ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

6. Topology

1991 MS Classification: 57 Skopenkov, A., Department of Mathematics, Kolmogorov Colledge, Kremenchugskaya, 11, Moscow, 117234, Russia A Generalization of The Haefliger-Weber Theorem On Embeddings of Manifolds in ℝ^{m} We improve the PL case of the Haefliger-Weber theorems on immersions and embeddings. Let N be a (3n−2m+2)-connected n-manifold and m−n ≥ 3. Theorem. Suppose that N is smooth and either closed or ∂N ≅ S^{n−1}, respectively. If there exists an equivariant map F : Ñ → S^{m−1}, then there exists either PL or smooth, respectively, embedding f : N → ℝ^{m}. Theorem. If N is closed PL, h : N → ℝ^{m} is a PL immersion and F : Ñ → S^{m−1} is an equivariant map such that h̃|_{OΔ−Δ} ≃_{eq} F |_{OΔ−Δ} for some neighborhood OΔ of the diagonal Δ in N × N, then there is an embedding f : N → ℝ^{m}. Here Ñ = {(x, y) ∈ N × N|x ≠ y}, for a PL immersion (i. e. local embedding) h : N → ℝ^{m} and sufficiently small neighborhood OΔ of the diagonal Δ in N × N, the equivariant map h̃ : OΔ − Δ → S^{m−1} is welldefined by h(x, y) = \frac{hx−hy}{|hx−hy|}. We also prove isotopy versions of the above theorems.