4-manifolds in 7-space
This page has been accepted for publication in the Bulletin of the Manifold Atlas. |
Contents |
1 Introduction
For notation and conventions see high codimension embeddings.
Denote by is the Hopf map.
2 Examples
There is The Hudson torus
.
Analogously to the case for an orientable 4-manifold
, an embedding
and a class
one can construct an embedding
. However, this embedding is no longer well-defined. We have
for the Whitney invariant (defined analogously to The Whitney invariant for
).
2.1 Embeddings of CP2 into R7
We follow [Boechat&Haefliger1970], p. 164. Recall that is the mapping cylinder of
. Recall that
. Define an embedding
by
, where
. In other words, the segment joining
and
is mapped onto the arc in
joining
to
. Clearly, the boundary 3-sphere of
is standardly embedded into
. Hence
extends to an embedding
.
Apriori this extension need not be unique (because it can be changed by a connected sum with an embedding ). Surprisingly, it is unique, and is the only embedding
(up to isotopy and a hyperplane reflection of
).
Theorem 2.1.
- There are exactly two isotopy classes of embeddings
(differing by a hyperplane reflection of
).
- For each embeddings
and
the embedding
is isotopic to
.
This follows by [Skopenkov2005], Triviality Theorem (a) or by general classification.
2.2 The Lambrechts torus and the Hudson torus
These two embeddings are defined [Skopenkov2006] as compositions
, where
,
is the projection onto the second factor,
is
the standard inclusion and maps
are defined below. We shall see that
are embeddings for each
, hence
and
are embeddings.
Define , where
is identified with the set of unit length quaternions and
with the set of unit
length complex numbers.
Define , where
is identified with the 2-sphere formed by unit
length quaternions of the form
.
Note that is PL isotopic to
the Hudson torus
.
Take the Hopf fibration . Take the standard embeding
. Its complement has the homotopy type of
. Then
. This is the construction of Lambrechts motivated by the following property:
![\displaystyle S^7-im\tau^1\simeq \eta^{-1}(S^2)\cong S^2\times S^3\not\simeq S^2\vee S^3\vee S^5\simeq S^7-im f_0,](/images/math/1/b/b/1bbdafdc2296d2623edc8f6b06241099.png)
where is the standard embedding.
2.3 The Haefliger torus
This is a PL embedding which is (locally flat but) not PL isotopic to a smooth embedding [Haefliger1962], [Boechat&Haefliger1970], p.165, [Boechat1971], 6.2. Take
the Haefliger trefoil knot
.
Extend it to a conical embedding
. By [Haefliger1962], the trefoil knot also extends to a smooth embedding
(see [Skopenkov2006], Figure 3.7.a). These two extensions together form the Haefliger torus
(see [Skopenkov2006], Figure 3.7.b).
3 The Boechat-Haefliger invariant
Let be a closed connected orientable 4-manifold. Fix an orientation on
and an orientation on
. A
for
is the image of the fundamental class
under the composition
of the Alexander and Poincar\'e duality isomorphisms. (This composition is an inverse to the composition
of the
boundary map
and the normal bundle map
, cf. [Skopenkov2008], the Alexander Duality Lemma; this justifies the name `homology Seifert surface'.)
Define to be the image of
under the composition
of the Poincar\'e-Lefschetz and Alexander duality isomorphisms. (This composition has a direct geometric definition
as above.)
This new definition is equivalent to the original one [Boechat&Haefliger1970] by [Crowley&Skopenkov2008], Section Lemma.
4 Classification
A classification of for a closed connected 4-manifold
such that
is presented in Embeddings of highly-connected manifolds. Here we work in the smooth category.
Theorem 4.1. . [Haefliger1966], [Skopenkov2005], [Crowley&Skopenkov2008].
