Embeddings of manifolds with boundary: classification
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Contents |
1 Introduction
Recall that some Unknotting Theorems hold for manifolds with boundary [Skopenkov2016c, 3], [Skopenkov2006,
2]. In this page we present results peculiar for manifold with non-empty boundary.
For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, 1,
3].
If the category is omitted, then a result stated below holds in both the smooth and piecewise-linear (PL) category.
We do not provide references to the original proofs of stated results.
Theorem 1.1.
Every -manifold
with nonempty boundary PL embeds into
.
This result can be found in [Horvatic1971, theorem 5.2]
2 Unknotting Theorems
Theorem 2.1.
Assume is a compact connected
-manifold and either
or
.
Then any two embeddings of
into
are isotopic.
The condition stands for General Position Theorem and the condition
stands for Whitney-Wu Unknotting Theorem, see theorems 2.1 and 2.2 respectivly of [Skopenkov2016c,
2].
Theorem 2.2.
Assume that is a compact connected
-manifold with non-empty boundary and one of the following conditions holds:
(a)
(b) is
-connected,
Then any two embeddings of into
are isotopic.
Part (a) of this theorem in case can be found in [Edwards1968,
4, corollary 5]. Case
is clear. Case
can be easily proved straightforward or can be deduced from Haefliger-Weber deleted square criterion [Skopenkov2006,
5]. Both condition are special cases of the Haefliger-Zeeman unknotting Theorem stated below, see [Penrose&Whitehead&Zeeman1961, Theorem 1.2b].
Theorem 2.3.
[The Haefliger-Zeeman unknotting Theorem]
For every ,
and closed
-connected
-manifold
, any two embeddings of
into
are isotopic.
Theorem 2.4.
For every and
and
-connected
-manifold
with non-empty boundary, any two embeddings of
into
are isotopic.
This result can be found in [Hudson1969, Theorem 10.3]
3 Construction and examples
...
4 Invariants
...
5 Classification
Theorem 5.1.[Becker-Glover]
Let be a closed homologically
-connected
-manifold
and
.
The cone map
is one-to-one
for
and is surjective
for
.
6 Further discussion
...
7 References
- [Becker&Glover1971] J. Becker and H. Glover, Note on the embedding of manifolds in Euclidean space, Proc. Am. Math. Soc. 27 (1971), 405-410. MR0268903 (42 #3800) Zbl 0207.22402
- [Edwards1968] Edwards, C. H. Unknotting polyhedral homology manifolds, Michigan Math. J. 15 (1968), 81-95. MR226629 Zbl 0167.52001
- [Horvatic1971] K. Horvatic, On embedding polyhedra and manifolds, Trans. Am. Math. Soc. 157 (1971), 417-436.
- [Hudson1969] J. F. P. Hudson, Piecewise linear topology, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR0248844 (40 #2094) Zbl 0189.54507
- [Penrose&Whitehead&Zeeman1961] R. Penrose, J. Whitehead and E. Zeeman, Imbedding of manifolds in Euclidean space., Ann. of Math. 73 (1961) 613–623. MR0124909 (23 #A2218) Zbl 0113.38101
- [Skopenkov2006] A. Skopenkov, Embedding and knotting of manifolds in Euclidean spaces, in: Surveys in Contemporary Mathematics, Ed. N. Young and Y. Choi, London Math. Soc. Lect. Notes, 347 (2008) 248-342. Available at the arXiv:0604045.
- [Skopenkov2016c] A. Skopenkov, Embeddings in Euclidean space: an introduction to their classification, to appear to Bull. Man. Atl.