Glossary of 3-Manifold Topology
 Level: 0 1 2 3 4 5 6 7
MSC Classification: 57 (Manifolds and cell complexes)

This glossary is for terms related to 3-manifold topology.

 Rough Guides to Topology
 General Topology Algebraic Topology Manifold Theory Knot Theory

# Definitions

\sep{$\del$}

$\del$-compressible surface (in a manifold)
A properly

embedded surface $S$ which is compressible when considered with part
of the manifold $M$'s boundary. Thus, there is a disk [[$D$, meeting
$S\cup\del M$ in its own boundary, without a corresponding disk $D'�14042�\subset S\cup\del M$ that has the same boundary as $D$.}

\entry{$\del$]]-compression disk (of a surface in a manifold)}{A disk
which meets the surface and the manifold's boundary in its own
boundary.}

\entry{$\del$]]-incompressible surface (in a manifold)}{A surface $S$ which is incompressible when considered with part of the manifold
$M$'s boundary. Thus, for any disk [[$D$ meeting $S\cup\del M$ in its
own boundary, there is a disk $D' \subset S\cup\del M$ with the same
boundary as $D$.}

\entry{$\del$]]-parallel surface (in a manifold)}{A properly embedded
surface which is isotopic to a subsurface of the boundary of the
manifold. Also called a peripheral surface.}

\sep{0123456789}

\entry{2-Sided surface (in a manifold)}{A surface whose neighborhood
is an interval bundle over that surface. Also defined for arbitrary
dimensions.}

\sep{Aa}

\entry{Alexander's Theorem}{Every embedded sphere in $R^3$ (or $S^3$)
bounds an embedded ball.}

\entry{Ambient isotopy}{A homeomorphism isotopic to the identity. Two
sets $J$ and $K$ are ambient isotopic if $h(J)=K$ for some
ambient isotopy $h$, and can be considered “the same” for almost all
topological purposes.}

\entry{Atoroidal (manifold)}{An irreducible manifold with no
non-peripheral incompressible tori. Thus, every incompressible torus
is parallel to the boundary of the manifold.
\par
A manifold can be split along a finite collection of disjoint
incompressible tori such that all the resulting components are
atoroidal (the torus decomposition of a manifold).}

\sep{Bb}

\entry{Base space (of a fiber bundle)}{The “beginning” manifold in
the fiber bundle construction.
See fiber bundle.}

Boundary-\_\_\_\_\_.
See $\del$-\_\_\_\_\_.

\sep{Cc}

\entry{Category}{How the manifold we are working with is defined. The
standard categories are topological, smooth, and piecewise linear
(PL). For dimensions $3$ and lower, the category is arbitrary.}

\entry{Classification of Surfaces}{Every closed surface can be
disjoint disks, and glue back either a handle (a torus with a disk
removed) or a M\:obius band. Alternately, every closed surface is
homeomorphic to either the sphere, the connected sum of the sphere
with several tori, or the connected sum of the sphere with several
projective planes. Every closed surface has a unique Euler number.}

\entry{Complexity (of a surface)}{Defined for a compact surface $S$ as
the sum $]]-\mathbb{C}hi(S)+n_c(S)+n_{S^2}(S)$, where $\mathbb{C}hi(S)$ is the Euler
characteristic of the surface, $n_c(S)$ is the number of components of
the surface, and $n_{S^2}(S)$ is the number of spherical
components. Decreases under a compression of the surface.
See compression disk, compressible surface, and
incompressible surface.}

\entry{Composite \thrm}{Can be expressed as the connected sum of two
other \thrms, both of which are non-trivial.}

\entry{Compression disk (of a surface in a manifold)}{A disk whose
boundary is on a given surface but does not bound a disk on that
surface. If such a disk exists, the surface is compressible.
See complexity (of a surface), compressible surface,
incompressible surface.}