Theorem 4.2. [Crowley&Skopenkov2008] Let be a closed connected 4-manifold such that
. There is the Bo\'echat-Haefliger invariant
![\displaystyle BH:E^7(N)\to H_2(N)](/images/math/8/8/2/882cbd54ce379ac5e01c783bdaeb36b2.png)
whose image is
![\displaystyle im(BH)=\{u\in H_2(N)\ |\ u\equiv PDw_2(N)\mod2,\ u\cap u=\sigma(N)\}.](/images/math/b/c/3/bc3b14153ffc1c11b3f2d59df714dda9.png)
For each there is an injective invariant called the Kreck invariant,
![\displaystyle \eta_u:BH^{-1}(u)\to\Zz_{GCD(u,24)}](/images/math/9/b/e/9bec43c7d22631125a26de9d98c18da1.png)
whose image is the subset of even elements.
Here is the maximal integer
such that both
and 24 are divisible by
. Thus
is surjective if
is not divisible by 2. Note that
is divisible by 2 (for some
or, equivalently, for each
) if and only if
is spin.
For definition of the Kreck invariant see [Crowley&Skopenkov2008].
Theorem 4.2 implies that
- There are exactly twelve isotopy classes of embeddings
if
is an integral homology 4-sphere (cf. Theorem 4.1).
- For each integer
there are exactly
isotopy classes of embeddings
with
, and the same holds for those with
. Other values of
are not in the image of
. (We take the standard basis in
.)
Theorem 4.2 implies the following examples (first proved in [Skopenkov2005]) of the triviality and the effectiveness of the connected sum action .
- Let
be a closed connected 4-manifold such that
and the signature
of
is not divisible by the square of an integer
. Then for each embeddings
and
the embedding
is isotopic to
(in other words,
is injective).
- If
is a closed connected 4-manifold such that
and
for an embedding
, then for each embedding
the embedding
is not isotopic to
.
The following can be obtained using [Crowley&Skopenkov2008] (but not using [Skopenkov2005]).
- Take an integer
and the Hudson torus
. If
, then for each embedding
the embedding
is isotopic to
. (For a general integer
the number of isotopy classes of embeddings
is
.)
Under assumptions of Theorem 4.2 for each pair of embeddings and
![\displaystyle BH(f\#g)=BH(f)\quad\text{and}\quad \eta_{BH(f)}(f\#g)\equiv\eta_{BH(f)}(f)+\eta_0(g)\mod GCD(BH(f),24).](/images/math/8/c/1/8c19288ea077253c166ddb0f7fbec8de.png)
5 References
- [Boechat&Haefliger1970] J. Boéchat and A. Haefliger, Plongements différentiables des variétés orientées de dimension
dans
, (French) Essays on Topology and Related Topics (Mémoires dédiés à Georges de Rham), Springer, New York (1970), 156–166. MR0270384 (42 #5273) Zbl 0199.27003
- [Boechat1971] J. Boéchat, Plongements de variétées différentiables orientées de dimension
dans
, Comment. Math. Helv. 46 (1971), 141–161. MR0295373 (45 #4439) Zbl 0218.57016
- [Crowley&Skopenkov2008] D. Crowley and A. Skopenkov, A classification of smooth embeddings of 4-manifolds in 7-space, II, Intern. J. Math., 22:6 (2011) 731-757. Available at the arXiv:0808.1795.
- [Haefliger1962] A. Haefliger, Knotted
-spheres in
-space, Ann. of Math. (2) 75 (1962), 452–466. MR0145539 (26 #3070) Zbl 0105.17407
- [Haefliger1966] A. Haefliger, Differential embeddings of
in
for
, Ann. of Math. (2) 83 (1966), 402–436. MR0202151 (34 #2024) Zbl 0151.32502
- [Skopenkov2005] A. Skopenkov, A classification of smooth embeddings of 4-manifolds in 7-space, Topol. Appl., 157 (2010) 2094-2110. Available at the arXiv:0512594.
- [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.
- [Skopenkov2008] A. Skopenkov, A classification of smooth embeddings of 3-manifolds in 6-space, Math. Z. 260 (2008), no.3, 647–672. Available at the arXiv:0603429MR2434474 (2010e:57028) Zbl 1167.57013
This page has not been refereed. The information given here might be incomplete or provisional. |