Isotopy
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[edit] 1 Introduction
We work in the smooth or piecewise-linear (PL) or topological (TOP) category. If a category is omitted, then the result holds (or a definition is given) in all the three categories.
All manifolds are tacitly assumed to be compact.
[edit] 2 Ambient and non-ambient isotopy
Definition 2.1 (Ambient isotopy). For manifolds two embeddings are called ambiently isotopic, if there exists a homeomorphism onto such that
- for each
- for each and
- for each
See [Skopenkov2006, Figure 1.1]. An ambient isotopy is the above homeomorphism , or, equivalently, a family of homeomorphisms generated by the map in the obvious manner. The latter family can be seen as a homotopy .
Evidently, ambient isotopy is an equivalence relation on the set of embeddings of into . Classification of embeddings up to ambient isotopy is a classical problem in topology, see [Skopenkov2016c, 1].
Ambient isotopy is a stronger equivalence relation than non-ambient isotopy, isoposition, concordance, bordism, etc., see below
The words ambient isotopy are often abbreviated to just isotopy. One should be careful because isotopy often stands for non-ambient isotopy or for homotopy in the class of embeddings.
Definition 2.2 (Non-ambient isotopy). For manifolds two embeddings are called non-ambient isotopic, if there exists an embedding such that
- ,
- for each and
- for each .
This is equivalent to the existence of a homotopy in the class of embeddings. (Recall that we tacitly work with compact manifolds. For counterexample involving non-compact manifolds see [Geiges2018].)
In the smooth category, or for in the PL or TOP category, non-ambient isotopy implies ambient isotopy [Hirsch1976], [Hudson&Zeeman1964], [Hudson1966], [Akin1969], [Edwards1975]. For this is not so: e.g., any knot is non-ambiently PL isotopic to the trivial one, but not necessarily ambiently PL isotopic to it.
In the smooth category, non-ambient isotopy is called diffeotopy.
[edit] 3 Isoposition and concordance
Definition 3.1 (Isoposition). For manifolds two embeddings are called (orientation preserving) isopositioned, if there is an (orientation preserving) homeomorphism such that .
For embeddings into PL orientation preserving isoposition is equivalent to PL isotopy (the Alexander-Guggenheim Theorem) [Rourke&Sanderson1972, 3.22]. It would be interesting to know if the smooth analogue of this result holds.
Definition 3.2 (Concordance). For manifolds two embeddings are called ambiently concordant, or just concordant, if there is a homeomorphism onto (which is called a concordance) such that
- for each and
- for each .
The definition of non-ambient concordance is analogously obtained from that of non-ambient isotopy by dropping the last condition of level-preservation. Note that in knot theory non-ambient concordance is called cobordism.
In the DIFF category or for in the PL or TOP category non-ambient concordance implies ambient concordance and ambient isotopy [Lickorish1965], [Hudson1970], [Hudson&Lickorish1971]. (This is not so in the PL or TOP category for codimension 2.) This result allows a reduction of the Knotting Problem to the relativized Embedding Problem, see [Skopenkov2016c, 1].
[edit] 4 References
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- [Geiges2018] H. Geiges, Isotopies vis-`a-vis level-preserving embeddings, Arch. Math. 110 (2018), 197–200. Available at the arXiv:1708.09703.
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- [Skopenkov2016c] A. Skopenkov, Embeddings in Euclidean space: an introduction to their classification, submitted to Bull. Man. Atl.