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Lattice-isotopic arrangements are topologically isomorphic. (English) Zbl 0681.57016
An arrangement \({\mathcal A}=\{H_ 1,H_ 2,...,H_ n\}\) is a finite collection of affine subspaces (of possibly varying dimensions) in \(K^{\ell}\) where \(K={\mathbb{R}}\) or \({\mathbb{C}}\), or a collection of linear subspaces of a projective space \(K{\mathbb{P}}^{\ell -1}\). The main result of this paper shows that if two arrangements are connected by a one- parameter family of arrangements which have the same lattice, the complements are diffeomorphic, hence of the same homotopy type.
Definition. A smooth 1-parameter family of arrangements \({\mathcal A}\) is a finite collection \(\{H_{it}\}\) of subspaces for each \(t\in {\mathbb{R}}\) and \(i=1,2,...,n\) so that \(H_{it}\) is the locus in \({\mathbb{P}}\times \{t\}\subset {\mathbb{P}}\times {\mathbb{P}}\) of a system of \(c_ i\) equations linear in the variables of \({\mathbb{P}}\) with coefficients smooth functions of t. Definition. Arrangements \({\mathcal A}_ 0=\{H_ 1,H_ 2,...,H_ n\}\) and \({\mathcal A}_ 1=\{G_ 1,G_ 2,...,G_ n\}\) have the same lattice if for all \(I\subset \{1,2,...,n\}\), dim\(\cap_{i\in I}H_ i=\dim \cap_{i\in I}G_ i\). Definition. A 1-parameter family \({\mathcal A}\) is a lattice isotopy provided that for any \(t_ 1\), \(t_ 2\), the arrangements \({\mathcal A}_{t_ 1}\) and \({\mathcal A}_{t_ 2}\) have the same lattice. The author proves the following theorem. Theorem. If \({\mathcal A}\) is a lattice-isotopy, then \(M_ 0\) is diffeomorphic to \(M_ 1\) and the pair \(({\mathbb{P}},N_ 0)\) is homeomorhic to \(({\mathbb{P}},N_ 1)\). Moreover three corollaries are stated.
Reviewer: G.Rassias

57R40 Embeddings in differential topology
57R52 Isotopy in differential topology
05B35 Combinatorial aspects of matroids and geometric lattices
57Q37 Isotopy in PL-topology
Full Text: DOI
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