\entry{Compressible surface (in a manifold)}{A surface properly
embedded in a manifold which has a compression disk. A
compression disk is a disk $D$ whose boundary is on the given
surface but does not bound a disk on the surface.
\par
By cutting a surface on the neighborhood of a compression disk, and
gluing in two disks to “cap” the surface (the ends of a bicollar of
the compression disk), we can create a new surface with decreased
complexity, and by iterating this process obtain an incompressible
surface.
See complexity (of a surface), incompressible surface.}

\entry{Connected sum}{Given two manifolds $M_1$ and $M_2$, their
connected sum is obtained by removing one $3$-ball from each, and
gluing the two together along the resulting boundary component
spheres. This sum is well-defined for oriented manifolds by requiring
the boundary homeomorphism to be orientation-reversing.
\par
In general, the connected sum of $n$-manifolds is obtained by removing
$n$-balls from two $n$-manifolds, and gluing them together along the
resulting boundary components.}

\entry{Covering space}{A covering space of a manifold $M$ is a map
$p:\tilde{M} \to M$ such that every point $x \in M$ has a neighborhood
with inverse image a disjoint union of homeomorphic neighborhoods.
\par
With respect to \thrms, a couple theorems follow. A manifold with an
irreducible covering space is itself irreducible.}

\sep{Dd}

\entry{Dehn's Lemma}{If a disk $D$ is mapped into a \thrmm,
such that its boundary is mapped injectively into $M$, then the map
can be adjusted on the interior of the disk to make it an embedding.}

\entry{Dehn twist}{A map defined on an annulus (often a subset of a
surface) which fixes one side of the annulus but rotates the other by
$2\pi$. Any self-homeomorphism of a compact orientable surface is
isotopic to the composition of a finite number of Dehn twists.}

\entry{*Dehn surgery}{The process of removing a solid torus from a
\thrm, then replacing it via a (non-trivial) boundary
homeomorphism. Any closed orientable \thrm can be obtained by surgery
along some link in $S^3$.}

\sep{Ee}

Eilenberg-MacLane space
See $K(\pi,1)$ and $K(G,n)$.

\entry{Embedding}{A map between manifolds $N \to M$ such that $N$ is
homeomorphic to its image in $M$. An embedding is proper if its
image is a submanifold of $M$, so the image of its boundary is a
subset of the boundary of $M$.}

\entry{Essential SCC (arc)}{A non-contractible SCC, or an arc properly
embedded in a surface which never bounds a disk when closed up with an
arc on the boundary of the surface. Otherwise, it is inessential.}

\entry{Essential surface (in a manifold)}{A surface which is both
incompressible and $\del$]]-incompressible in the
manifold.
\par
For example, the complement of a satellite knot has an essential
torus, that used in its construction.}

\entry{Exceptional fiber (of a Seifert fiber space)}{A fiber of a
Seifert fiber space whose neighborhood is a non-trivial circle
bundle. Also called a singular or multiple fiber.}

\sep{Ff}

\entry{Fiber}{A fiber consists of all the points mapping to a given
point under the fiber bundle construction.
See fiber bundle.}

\entry{Fiber bundle}{Basically, a generalization of a covering
space. Given a manifold $M$ (the base space) and another $F$ (the fiber), a fiber bundle is a map $p:B\to M$ such that
each point in $M$ has a neighborhood $U$ with inverse image
$p^{-1}(U)$ expressible as the product $U'\cross F$.
\par
$B$ is known as the total space. A section of the fiber
bundle is a copy of $M$ lying in $B$ on which $p$ is the identity.
\par
Two fiber bundles over a manifold $M$ are equivalent if there is a
homeomorphism over their total spaces commuting with the bundle
maps. Given a fiber bundle $B\to M\cross I$, the associated
(inclusion) bundles $B\to M\cross\{0\}$ and $B\to M\cross\{1\}$ are equivalent.
\par
Given a map $M'\to M$, one can define a pull-back bundle
$p':B'\to M'$, which looks locally like the originally fiber bundle.
\par
The pull-back with respect to a constant map is a \emph{product
bundle}, and looks the same everywhere. It can be written as
$p':M'\times F\to M'$. Bundles over contractible spaces are always
product bundles.}

