Combinatorial infinite-dimensional manifolds and \({\mathbb{R}}^{\infty}\)- manifolds
type: Item
instance of: scholarly article
title: Combinatorial infinite-dimensional manifolds and \({\mathbb{R}}^{\infty}\)- manifolds
zbMATH Open document ID: 0622.57011
DOI: 10.1016/0166-8641(87)90025-3
author: K. Sakai
published in: Topology and its Applications
publication date: 1987-01-01T00:00:00Z
review text: Generalizing combinatorial manifolds to the infinite-dimensional case, we can define combinatorial \(\infty\)-manifolds as countable simplicial complexes such that the star of each vertex is combinatorially equivalent to \(\Delta^{\infty}\), the countably infinite full complex, namely the \(\infty\)-simplex. A separable paracompact (topological) manifold modelled on \({\mathbb{R}}^{\infty}\) \(=\) dir lim \({\mathbb{R}}^ n\), the direct limit of Euclidean spaces, is called an \({\mathbb{R}}^{\infty}\)-manifold. We can observe that each \({\mathbb{R}}^{\infty}\)-manifold can be triangulated by a combinatorial \(\infty\)-manifold. The main purpose of this paper is to prove the Hauptvermutung for combinatorial \(\infty\)-manifolds: Any two homeomorphic combinatorial \(\infty\)-manifolds are combinatorially equivalent, that is, they have simplicially isomorphic subdivisions. As an application, we have the stable Hauptvermutung: For any two countable homeomorphic complexes K and L, the product complexes \(K\times \Delta^{\infty}\) and \(L\times \Delta^{\infty}\) are combinatorially equivalent. In fact, it is proved that the product complex \(K\times \Delta^{\infty}\) is a combinatorial \(\infty\)-manifold for any countable simplicial complex K. As the main tool of the proof, the following theorem is established: Let K and L be finite simplicial complexes, J a subcomplex of K and \(f: K\to L\) a p.l. embedding such that \(f| J\) is simplicial. Then f is homotopic to an embedding f': \(K\to L\) which is simplicial with respect to subdivisions K' and L' of K and L containing J and f(J) as subcomplexes, respectively. In a forthcoming paper [''Simplicial complexes triangulating infinite- dimensional manifold'', ibid. (to appear)], the author has characterized combinatorial \(\infty\)-manifolds as countable simplicial complexes such that each simplex has nonempty contractible link and shown that any simplicial complex triangulating an \({\mathbb{R}}^{\infty}\)-manifold is a co…
Mathematics Subject Classification ID: 57N20
Mathematics Subject Classification ID: 57Q05
Mathematics Subject Classification ID: 57Q25
Mathematics Subject Classification ID: 57Q15
zbMATH DE Number: 4009338
zbMATH Keywords: triangulation
zbMATH Keywords: combinatorial \(\infty \)-manifolds
zbMATH Keywords: countably infinite full complex
zbMATH Keywords: \(\infty \)-simplex
zbMATH Keywords: manifold modelled on \({bbfR}^{\infty }\)
zbMATH Keywords: \({bbfR}^{\infty }\)-manifold
zbMATH Keywords: Hauptvermutung for combinatorial \(\infty \)-manifolds
zbMATH Keywords: stable Hauptvermutung
zbMATH Keywords: p.l. embedding
MaRDI profile type: Publication
full work available at URL: 0166-8641(87)90025-3
OpenAlex ID: W2039543448
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description: scientific article; zbMATH DE number 4009338