Embeddings just below the stable range: classification
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Contents |
1 Introduction
Most of this page is intended not only for specialists in embeddings, but also for mathematician from other areas who want to apply or to learn the theory of embeddings.
Recall the Whitney-Wu Unknotting Theorem: if is a connected manifold of dimension
, and
, then every two embeddings
are isotopic [Skopenkov2016c, Theorem 3.2], [Skopenkov2006, Theorem 2.5]. In this page we summarize the situation for
and
is a connected, as well as in some more general situations.
For the classification of embeddings of some disconnected manifolds see [Skopenkov2016h].
For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, 1,
3]. Denote
.
2 Classification
For the next theorem, the Whitney invariant is defined in
5 below.
Theorem 2.1. Assume that is a closed connected
-manifold, and either
or
and we are in the PL category.
(a) If is oriented, the Whitney invariant,
![\displaystyle W:E^{2n}(N)\to H_1(N;\Zz_{\varepsilon(n-1)}),](/images/math/0/c/0/0c01fbf8f4f9771a7e251feb59559526.png)
is a 1-1 correspondence.
(b) If is non-orientable, then there is a 1-1 correspondence
![\displaystyle E^{2n}(N)\to \begin{cases} H_1(N;\Zz_2) & n\text{ is odd}\\ \Zz\oplus\Zz_2^{s-1}\quad\mbox{where}\quad H_1(N;\Zz_2)\cong\Zz_2^s & n\text{ is even}\end{cases}.](/images/math/2/4/e/24e59ff646a210203da373f9a602a0bf.png)
Remark 2.2 (Comments on the proof). Part (a) is proved in [Haefliger1962b, 1.3.e], [Haefliger1963], [Haefliger&Hirsch1963, Theorem 2.4], [Bausum1975, Theorem 43] in the smooth category, and in [Weber1967], [Vrabec1977, Theorems 1.1 and 1.2] in the PL category.
Part (b) is proved in [Bausum1975, Theorem 43] in the smooth category. According to [Weber1967], [Skopenkov1997] the proof works also in the PL category.
In part (b) we replaced the kernel from [Bausum1975, Theorem 43] by
.
This is possible because, as a specialist could see,
is the multiplication with
, so
.
The 1-1 correspondence from (b) can presumably be defined as a (generalized) Whitney invariant, see [Vrabec1977], but the proof used 'the Haefliger-Wu invariant' whose definition can be found e.g. in [Skopenkov2006, 5].
It would be interesting to check if part (b) is equivalent to different forms of description of
[Haefliger1962b, 1.3.e], [Haefliger1963], [Vrabec1977, Theorem 1.1].
The classification of smooth embeddings of 3-manifolds in is more complicated, see Theorem 6.3 below for
or [Skopenkov2016t].
Concerning embeddings of connected -manifolds in
see [Yasui1984] for
, [Skopenkov2016f] for
, and [Saeki1999], [Skopenkov2010], [Tonkonog2010] for manifolds with boundary.
Theorem 2.1 is generalized to a description of for closed
-connected
-manifolds
, see Theorem 2.1.
3 Hudson tori
Together with the Haefliger knotted sphere [Skopenkov2016t, Example 2.1], [Skopenkov2006, Example 3.4], the examples of Hudson tori presented below were the first examples of embeddings in codimension greater than 2 which are not isotopic to the standard embedding defined below. (Hudson's construction [Hudson1963] was not as explicit as those below.)
Take the standard embedding![S^{n-1}\subset\Rr^n\subset\Rr^{2n}](/images/math/5/b/7/5b7da8b0b8016ef6769594ce2cc1ba27.png)
Tex syntax error.
Example 3.1. Let us construct, for any and
, a smooth embedding
![\displaystyle \Hud_n(a):S^1\times S^{n-1}\to\Rr^{2n}.](/images/math/0/e/9/0e91c3bf0d283c4df6842ac2b40fcaf6.png)
We start with the cases .
Take the standard inclusion .
