Haefliger-Zeeman unknotting theorem

The Haefliger-Zeeman unknotting theorem (German about: Entknotungssatz von Haefliger and Zeeman ) is a theorem from the mathematical field of differential topology . It gives easily verifiable conditions when two embeddings of a manifold in a Euclidean space can be deformed into one another (i.e. are isotopic to one another ). It is named after André Haefliger and EC Zeeman . ${\displaystyle \mathbb {R} ^{n}}$

requirements

An isotopy of embeddings of the interval in the plane.

Let it be a differentiable manifold . It is called -contiguous if the homotopy groups are trivial for everyone . An embedding ${\displaystyle M}$ ${\displaystyle k}$ ${\displaystyle \pi _{i}M}$ ${\displaystyle i\leq k}$

${\displaystyle f\colon M\to \mathbb {R} ^{n}}$

in Euclidean space is a differentiable mapping, which is an immersion and a topological embedding, i.e. H. is a homeomorphism on their image (especially injective ). ${\displaystyle \mathbb {R} ^{m}}$

Two embeddings are called isotopic if there is a smooth homotopy ${\displaystyle f_{0},f_{1}}$

${\displaystyle H\colon M\times \left[0,1\right]\to \mathbb {R} ^{n}}$

with gives, so that the figure is an embedding for each . ${\displaystyle H(.,)=f_{0},H(.,1)=f_{1}}$ ${\displaystyle t\in \left[0,1\right]}$ ${\displaystyle H(.,t)\colon M\to \mathbb {R} ^{n}}$

Haefliger-Zeeman theorem

For and are all embeddings of -contiguous -dimensional manifolds in the isotopic to one another. ${\displaystyle m\geq 2k+2}$ ${\displaystyle n\geq 2m-k+1}$ ${\displaystyle k}$ ${\displaystyle m}$ ${\displaystyle \mathbb {R} ^{n}}$

Special cases

This embedding of the circle is not isotopic to the unknot .

Connected manifolds

In the case one obtains: for and are all embeddings of connected -dimensional manifolds in the isotopic to one another. ${\displaystyle k=0}$ ${\displaystyle m\geq 2}$ ${\displaystyle n\geq 2m+1}$ ${\displaystyle m}$ ${\displaystyle \mathbb {R} ^{n}}$

This theorem does not hold for : there are numerous non-isotopic nodes in the . ${\displaystyle m=1}$ ${\displaystyle \mathbb {R} ^{3}}$

Simply connected manifolds

In the case one obtains: for and are all embeddings of simply connected -dimensional manifolds in the isotopic to one another. ${\displaystyle k=1}$ ${\displaystyle m\geq 4}$ ${\displaystyle n\geq 2m}$ ${\displaystyle m}$ ${\displaystyle \mathbb {R} ^{n}}$

literature

• Roger Penrose , JHC Whitehead , EC Zeeman , Imbedding of manifolds in Euclidean space , Ann. of Math. 73 (1961) 613-623.
• A. Haefliger , Plongements différentiables de variétés dans variétés , Comment. Math. Helv. 36 (1961), 47-82.
• EC Zeeman, Isotopies and knots in manifolds , Topology of 3-manifolds and related topics (Proc. The Univ. Of Georgia Institute, 1961), Prentice-Hall (1962), 187-193.
• M. Irwin, Embeddings of polyhedral manifolds , Ann. of Math. (2) 82 (1965) 1-14.
• JFP Hudson, Piecewise linear topology , WA Benjamin, Inc., New York-Amsterdam, 1969.