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Essential coverings and triangulations. (English) Zbl 0631.57018

Let V be an m-dimensional differential manifold without boundary embedded in a Euclidean space \(R^ n\). Let \({\mathfrak U}=\{U_ i\}_{i\in I}\) be a locally finite open covering of V. The author defines:
(a) \({\mathfrak U}\) is a simple covering if the following holds: (i) The closure of each \(U_ i\) is diffeomorphic to \(D^ m\), (ii) The family \(\{S_ i=\partial U_ i\}_{i\in I}\) is transversal (for each \(F\subset I\) finite \(\{S_ i/i\in F\}\) is transversal), (iii) If \(U_{i_ 1...i_ r}=U_{i_ 1}\cap...\cap U_{i_ r}\neq \emptyset\), then the manifold obtained from \(\bar U{}_{i_ 1...i_ r}\) by “smoothing its corners” is diffeomorphic to \(D^ m;\)
(b) \({\mathfrak U}\) is a ball covering if it is simple and if \(U_{i_ 1...i_ r}\cap S_{j_ 1}\cap...\cap S_{j_ s}\neq \emptyset\) then the manifold obtained of \(\bar U{}_{i_ 1...i_ r}\cap S_{j_ 1}\cap...\cap S_{j_ s}\) by “smoothing its corners” is diffeomorphic to \(D^{m-s};\)
(c) \({\mathfrak U}\) is an essential covering if it is a ball covering of order \(m+1\) such that \(U_{i_ 1...i_ r}\not\supset \cup \{U_ j/j\neq i_ 1,...,i_ r\}\) for any \(i_ 1,...,i_ r\) with \(U_{i_ 1...i_ r}\neq \emptyset.\)
The author proves that there exist essential coverings refining any given open covering of V. Moreover, if K is the nerve of an essential covering, then K is P.L. isomorphic to any smooth triangulation of V. The results are true also if V has a boundary provided that definitions and proofs are slightly modified.
A. Weil [Comment. Math. Helvet. 26, 119-145 (1952; Zbl 0047.167)] proved that the nerve of a simple covering has the homotopy type of V.
Reviewer: E.Outerelo

MSC:

57R05 Triangulating
57Q15 Triangulating manifolds

Citations:

Zbl 0047.167
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References:

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