3-manifolds
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So let us restrict to the case with only one solid torus $L':S^1\times D^2\rightarrow S^3$. | So let us restrict to the case with only one solid torus $L':S^1\times D^2\rightarrow S^3$. | ||
Choose any self-homeomorphism $f$ of the torus $S^1\times S^1$. The result of the Dehn surgery is defined as | Choose any self-homeomorphism $f$ of the torus $S^1\times S^1$. The result of the Dehn surgery is defined as | ||
− | $$M_f:=S^3\setminus L'(S^1\times D^2) \cup_f S^1\times D^2$$ | + | $$M_f:=S^3\setminus L'(S^1\times \mathring{D}^2) \cup_f S^1\times D^2$$ |
− | + | If $f$ is the coordinate flipping and if the embedding $S^1\rightarrow S^3$ is the unknot, the result of Dehn-surgery would be $S^1\times S^2$. (TODO CHECK is the framing important ?). | |
− | If $f$ is the coordinate flipping and if the embedding $S^1\rightarrow S^3$ is the unknot, the result of Dehn-surgery would be $S^1\times S^2$. (TODO CHECK is | + | |
Changing $f$ by an isotopy won't change the homeomorphism type of $M$ (TODO proof, reference, ). Every self-homeomorphism of $T^2$ is isotopic to exactly one self-homeomorphism $f_A\in \Homeo(T^2)$ of the form: | Changing $f$ by an isotopy won't change the homeomorphism type of $M$ (TODO proof, reference, ). Every self-homeomorphism of $T^2$ is isotopic to exactly one self-homeomorphism $f_A\in \Homeo(T^2)$ of the form: | ||
− | $$f_A:\Rr^2/\Zz^2 \rightarrow \Rr^2 /\Zz^2 \qquad (x | + | $$f_A:\Rr^2/\Zz^2 \rightarrow \Rr^2 /\Zz^2 \qquad \left(\begin{array}{c}x\\y\end{array}\right)\mapsto A\cdot\left(\begin{array}{c}x\\y\end{array}\right)$$ |
− | + | , where $A\in SL_2(\Zz)$ (reference of proof). | |
− | , where $A\in SL_2(\Zz)$ (reference of proof). | + | TODO composition corresponds to two successive Dehn- surgeries. We have to find out, for self-homeomorphisms of the torus Dehn surgery gives $S^3$ back. Consider a matrix of the form $\left( \begin{array}{cc} 1 & 0 \\ k & 1 \end{array}\right)$. The homeomorphism $f_A \in \Homeo(T^2)$ extends to a homeomorphism of $\bar{f_A}\in \Homeo(S^1\times D^2)$: |
− | + | ||
− | TODO composition corresponds to two successive Dehn- surgeries | + | |
$$\bar{f_A}(x,y):=(x,x^ky), $$ | $$\bar{f_A}(x,y):=(x,x^ky), $$ | ||
where $x\in S^1 = \{z\in \Cc| |z|=1\}, y\in D^2=\{y\in\Cc||y|\le 1\}$. | where $x\in S^1 = \{z\in \Cc| |z|=1\}, y\in D^2=\{y\in\Cc||y|\le 1\}$. | ||
Using this homeomophism one can define a homeomorphism from $M_f$ to $S^3$: | Using this homeomophism one can define a homeomorphism from $M_f$ to $S^3$: | ||
− | $$S^3 = S^3\setminus L(S^1\times D^2)\cup_1 S^1\times D^2 \rightarrow M_{f_A}=S^3\setminus L(S^1 \times D^2)\cup_{f_A} S^1\times D^2$$ | + | $$S^3 = S^3\setminus L(S^1\times \mathring{D}^2)\cup_1 S^1\times D^2 \rightarrow M_{f_A}=S^3\setminus L(S^1 \times \mathring{D}^2)\cup_{f_A} S^1\times D^2$$ |
given by the identity on the left component and $\bar{f_A}$ on the right component. | given by the identity on the left component and $\bar{f_A}$ on the right component. | ||
Together with (link to comment about composition), this tells us, that $M_{f_A}$ really only depends on the coset $A\cdot \left(\begin{array}{cc}1&*\\0&1\end{array}\right)$ (TODO check right or left coset). This coset is uniquely determined by the image $(p,q)$ of $(1,0)$ with $p$ and $q$ coprime. | Together with (link to comment about composition), this tells us, that $M_{f_A}$ really only depends on the coset $A\cdot \left(\begin{array}{cc}1&*\\0&1\end{array}\right)$ (TODO check right or left coset). This coset is uniquely determined by the image $(p,q)$ of $(1,0)$ with $p$ and $q$ coprime. | ||
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Revision as of 14:50, 8 June 2010
This page has not been refereed. The information given here might be incomplete or provisional. |
Contents |
1 Introduction
In the 3-dimensional setting there is no distinction between smooth, PL and topological manifolds neccesary; the categories of smooth, PL and topological manifolds are equivalent (TODO ref). A lot of techniques have been developed in the last century to study 3-manifolds but most of them are very special and don't generalise to higher dimensions. One key idea is to decompose manifolds along incompressible surfaces into smaller pieces, to which certain geometric models apply. A great progress was made in with the proof of the Poincaré conjecture and Thurton's geometrization conjecture by Perelman in 2003.