The 'standard embedding'
is given by the standard inclusions
Tex syntax error
Tex syntax erroranalogously to
Tex syntax error, where
![2](/images/math/0/3/3/0339b8cd4613c74a86d715439a3c09f7.png)
Take the embedding given by
Tex syntax error
Tex syntax errorjoins the images of
![\Hud_n(0)](/images/math/4/a/f/4af6f5ce195d8f168b5b889e4f281f1c.png)
![g_1](/images/math/4/4/b/44be6ace5668ee8cd97c7accc925aab4.png)
![\Hud_n(1)](/images/math/5/2/d/52d8f018b81e36930436db3c1454a796.png)
![\Hud_n(0)](/images/math/4/a/f/4af6f5ce195d8f168b5b889e4f281f1c.png)
![g_1](/images/math/4/4/b/44be6ace5668ee8cd97c7accc925aab4.png)
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
For we repeat the above construction of
replacing
by
copies
of
,
.
The copies are outside
and are `parallel' to
.
The copies have the standard orientation for
or the opposite orientation for
.
Then we make embedded connected sum along natural segments joining every
-th copy to the
-th
copy.
We obtain an embedding
which has disjoint images with
.
Let
be the linked embedded connected sum of
and
.
The original motivation for Hudson was that is not isotopic to
for any
(this is a particular case of Proposition 3.2 below). One guesses that
is not isotopic
to
for
.
And that a
-valued invariant exists and is `realized' by the homotopy class of the map
![\displaystyle S^n\overset g\to S^{2n}-i(D^{n+1}\times S^{n-1})\sim S^{2n}-S^{n-1}\sim S^n \quad\text{which is}\quad a\in\pi_n(S^n)\cong\Zz.](/images/math/e/c/4/ec488593f70dd217587b7fccbd7e195f.png)
However, this is only true for odd.
Proposition 3.2.
For odd
is isotopic to
if and only if
.
For even
is isotopic to
if and only if
.
Proposition 3.2 follows by calculation of the Whitney invariant (Remark 5.3.d below) and, for even, by Theorem 2.1.
This proposition holds, with the same proof, in the piecewise smooth category (see [Skopenkov2016f, Remark 1.2]) and in the PL category (with an analogous construction of
for the PL category).
It would be interesting to find an explicit construction of an isotopy between
and
, cf. [Vrabec1977,
5].
Analogously,
is not isotopic to
if
.
It would be interesting to know if the converse holds, e.g. is
(PS or smoothly) isotopic to
.
Example 3.3.
Take any .
Take a map
of degree
(so
).
Recall that
.
Define the smooth embedding
to be the composition
Tex syntax error
Let us present a geometric description of this embedding.
Define a map by
.
This map gives an embedding
Tex syntax error
See [Skopenkov2006, Figure 2.2].
The image of is the union of the graphs of the maps
and
.
For any the disk
intersects the image of this embedding at two points
lying in
, i.e., at the image of an embedding
.
The embedding
is obtained by extending the latter embeddings to embeddings
for all
.
Cf. [Skopenkov2006, Figure 2.3].
Remark 3.4. (a) The analogue of Proposition 3.2 for replaced to
holds, with an analogous proof.
(b) The embeddings and
are smoothly isotopic for
and are PS isotopic for
[Skopenkov2006a] (see [Skopenkov2016f, Remark 1.2]).
This follows by calculation of the Whitney invariant (Remark 5.3.d below).
It would be interesting to know if they are smoothly isotopic for
.
It would be interesting to know if they are PS isotopic for
.
(c) For Example 3.3 gives what we call the 'left' Hudson torus.
The 'right' Hudson torus is constructed analogously.
It is the composition of the left Hudson torus and the exchanging factors autodiffeomorphism of
.
The right and the left Hudson tori are not isotopic by Remark 5.3.d below.
(d) Analogously one constructs the Hudson torus for
and
or, more generally,
for
and
.
There are versions
of these constructions corresponding to Definition 3.3.
For
this corresponds to the Zeeman map
[Skopenkov2016h, Definition 2.2] and its composition with 'the unframed second Kirby move' [Skopenkov2015a,
2.3].
It would be interesting to know if the links
are isotopic, cf. [Skopenkov2015a, Remark 2.7.b].
These constructions can be further generalized [Skopenkov2016k].
4 Action by linked embedded connected sum
In this section we generalize the construction of the Hudson torus .
Let
be a closed connected oriented
-manifold.
We work in the smooth category which we omit.
Apparently analogous results hold for
in the PL and PS categories (see [Skopenkov2016f, Remark 1.2]).
Example 4.1.
For any , an embedding
and
, we shall construct an embedding
.
