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 (which is 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 Theorem 4.2.
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], 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-Lefschetz 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
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. |