Sigma$ is itself a poset, and we can take its nerve as a simplicial set. The first claim is that the realization of this nerve is homeomorphic to the realization of the simplicial complex. This I believe is or should be a basic result in the technique of subdivision. Hence realizations of simplicial sets subsume triangulable spaces.

   For the other (harder) direction, showing that realizations of simplicial sets are triangulable, I want a lemma: that the realization of a nerve of a poset is triangulable. Basically the idea is that we use the simplicial complex whose vertices are elements of the poset and whose simplices are subsets $\{x_1, x_2, \ldots, x_n\}$ for which we have a strictly increasing chain $x_1 \lt x_2 \lt \ldots \lt x_n$. Then, the next step would use the following construction: given a simplicial set $X$, construct the poset whose elements are nondegenerate simplices (elements) of $X$, ordered $x \lt y$ if $x$ is some face of $y$. The claim would be that the realization of $X$ is homeomorphic to the realization of the nerve of this poset.

   All of this could very well be completely standard, but it’s hard for me to find an account of this in one place. Alternatively, my intuitions might be wrong here. Or, perhaps I’m going about it in a clumsy way.