This embedding is said to be obtained by linked embedded connected sum of
with an
-sphere representing the `homology Alexander dual'
of
(defined in [Skopenkov2005, Alexander Duality Lemma 4.6]).
Represent by an embedding
.
By definition, the class
is represented by properly oriented
.
Since any orientable bundle over
is trivial,
.
Take an embedding
whose image is
and which represents
.
By embedded surgery on
we obtain an embedding
representing
(see details in Proposition 4.2 below).
Define
to be the linked embedded connected sum of
and
, along some arc joining their images.
Proposition 4.2 (embedded surgery).
For any , a neighborhood
of a codimension at least 3 subpolyhedron in
and an embedding
there is an embedding
homologous to
.
Proof. Take a vector field on normal to
.
Extend
along this vector field to a map
.
Since
and
is a neighborhood
of a codimension at least 3 subpolyhedron, by general position we may assume that
is an embedding and that
misses
.
Since
, we have
.
Hence the standard
-framing of
in
extends to an
-framing on
in
.
Thus
extends to an embedding
![\displaystyle \widehat b:D^2\times D^{n-1}\to\R^{2n}-U\quad\text{such that}\quad \widehat b(\partial D^2\times D^{n-1})\subset g(S^1\times S^{n-1}).](/images/math/a/a/3/aa3cea2126097ada817f3b858e41c7c0.png)
Take an embedding such that
![\displaystyle g_1(S^n)\ =\ g(S^1\times S^{n-1})-\widehat b(\partial D^2\times Int D^{n-1}) \bigcup\limits_{\widehat b(\partial D^2\times\partial D^{n-1})} \widehat b(D^2\times\partial D^{n-1})\ \cong\ S^n.](/images/math/f/9/8/f98027e1247f5e3c1806730cd534c0b5.png)
with proper orientation so that is homologous to
. QED
The isotopy class of the embedding is independent of the choises in the construction.
The independence of the arc and of the maps
follows by
and by Proposition 4.3 below, respectively.
By Definition 5.1 of the Whitney invariant, is
for
odd and
for
even.
Thus by Theorem 2.1.a for
all isotopy classes of embeddings
can be obtained from any chosen embedding
by the above construction.
Proposition 4.3. For any both the linked embedded connected sum and parametric connected sum (introduced in [Skopenkov2006a], [Skopenkov2015a]) define free transitive actions of
on
.
This follows by Theorem 2.1.a and by [Skopenkov2014, Remark 18.a].
5 The Whitney invariant
Let be a closed
-manifold.
Take an embedding
.
Fix an orientation on
.
For any other embedding
we define the Whitney invariant
![\displaystyle W(f, f_0)=W_{f_0}(f)=W(f)\in H_{2n-m+1}(N;\Zz_N).](/images/math/5/2/4/5241adb18efe69df96fbf1725c71c755.png)
Here the coefficients are
if
is oriented and
is odd, and are
otherwise.
Tex syntax erroris defined as the homology class of the closure of the self-intersection set of a general position homotopy
![H](/images/math/2/f/b/2fbada10033dab2ef3330c6cb17a3a0c.png)
![f](/images/math/6/b/6/6b6e98cde8b33087a33e4d3a497bd86b.png)
![f_0](/images/math/0/7/e/07ed23691bfe5279d8b1eb9c83119baa.png)
This is formalized in Definition 5.2 in the smooth category, following [Skopenkov2010], see
also [HaefligerHirsch1963].
The definition in the PL category is analogous [Hudson1969, 11], [Vrabec1977, p. 145],
[Skopenkov2006,
2.4 `The Whitney invariant'].
We begin by presenting a simpler definition, Definition 5.1, for a particular case.
For Theorem 2.1 only the case is required.
Definition 5.1.
Assume that is
-connected and
.
Then restrictions of
and
to
are regular homotopic (see
[Koschorke2013, Definition 2.7], [Hirsch1959]).
Since
is
-connected,
retracts to an
-dimensional polyhedron.
Therefore these restrictions are isotopic, cf. [Haefliger&Hirsch1963, 3.1.b], [Takase2006, Lemma 2.2].
So we can make an isotopy of
and assume that
on
.
Take a general position homotopy
relative to
between the
restrictions of
and
to
.
Let
(`the intersection of this homotopy with
').