2 Construction and examples
Basic examples are with
any surface.
Important types of 3-manifolds are Haken-Manifolds, Seifert-Manifolds, 3-dimensional lens spaces, Torus- and Torus semi-bundles.
There are two topological processes to join 3-manifolds to get a new one.
The first is the connected sum of two manifolds and
. Choose embeddings
and
, remove the interior of
and
and glue
and
together along the boundaries
and
.
The second uses incompressible surfaces. Let
be manifold and
a surface.
is incompressible, if there is no disk
in
with
.
The torus sum is the process which glues incompressible tori boundary components together.
(TODO What is incompressibility needed? / What is is good for/ What happen if one takes a compressible surface ?)
3 Invariants
In the 3-dimensional world the fundamental group is a powerful invariant to distiguish manifolds. It determines already all homology groups:
-
= abelization of
.
-
torsion
-
-
for
4 Classification/Characterization
By reversing the process of connected and torus sum every 3-manifold can be decomposed into pieces which admit a geometric structure. We describe the details in the following.
4.1 Prime decomposition
One can at first decompose as a connected sum of wikiref prime 3-manifolds TODO cite. A prime 3-manifold is either
or every embedded 2-sphere bounds a ball, in which case the manifold is called irreducible.
Prime 3-manifolds can be distinguished by their fundamental groups into 3 types.
Theorem 4.1 Kneser.
Every orientable, compact 3-manifold has a unique decomposition
into prime manifolds
.
A manifold is called prime, if it can't be written as a non-trivial connected sum, i.e.
implies
or
.
Van Kampen's theorem tells you, that
. Hence any 3-manifold, whose fundamental group cannot be written as a free product of two nontrivial subgroups, can only be written as the connected sum of another 3-manifold with a simply connected 3-manifold. By the Poincaré conjecture a simply connected 3-manifold is already homeomorphic to
. Hence each such manifold is prime.
Examples are
.
A manifold
is called irreducible if every embedded
bounds a ball, i.e. the embedding extends to an embedding of
(TODO: check).
Irreducibility is only slightly stronger than being prime.
is the only orientable not irreducible prime manifold.
4.1.1 Type I: finite
The universal cover is a simply connected 3-manifold. As the fundamental group already determines the homology of a oriented, closed compact 3-manifold, it has to be a homology sphere. Using the Hurewicz-theorem, its fundamental class is represented by a degree 1 map
. This map induces isomorphisms on the homology and on the fundamental group. Hence it is a weak homotopy equivalence, and hence a homotopy equivalence by Whitehead's theorem (ref?).
Hence every prime
-manifold with finite fundamental group arises as the quotient of a homotopy sphere by a free action of a finite group.
With the use of the Poincaré conjecture every homotopy 3-sphere is homeomorphic to
and we can write
. If
is cyclic
is known as lens space (ref).
4.1.2 Type II: infinite cyclic
is the only orientable closed prime 3-manifold of this type. Futhermore it is the only not irreducible prime manifold. (TODO: proof/ref)
4.1.3 Type III: infinite non-cyclic
Such a manifold is always aspherical (TODO ref). The sphere theorem states, that every map
is homotopic to an embedding; and - as
is irrecudible - it is nullhomotopic. Hence
. Consider the universal convering
of
. Its first homology vanishes, as it is simply connected. The long exact sequence of homotopy groups of the fibration
gives a isomorphism
. Hence by Hurewicz' theorem
. Furthermore
, as
is noncompact. Applying Hurewicz theorem again, we get, that all homotopy groups of
vanish and hence by Whiteheads theorem
is contractible. This means, that
is apherical.
Hence the homotopy type of a prime 3-manifold with infinite non-cyclic fundamental group is uniquely determined by its fundamental group.
Furthermore not every group can occur as a fundamental group of a prime 3-manifold. The equivariant cellular chain complex of
is a projective resolution of the trivial
-module
. Hence ....
For any subgroup
the space
is a finite-dimensional model for
. For example a finite group cannot have such a model (by group homology ref) and hence
must be torsionfree. Furthermore it is a Poincaré duality group (link).
4.2 Torus decomposition
According to the previous section it remains to classify irreducible prime 3-manifolds. After cutting along spheres which don't bound balls as far as possible the next step is to consider incompressible tori, i.e. which don't bound solid tori and are disjoint from the boundary. It turns out that this is possible in a certain unique way:
Theorem 4.2 Jacob-Shalen, Johannson.
If is an irreducible compact orientable manifold, then there is a collection of disjoint incompressible tori
in
such that splitting
along the union of these tori produces manifolds
which are either Seifert-fibered or atoroidal — every incompressible torus in
is isotopic to a torus component of
. Furthermore, a minimal such collection of tori
is unique up to isotopy in
.
