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\(k\)-regular maps into Euclidean spaces and the Borsuk-Boltyanskij problem. (English. Russian original) Zbl 1041.54018

Sb. Math. 193, No. 1, 73-82 (2002); translation from Mat. Sb. 193, No. 1, 73-82 (2002).
A map \(f:X \to \mathbb R^m\) is said to be \(k\)-regular if for any \(k+1\) points \(x_0,x_1, \cdots ,x_k\), the images \(f(x_0),f(x_1), \cdots ,f(x_k)\) are vertices of a \(k\)-dimensional simplex. Then, 1-regular maps are clearly embeddings; 2-regular maps are embeddings such that any three points of the image-set are non-collinear, etc. Starting with the classical Noebling-Pontrjagin-Lefschetz theorem, viz., every compactum \(X\) of dimension \(n\) can be embedded into \((2n+1)\)-dimensional Euclidean space \(\mathbb R^{2n+1}\), the author studies the Borsuk-Boltyanskii problem on \(k\)-regular embeddings. He solves that problem by determining the minimum dimension of a Euclidean space into which any \(n\)-dimensional polyhedron can be \(k\)-regularly embedded. A new lower bound for even \(k\) is also obtained.

MSC:

54C25 Embedding
54B10 Product spaces in general topology
57N35 Embeddings and immersions in topological manifolds
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