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In this paper we prove the existence of a closed neat embedding of a Hausdorff paracompact Hilbert manifold with smooth boundary into $H \times [0, + \infty)$ , where $H$ is a Hilbert space, such that the normal space in each point of a certain neighbourhood of the boundary is contained in $H \times {0}$ . Then, we give a neccesary and sufficient condition that a Hausdorff paracompact topological space could admit a differentiable structure of class $\infty$ with smooth boundary.

Embedding products of graphs into Euclidean spaces

Mikhail Skopenkov (2003)

Fundamenta Mathematicae

For any collection of graphs $G ₁, . . ., G_{N}$ we find the minimal dimension d such that the product $G ₁ \times . . . \times G_{N}$ is embeddable into $ℝ^{d}$ (see Theorem 1 below). In particular, we prove that (K₅)ⁿ and $(K_{3, 3}) ⁿ$ are not embeddable into $ℝ^{2 n}$ , where K₅ and $K_{3, 3}$ are the Kuratowski graphs. This is a solution of a problem of Menger from 1929. The idea of the proof is a reduction to a problem from so-called Ramsey link theory: we show that any embedding $L k O \to S^{2 n - 1}$ , where O is a vertex of (K₅)ⁿ, has a pair of linked (n-1)-spheres.

Embedding proper homotopy types

M. Cárdenas, T. Fernández, F. F. Lasheras, A. Quintero (2003)

Colloquium Mathematicae

We show that the proper homotopy type of any properly c-connected locally finite n-dimensional CW-complex is represented by a closed polyhedron in $ℝ^{2 n - c}$ (Theorem I). The case n - c ≥ 3 is a special case of a general proper homotopy embedding theorem (Theorem II). For n - c ≤ 2 we need some basic properties of “proper” algebraic topology which are summarized in Appendices A and B. The results of this paper are the proper analogues of classical results by Stallings [17] and Wall [20] for finite CW-complexes;...

Embedding punctured lens spaces in four-manifolds.

A.L. Edmonds, Ch. Livingston (1996)

Commentarii mathematici Helvetici

Embedding Semi-Algebraic Spaces.

Robert Robson (1983)

Mathematische Zeitschrift

Embedding solenoids

Alex Clark, Robbert Fokkink (2004)

Fundamenta Mathematicae

A generalized solenoid is an inverse limit space with bonding maps that are (regular) covering maps of closed compact manifolds. We study the embedding properties of solenoids in linear space and in foliations.