![n+2(n+1)<2m](/images/math/4/9/2/492957e7b3a24f7196e54cb98a8a21b0.png)
Tex syntax erroris a compact
![(2n+1-m)](/images/math/0/6/3/063e854d764c6b9c09f9df1bcf76a85b.png)
![\partial N_0](/images/math/f/9/6/f96c15bf3a49cdf2051f311be0c68a80.png)
So carries a homology class with
coefficients.
If
is odd and
is oriented, then
has a natural orientation defined below,
and so carries a homology class with
coefficients.
Define
to be the homology class:
Tex syntax error
![f\cap F](/images/math/6/a/3/6a3602509e8d9d182f7d67881eff9fc0.png)
Tex syntax error) is defined (for
![m-n](/images/math/b/6/e/b6e54703653c23e61afe266a3bdbdd5f.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
as follows.
For any point take a vector at
tangent to
.
Complete this vector to a positive base tangent to
.
Since
, by general position there is a unique point
such that
.
The tangent vector at
thus gives a tangent vector at
to
.
Complete this vector to a positive base tangent to
, where the orientation on
comes
from
.
The union of the images of the constructed two bases is a base at
of
.
If the latter base is positive, then call the initial vector of
'positive'.
Since a change of the orientation on
forces a change of the orientation of the latter base
of
, this condition indeed defines an orientation on
.
Definition 5.2.
Assume that .
Take a general position homotopy
between
and
.
Tex syntax errorof the self-intersection set carries a cycle mod 2.
If is oriented and
is odd, the closure also carries an integer cycle.
See [Hudson1967,
11], [Skopenkov2006,
2.3 `The Whitney obstruction'].
(Let us present informal explanations of these facts.
For![2m\ge3n+2](/images/math/a/c/4/ac42ffc32fee13f8930d2fe0bef58252.png)
Tex syntax errorcan be assumed to be a submanifold.
In general, since , by general position the closure has codimension 2 singularities, see definition in
7.
So the closure carries a cycle mod 2.
The closure also has a natural orientation, see
Definition 7.1 and remark below.
So the closure carries an integer cycle.)
Define the Whitney invariant to be the homology class:
Tex syntax error
Clearly, if
is isotopic to
.
Hence the Whitney invariant defines a map
![\displaystyle W:E^m(N)\to H_{2n-m+1}(N;\Zz_N),\quad [f] \mapsto W(f).](/images/math/c/3/0/c30f51328f41351aac3861e4f870de3e.png)
Clearly, (for both definitions).
The definition of depends on the choice of
, but we write
not
for brevity.
Remark 5.3.
(a) The Whitney invariant is well-defined by Definition 5.2,
i.e. is independent of the choice of a general position homotopy
from
to
.
Tex syntax errorfor a general position homotopy
![H_{01}:N\times I\times I\to\Rr^m\times I\times I](/images/math/7/d/9/7d974d2c2ed915a105323b69b287d03e.png)
![H_0,H_1:N\times I\to\Rr^m\times I](/images/math/0/8/7/0874f958e49857acc9c56909b71752e6.png)
![f_0](/images/math/0/7/e/07ed23691bfe5279d8b1eb9c83119baa.png)
![f](/images/math/6/b/6/6b6e98cde8b33087a33e4d3a497bd86b.png)
See details in [Hudson1969, 11].
(b) Definition 5.1 is a particular case of Definition 5.2.
(Indeed, if on
, we can take
to be fixed on
. See details in [Skopenkov2010, Difference
Lemma 2.4].)
Hence the Whitney invariant is well-defined by Definition 5.1, i.e. independent of the choice
of
and of the isotopy making
outside
.
![W(f)](/images/math/2/2/1/221f32285325c78c47de94fd7c7f75ab.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![f\cap F](/images/math/6/a/3/6a3602509e8d9d182f7d67881eff9fc0.png)
Tex syntax error). For the reflection
![\sigma:\Rr^m\to\Rr^m](/images/math/6/1/9/6193d98c9ab0422c651027e2264dd25a.png)
![W(\sigma\circ f)=-W(f)](/images/math/1/9/2/1924babe23e26ae0ce7bf8327aff3f63.png)
![\Rr^m](/images/math/7/f/2/7f261d3a1d5315d963aab7981fccbd80.png)
![f\cap F](/images/math/6/a/3/6a3602509e8d9d182f7d67881eff9fc0.png)
Tex syntax error; for Definition 5.1 also observe that we may assume that
![f=f_0=\sigma\circ f](/images/math/c/6/c/c6cf229f35059deca4f1195f751730d9.png)
![N_0](/images/math/a/e/7/ae77718b7e730be439983a5bf2348800.png)
(d) For the Hudson tori is
or
for
,
and
.