Thurston's geometrization conjectures states that all the pieces you get by this decomposition admit one of eight possible geometric structures:
There is a list of eight simply connected Riemannian manifolds - the so called model geometries. A geometric structure on is the choice of a Riemannian metric on
, with the property that its universal covering
equipped with the pull-back metric is isometric to one of the eight model geometries. It might a priori be easier to classify all cocompact actions on the several model geometries.
The Seifert-fibered pieces are well understood since the work of Seifert in the 30s. For the atoroidal pieces are described by the following Hyperbolization theorem which was stated by Thurston (ref) and proven by Perelman. \begin{thm} Every irreducible atoroidal closed 3-manifold that is not Seifert-fibred is hyperbolic. \end{thm}
4.3 Dehn surgery
Dehn surgery is a way of constructing (TODO oriented ? neccesary) 3-manifolds. Given a link
![\displaystyle L: \coprod_{i=1}^n S^1\rightarrow S^3,](/images/math/d/3/0/d30853002b621116b7a9d8b947460542.png)
and a choice of a tubular neighborhood of
![\displaystyle L': \coprod_{i=1}^n S^1\times D^2\rightarrow S^3](/images/math/3/5/9/359b421fbd348b9e879a5da25889aed4.png)
(This choice essentially is the choice of a trivialization of the normal bundle; TODO find a correct formulation for this).
This gives us a family of embedded, disjoint, full tori. The idea of Dehn-surgery is to remove these Tori and glue them back in using a twist.\\
So let us restrict to the case with only one solid torus .
Choose any self-homeomorphism
of the torus
. The result of the Dehn surgery is defined as
![\displaystyle M_f:=S^3\setminus L'(S^1\times \mathring{D}^2) \cup_f S^1\times D^2](/images/math/6/4/9/649d0ad7a02449c5c996b372dfa39b5c.png)
If is the coordinate flipping and if the embedding
is the unknot, the result of Dehn-surgery would be
. (TODO CHECK is the framing important ?).
![f](/images/math/6/b/6/6b6e98cde8b33087a33e4d3a497bd86b.png)
![M](/images/math/8/9/f/89f5942bc37eb2131578d77a79571c59.png)
Tex syntax erroris isotopic to exactly one self-homeomorphism
![f_A\in \Homeo(T^2)](/images/math/8/4/4/8447244bd28cbdf9e1e71ebc99ba091b.png)
![\displaystyle f_A:\Rr^2/\Zz^2 \rightarrow \Rr^2 /\Zz^2 \qquad \left(\begin{array}{c}x\\y\end{array}\right)\mapsto A\cdot\left(\begin{array}{c}x\\y\end{array}\right)](/images/math/8/6/c/86c89df597870e4fe1b0aefea1d02f63.png)
, where (reference of proof).
TODO composition corresponds to two successive Dehn- surgeries. We have to find out, for self-homeomorphisms of the torus Dehn surgery gives
back. Consider a matrix of the form
. The homeomorphism
extends to a homeomorphism of
:
![\displaystyle \bar{f_A}(x,y):=(x,x^ky),](/images/math/e/8/e/e8efe87755bf61c475091f19650b9e0e.png)
where .
Using this homeomophism one can define a homeomorphism from
to
:
![\displaystyle S^3 = S^3\setminus L(S^1\times \mathring{D}^2)\cup_1 S^1\times D^2 \rightarrow M_{f_A}=S^3\setminus L(S^1 \times \mathring{D}^2)\cup_{f_A} S^1\times D^2](/images/math/7/3/2/7325e67487c8a4d9e1a2c7a44057b1fe.png)
given by the identity on the left component and on the right component.
Together with (link to comment about composition), this tells us, that
really only depends on the coset
(TODO check right or left coset). This coset is uniquely determined by the image
of
with
and
coprime.
The ratio is called the surgery coefficient. (TODO what is the quotient good for ?)<++>
TODO does the result give different manifolds.
TODO does the result only depend on the isotopy class of the link.
Every compact (oriented /able, neccesary ?) 3-manifold might be obtained from by a Dehn surgery along a link (TODO ref). Of course this does not satisfy to classify 3-manifolds without having a good classification of links in
.
5 References
[Scott1983], [Thurston1997], [Hatcher2000], [Hempel1976]
- [Hatcher2000] A. Hatcher, Basic Topology of 3-Manifolds, Unpublished notes.
- [Hempel1976] J. Hempel,
-Manifolds, Princeton University Press, Princeton, N. J., 1976. MR0415619 (54 #3702) Zbl 1058.57001
- [Scott1983] P. Scott, The geometries of
-manifolds, Bull. London Math. Soc. 15 (1983), no.5, 401–487. MR705527 (84m:57009) Zbl 0662.57001
- [Thurston1997] W. P. Thurston, Three-dimensional geometry and topology. Vol. 1, Princeton University Press, Princeton, NJ, 1997. MR1435975 (97m:57016) Zbl 0873.57001