# zbMATH — the first resource for mathematics

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

##### MSC:
 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:
##### References:
 [1] Mark Goresky and Robert MacPherson, Stratified Morse theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 14, Springer-Verlag, Berlin, 1988. · Zbl 0639.14012 [2] J. Mather, Notes on topological stability, Harvard University, 1970, mimeographed notes. · Zbl 0207.54303 [3] Peter Orlik, Introduction to arrangements, CBMS Regional Conference Series in Mathematics, vol. 72, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1989. · Zbl 0722.51003 [4] Peter Orlik and Louis Solomon, Combinatorics and topology of complements of hyperplanes, Invent. Math. 56 (1980), no. 2, 167 – 189. · Zbl 0432.14016 · doi:10.1007/BF01392549 · doi.org
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.