\sep{Gg}

\entry{*Genus (of a surface)}{A sphere has genus $0$, while a sphere
with $n$ handles has genus $n$.}

\entry{Genus (of a manifold)}{The minimum genus taken over all
Heegaard splittings (handlebody decompositions) of a \thrm
$M$.
\par
It exists for all closed, orientable $M$. Additive for connected sums
of \thrms. The only manifold with genus $0$ is $S^3$, and the
only orientable ones with genus $1$ are the lens spaces. The
nonorientable sphere bundle over $S^1$ is the only nonorientable
manifold with genus $1$. See Heegard splitting/surface and
Heegaard diagram.}

\entry{*Group homology}{The homology of a group $G$ is the homology of
$K(G,1)$. Thus, it is the homology of a manifold with first
fundamental group $G$ and all other fundamental groups vanishing.}

\sep{Hh}

Handlebody
The closure of one side of an orientable genus $g$ surface embedded in $S^3$.

\entry{Haken \thrm}{A compact orientable prime \thrm with a connected
orientable properly embedded incompressible surface (other than the
sphere). Such manifolds having boundaries are Haken. The $3$-sphere,
however, is not Haken, as the sphere is its only incompressible
surface.
\par
Haken manifolds are well understood— they are [[$K(\pi,1)$'s. Moreover,
if a Haken \thrm is homotopy equivalent to a prime \thrm, both
being closed and orientable, then they are homeomorphic.
}

Haken Number
See Jaco III.23, p. 49.

\entry{Heegaard diagram}{A handlebody of genus $n$ together with a
collection of $n$ SCC's on its boundary. It describes a \thrm,
which is obtained by gluing on another handlebody of genus $n$ such
that each of its handles has one of the curves on the boundary (a
meridian curve for the torus, for example). The boundary surface of
the handlebodies is then a Heegaard Splitting for the manifold.
\par
Using Van Kampen's theorem, a presentation for the fundamental group
$\pi_1(M)$ of the manifold can be constructed, as the curves relate
the $pi_1$ generators of one of the handlebodies to the $\pi_1$ generators of the other. See Heegaard Splitting/surface and
Genus (of a manifold).}

\entry{Heegaard splitting/surface}{A closed orientable
surface in a \thrm which separates the manifold into two
handlebodies. The genus of the splitting is the genus of the
surface. See Heegaard diagram and \emph{Genus (of a
manifold).}}

\entry{Heirarchy (for a surface)}{A way to cut up a surface into
disks. Specifically, a sequence of pairs $(S_0,\alpha_0),�14183�(S_1,\alpha_1), \ldots, (S_n,\alpha_n)$ where $S_0=S$, $\alpha_i$ is
essential in $S_i$, $S_{i+1}$ is obtained from $S_i$ by splitting
along $\alpha_i$, and each component of $S_{n+1}$ is a disk.
\par
The number of components of $S_{n+1}$ will be strictly less than the
number of boundary components of the original surface.}

\entry{Heirarchy (for a \thrmm)}{A sequence $M_0,M_1,\ldots$ where
$M_0=M$ and $M_{i+1}$ is obtained from $M_i$ by cutting along a
properly embedded incompressible surface. If $M$ is a \emph{Haken
manifold}, this process always terminates at some union of balls
$M_n$.}

Homology (of a group)
See Group homology.

\entry{Horizontal surface (in a Seifert fiber space)}{A surface in the
given manifold which is transverse to all fibers.}

\sep{Ii}

\entry{Incompressible surface (in a manifold)}{A surface properly
embedded in a manifold is incompressible if it has no compression
disks. A compression disk is a disk $D$ whose boundary is on
the given surface but does not bound a disk on the surface.
\par
Properly embedded spheres and disks in a manifold are always
incompressible. A properly embedded surface $S$ is incompressible if
and only if it is $\pi_1$-injective, so that $ker(\pi_1(S) \to�14201�\pi_1(M)$ is trivial.
\par
Some examples: (1) Seifert surfaces are incompressible in the knot
complement. (2) Boundaries of nontrivial knot neighborhoods are
incompressible in the knot complement. (3) A torus separating a
manifold into two nontrivial knot complements is incompressible.
See complexity (of a surface), compression disk,
compressible surface.
}