For and
this was proved in [Hudson1963] (using and proving a particular case
of Remark 5.3.f).
For
the proof is analogous.
For
this is clear by Definition 5.1.
(e) for any pair of embeddings
and
.
![W(f\#g)-W(f)=W(f\#g,f_0)-W(f,f_0)=W(f\#g,f)=0](/images/math/d/b/d/dbdd409aa4c1f3c7c9265288b0c274ba.png)
![H_f](/images/math/0/4/b/04bcba99b75cca1cc5abd4e5fddb95d8.png)
![f](/images/math/6/b/6/6b6e98cde8b33087a33e4d3a497bd86b.png)
![H_g](/images/math/f/7/b/f7b9ec248e54e11f3b2ed76020fd35a5.png)
![g](/images/math/f/4/6/f46271e5c04cf1146670e9315ac9713d.png)
![H_f\sharp H_g](/images/math/2/8/e/28e29f776c97e9d5df24c5584b361f57.png)
![f\#g](/images/math/0/a/a/0aabaf11c5162ade8ba1ee6647598003.png)
![f](/images/math/6/b/6/6b6e98cde8b33087a33e4d3a497bd86b.png)
Tex syntax erroris null-homologous in
![S^n](/images/math/a/f/d/afd38444b95e8b5abbf51c458ea39cbc.png)
![N\cong N\#S^n](/images/math/2/f/1/2f18f52244a27abfdb750b1a4c8861ba.png)
(f) For and
the Whitney invariant equals to the pair of linking coefficients [Skopenkov2016h].
(g) The Whitney invariant need not be a bijection for (see
\ref{s:begl}) or for
,
even and
non-orientable (by Theorem 2.1.b).
6 A generalization to highly-connected manifolds
In this section let be a closed orientable homologically
-connected
-manifold,
. Recall the unknotting theorem [Skopenkov2016c] that all embeddings
are isotopic when
and
. In this section we generalize Theorem 2.1 to a description of
and further to
for
.
6.1 Examples
Some simple examples are the Hudson tori .
Example 6.1 (Linked embedded connected sum, cf. [Skopenkov2010, Definition 1.4]).
Assume that is
-connected and
.
Then for an embedding
and a class
one can construct an embedding
by linked connected sum analogously to the case
presented in Example 4.1.
We have for the Whitney invariant. Hence by Theorem 6.2 below this construction gives a free transitive action of
on
, provided
or
in the PL or smooth categories, respectively.
The embedding has an alternative construction using parametric connected sum [Skopenkov2014, Remark 18.a].
6.2 Classification
Theorem 6.2. Let be a closed oriented homologically
-connected
-manifold,
. Then the Whitney invariant
![\displaystyle W:E^{2n-k}(N)\to H_{k+1}(N;\Zz_{\varepsilon(n-k-1)})](/images/math/1/f/9/1f9ca0866bacc6e8147523f65e9df3c8.png)
is a bijection, provided in the smooth category or
in the PL category.
This was proved for -connected manifolds in the smooth category [Haefliger&Hirsch1963], and in the PL category in [Hudson1969,
11], [Boechat&Haefliger1970], [Boechat1971], cf. [Vrabec1977]. The proof actually used the homological
-connectedness assumption (see some details in [Skopenkov2016c, the text after the Haefliger-Zeeman Unknotting Theorem 2.4]).
For Theorem 6.2 is covered by Theorem 2.1; for
it is not.
For
the PL case of Theorem 6.2 gives nothing but the Zeeman Unknotting Spheres Theorem [Skopenkov2016c].
For the case of knotted tori see [Skopenkov2016k, Theorem 3.1].
An inverse to the map of Theorem 6.2 is given by Example 6.1.
Because of the existence of knotted spheres the analogues of Theorem 6.2 for in the PL case, and for
in the smooth case are false.
