Combinatorial infinite-dimensional manifolds and \({\mathbb{R}}^{\infty}\)- manifolds
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 combinatorial \(\infty\)-manifold.
manifold modelled on \({bbfR}^{\infty }\)
Hauptvermutung for combinatorial \(\infty \)-manifolds
scientific article; zbMATH DE number 4009338