\entry{Inessential SCC/arc (in a manifold)}{A contractible SCC, or an
arc properly embedded in a surface which, taken with an arc on the
boundary of the surface, bounds a disk. Otherwise, it is essential.}

\entry{Irreducible manifold}{Contains no incompressible spheres, so
that all spheres in the manifold bound balls. Such manifolds are
prime.}

\entry{Isotopy}{A homeomorphism $H:M\cross I \to M\cross I$ such that
$M|_{M\cross\{i\}}$ is a homeomorphism onto $M\cross\{i\}$. The
homeomorphisms $H|_{M\cross\{0\}}$ and $H|_{M\cross\{1\}}$ are
isotopic.}

\sep{JjKk}

\entry{$K(\pi,1)$}{A manifold whose only nontrivial fundamental group
is the first: $\pi_1 \neq 0$. A special case of a $K(G,n)$, the
Eilenberg-MacLane spaces.}

\entry{$K(G,n)$}{A manifold with fundamental group $\pi_n=G$ and
$\pi_i=0$ for $i\neq n$. They are known as \emph{Eilenberg-MacLane
spaces}, and are sometimes denoted $K(\pi,n)$.}

\sep{Ll}

\entry{*Lens Space}{The space obtained by gluing two solid tori
together by a homeomorphism along their boundaries.}

\entry{Loop Theorem}{A homotopy element nontrivial in $\del M$ but
trivial in $M$ can be represented by a simple closed curve in $\del�14223�M$. This statement can be made slightly more general.}

\sep{Mm}

\entry{Manifold ($n$]]-dimensional, topological)}{A Hausdorff
(separable) topological space with a countable basis of open sets for
which each point has a neighborhood homeomorphic to $\mathbb{R}^n$ (interior points) or $\mathbb{R}^n_+$ (boundary points). A
compact, connected $n$-manifold is orientable if and only if
$H_n(M,\del M)=\mathbb{Z}$, and an oriented manifold is a manifold
together with a choice of orientation (generator of $H_n(M,\del�14229�M)$).}

\entry{Mapping cylinder}{The \thrm obtained from a given surface $S$ by gluing the ends of $S\times I$ together homeomorphically. Isotopic
homeomorphisms give homeomorphic manifolds.}

\entry{Model Seifert fibering (of a solid torus)}{A decomposition of
the solid torus into disjoint circles (fibers) which is constructed by
attaching $\{0\}\cross D^2$ and $\{1\}\cross D^2$ by a $2\pi p/q$ rotation. Thus, a Seifert fiber space can be decomposed into disjoint
circles each of which has a neighborhood diffeomorphic to a model
Seifert fibering.}

\entry{Multiple fiber (of a Seifert fiber space)}{A fiber of a Seifert
fiber space whose neighborhood is a non-trivial circle bundle. Also
called a singular or exceptional fiber.}

\entry{Multiplicity (of a Seifert fiber)}{The number of times a small
disk transverse to the fiber meets each nearby fiber. Regular fibers
have multiplicty $1$ while singular fibers with a $2\pi p/q$ rotation
have multiplicity $q$.}

\sep{Nn}

\entry{Non-separating surface (in a manifold)}{A surface properly
embedded in a \thrm whose complement (in that manifold) is still
connected. A surface is non-separating if and only if there is a loop
properly embedded in the manifold intersecting the surface
transversely one (or an odd number of) times. For example, a Seifert
surface in the knot complement is non-separating.}

\sep{Oo}

\entry{Orientation preserving (reversing) loop (in a manifold).}{Start
with a neighborhood of a point on the loop, and translate it around
the whole loop, giving a homeomorphism from that neighborhood to
itself. The loop is orientation preserving (reversing) if the
homeomorphism is orientation preserving (reversing).
\par
For nonorientable manifolds, the set of orientation preserving loops
is a subgroup of $\pi_1$ of index two, and there is a corresponding
orientable double cover of the manifold.}

