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A new invariant and parametric connected sum of embeddings

A. Skopenkov (2007)

Fundamenta Mathematicae

We define an isotopy invariant of embeddings N m of manifolds into Euclidean space. This invariant together with the α-invariant of Haefliger-Wu is complete in the dimension range where the α-invariant could be incomplete. We also define parametric connected sum of certain embeddings (analogous to surgery). This allows us to obtain new completeness results for the α-invariant and the following estimation of isotopy classes of embeddings. In the piecewise-linear category, for a (3n-2m+2)-connected...

An unknotting theorem for tori in 4-dimensional spheres.

Akiko Shima (1998)

Revista Matemática Complutense

Let T be a torus in S4 and T* a projection of T. If the singular set Gamma(T*) consists of one disjoint simple closed curve, then T can be moved to the standard position by an ambient isotopy of S4.

Embedding products of graphs into Euclidean spaces

Mikhail Skopenkov (2003)

Fundamenta Mathematicae

For any collection of graphs G , . . . , G N we find the minimal dimension d such that the product G × . . . × G N is embeddable into d (see Theorem 1 below). In particular, we prove that (K₅)ⁿ and ( K 3 , 3 ) are not embeddable into 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 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 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;...

Hardness of embedding simplicial complexes in d

Jiří Matoušek, Martin Tancer, Uli Wagner (2011)

Journal of the European Mathematical Society

Let 𝙴𝙼𝙱𝙴𝙳 k d be the following algorithmic problem: Given a finite simplicial complex K of dimension at most k , does there exist a (piecewise linear) embedding of K into d ? Known results easily imply polynomiality of 𝙴𝙼𝙱𝙴𝙳 k 2 ( k = 1 , 2 ; the case k = 1 , d = 2 is graph planarity) and of 𝙴𝙼𝙱𝙴𝙳 k 2 k for all k 3 . We show that the celebrated result of Novikov on the algorithmic unsolvability of recognizing the 5-sphere implies that 𝙴𝙼𝙱𝙴𝙳 d d and 𝙴𝙼𝙱𝙴𝙳 ( d - 1 ) d are undecidable for each d 5 . Our main result is NP-hardness of 𝙴𝙼𝙱𝙴𝙳 2 4 and, more generally, of 𝙴𝙼𝙱𝙴𝙳 k d for all k , d with...

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