So for the smooth category,
and
closed connected, a classification of
is much harder: for 40 years the only known complete readily calculable classification results were for homology spheres
. E.g.
for any
[Haefliger1966], [Skopenkov2016s].
The following result for
was obtained using the Boéchat-Haefliger formula for the smoothing obstruction [Boechat&Haefliger1970], [Boechat1971].
Using that formula one can define the higher-dimensional Kreck invariant [Skopenkov2008].
Theorem 6.3 [Skopenkov2008]. Let be a closed orientable homologically
-connected
-manifold. Then the Whitney invariant
![\displaystyle W:E^{6l}_D(N)\to H_{2l-1}(N)](/images/math/b/7/6/b760d141e7e83636033b5274cd238183.png)
is surjective and for any the Kreck invariant
![\displaystyle \eta_u:W^{-1}u\to\Zz_{d(u)}](/images/math/7/f/f/7ff2f8eade12bc02562772d968d484ce.png)
is a 1-1 correspondence, where is the divisibility of the projection of
to the free part of
.
Recall that the divisibility of zero is zero and the divisibility of is
.
How does one describe when
is not
-connected?
For general
see the sentence on
at the end of
2.
We can say more as the connectivity
of
increases.
Some estimations of
for a closed
-connected
-manifold
are presented in [Skopenkov2010]. For
one can go even further:
Theorem 6.4 [Becker&Glover1971]. Let be a closed
-connected
-manifold embeddable into
,
and
. Then there is a 1-1 correspondence
![\displaystyle E^m(N)\to [N_0, V_{m,n+1}].](/images/math/6/4/3/643341df90bc99621c0740a72f5ab4ef.png)
The 1-1 correspondence is constructed using `the Haefliger-Wu invariant' involving the configuration space of distinct pairs [Skopenkov2006, 5].
For
Theorem 6.4 is the same as General Position Theorem 2.1 [Skopenkov2016c] (because
is
-connected).
For
Theorem 6.4 is covered by Theorem 6.2; for
it is not.
For application to knotted tori see [Skopenkov2016k, Theorem 5.1].
For generalization to arbitrary manifolds see [Haefliger1963], [Weber1967], [Skopenkov2002] and the survey [Skopenkov2006,
5]. Observe that in Theorem 6.4
can be replaced by
for any
.
7 An orientation on the self-intersection set
Let be a smooth map of an oriented
-manifold
.
![m\ge n+2](/images/math/1/f/2/1f2b3a0c2d4e3f11b684651d72a3e81d.png)
Tex syntax errorof the self-intersection set of
![f](/images/math/6/b/6/6b6e98cde8b33087a33e4d3a497bd86b.png)
Tex syntax errorsuch that both
![P](/images/math/6/b/5/6b52835f794dc38160c3157e48761ad3.png)
Tex syntax errorare subpolyhedra of some triangulation of
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
Tex syntax errorand
Tex syntax erroris an open manifold consisting of self-transverse double points of
![f](/images/math/6/b/6/6b6e98cde8b33087a33e4d3a497bd86b.png)
(The latter property of defines a dense subset of the set of all smooth maps
.)
Definition 7.1 (A natural orientation on ).
Take points
away from
and such that
. Then a
-base
tangent to
at
gives a
-base
tangent to
at
. Since
is oriented, we can take positive
-bases
and
at
and
normal to
and to
. If the base
of
is positive, then call the base
positive. This is well-defined because a change of the sign of
forces changes of the signs of
and
.
We remark that
- a change of the orientation of
forces changes of the signs of
and
and so does not change the orientation of
.
- the natural orientation on
need not extend to
Tex syntax error
: take the `smooth cone'over a general position map
having only two transverse self-intersection points. (We define
, where
and
.)
- the natural orientation on
extends to
Tex syntax error
ifis odd [Hudson1969, Lemma 11.4].
Definition 7.2 (A natural orientation on for
even).
Take a
-base
at a point
. Since
is oriented, we can take a positive
-base
normal to
in one sheet of
. Analogously construct an
-base
for the other sheet of
.
Since
is even, the orientation of the base
of
does not depend on choosing the first and the other sheet of
at
.
If the base
is positive, then call the base
positive.
This is well-defined because a change of the sign of
forces changes of the signs of
and so of
.
We remark that a change of the orientation of forces changes of the signs of
and so does not change the orientation of
.
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