\entry{Oriented manifold}{An orientable manifold together with a
choice of orientation. How orientation is defined depends on the
category. In the $PL$ category, for example, an orientation is
specified by orienting the simplices so that the orientations match up
on their intersections. For compact manifolds, the orientation is a
generator of $H_n(M,\del M)=\mathbb{Z}$.
\par
An $n$-manifold is not orientable if and only if it contains a
non-orientable $I^{n-1}$-bundle over $S^1$ (a M\:obius band in the
case of a surface, for example).
\par
A surface $S$ in a \thrm is orientable if and only if its neighborhood
is homeomorphic to $S\cross I$.}

\sep{PpQq}

\entry{Peripheral torus (in a manifold)}{A $\del$-parallel torus,
thus isotopic to one of the manifold's boundary components.}

\entry{Piecewise linear category}{In this category, manifolds are
defined as simplicial complexes in which every point has a
neighborhood homeomorphic to the $n$-ball. Often the category of
choice for \thrm theory.}

PL
Piecewise linear.

\entry{Poincar\'e Conjecture}{Every closed, connected,
simply-connected \thrm is homeomorphic to $S^3$.}

\entry{Prime Decomposition Theorem (for manifolds)}{Any compact,
connected, orientable \thrm can be written uniquely (up to insertion
and deletion of $3$-spheres) as a decomposition $M_1 \# M_2 \# \cdots�14272�\# M_n$ of prime manifolds $M_i$.}

Prime manifold
Can be written as a connected sum $M_1 \# M_2$ only if either $M_1$ or $M_2$ is $S^3$.

\entry{Proper embedding}{An embedding $N \to M$ which takes the
boundary of $N$ into the boundary of $M$.}

\entry{Pull-back bundle}{A fiber bundle $p:B'\to M'$ constructed
using a fiber bundle $p:B\to M$ and a map $M'\to M$.
See fiber bundle.}

\sep{Rr}

\sep{Ss}

SCC
A simple closed curve.

\entry{Sch\:onflies Theorem}{An embedded circle in $\mathbb{R}^2$ ($S^2$]])
bounds an embedded disk.}

\entry{Section (of a fiber bundle)}{A copy of the base space embedded
in the total space of a fiber bundle.
See fiber bundle.}

\entry{*Seifert fibering/fiber space/manifold}{A \thrmm which can be
expressed as the union of circles, in which each circle has a solid
torus neighborhood obtained by gluing the top and bottom of a cylinder
$D\times I$ with a $2\pi p/q$ rotation. $q$ is sometimes called the
multiplicity of the fiber. Circles with $|q|=1$ are
regular fibers, and the rest are singular,
multiple, or exceptional fibers. Essentially, an SFS is
obtained from a circle bundle over a compact surface by gluing in
solid tori (any gluing map will do, except the one taking a meridian
disk of the solid torus to the regular fibers). The singular fibers
are at the centers of the tori.
\par
Seifert Fiber Spaces are a very well-understood class of \thrms.
Any compact, orientable, irreducible \thrm whose fundamental
group contains an infinite cyclic normal subgroup is an SFS. Any such
\thrm whose fundamental group has a $\mathbb{Z}\cross\mathbb{Z}$ subgroup either has
an embedded incompressible torus or is an SFS.}

\entry{*Seifert surface (of a knot)}{A surface in $S^3$ whose boundary
is a knot (this exists for every knot).}

\entry{Seifert surface (of an SFS)}{The base space of the circle
bundle obtained from an SFS by deleting neighborhoods of the singular
fibers.}

\entry{Separating (surface)}{A surface properly embedded in a \thrm
whose complement (in that manifold) is not connected. See
non-separating surface.}

\entry{SFS}{A Seifert Fiber Space.}

\entry{Singular fiber (of a Seifert fiber space)}{A fiber of a Seifert
fiber space whose neighborhood is a non-trivial circle bundle. Also
called a multiple or exceptional fiber.}

\entry{Sphere Theorem}{If $\pi_2(M) \neq 0$, then there is a
noncontractible sphere embedded in $M$.}

\entry{Splitting}{The process of removing a (tubular) neighborhood of
a surface from a manifold. See Heegaard splitting.}

\entry{Submanifold}{A subset of an $n$-manifold $M$ is a $k$-manifold
$N$ such that any point $x$ of $N$ has a neighborhood $U$ for which
$(U,U\cap N)$ is homeomorphic to $(D^n,D^k)$. Basically, at every
point it looks like the embedding of standard disks. A submanifold is,
by this definition, a proper embedding (its boundary is in the
boundary of $M$). A submanifold has a neighborhood with a fiber bundle
construction, with fibers $I^n$ and the submanifold a section.}

\entry{Surface}{A $2$-manifold.}

\entry{Surface Heirarchy}{See heirarchy (for a surface).}

\entry{Surgery}{Surgery is the process of removing a piece of a
manifold and replacing it in some way. Examples include the connected
sum, adding handles to a manifold, and Dehn surgery.}

\sep{Tt}

\entry{Topological rigidity}{The notion that, if two manifolds satisfy
certain conditions, they are homeomorphic.}

\entry{Torus decomposition (of a \thrm)}{Any compact connected
irreducible manifold can be split into a number of atoroidal
components by a finite collection of disjoint incompressible tori.}

\entry{Total space (of a fiber bundle)}{The resulting space in a fiber
bundle construction (like a covering space but continuous rather than
discrete).
See fiber bundle.}

\entry{Trivial manifold}{Homeomorphic to $S^3$.}

\entry{Two-sided surface (in a manifold)}{A surface whose neighborhood
is an interval bundle over that surface. Also defined for arbitrary
dimensions.}

\sep{Uu}

\sep{VvWw}

\entry{Vertical surface (in a Seifert fiber space)}{A surface in the
given manifold which is a union of regular fibers.}

\sep{XxYyZz}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%% SECTION: THEOREMS %%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newpage

# Theorem Summaries

{\bf Prime Decomposition Theorem:} Every manifold decomposes uniquely
(plus or minus a few spheres) into prime manifolds.

\newpage

# Important Theorems

In this section, we give some important theorems used in
three-manifold geometry and topology. Where possible, the statements
are given in language rather than notation. In some cases we give two
versions of the theorem, one “intuitive” and the other “semantic,”
the second usually including extra information. An indication of the
proof is also given or, if not, a reference.

\sep{Embedding Theorems}

\describe{The following theorems give ways to translate
algebraic information into geometric information. Basically, if the
fundamental group is nontrivial (in dimensions one or two), they tell
how to find a representative of the group which is an
embedding, namely one-to-one.}

\thmtit{Dehn's Lemma}
If a disk $D$ is mapped into a
\thrmm, such that its boundary is mapped injectively into
$M$, then the map can be adjusted on the interior of the disk to make
it an embedding.
\begin{proof}Corollary of the Loop Theorem.\end{proof}

\thmtit{Loop Theorem}
\parIntuitive: A homotopy element nontrivial in $\del M$ but
trivial in $M$ can be represented by a simple closed curve in $\del�14340�M$.
\parSemantic: If $N$ is a normal subgroup of $\pi_1(S)$ for a
surface $S \subset \del M$ and $\ker(\pi_1(S)\to\pi_1(M)) \not�14343�\subset N$, then there is a proper embedding of the disk in $M$ whose
boundary is not an element of $N$.

\thmtit{Sphere Theorem}
\parIntuitive: If $\pi_2(M) \neq 0$, then there is a
noncontractible sphere embedded in $M$.
\parSemantic: If $N$ is a $\pi_1(M)$ invariant subgroup of
$\pi_2(M)$ with $\pi_2(M) \not \subset N$, then there is an embedding
of the sphere in $M$ which is not an element of $N$.

\sep{Heegaard Splittings}

\thmtit{Heegaard splittings}
Every closed, connected \thrm has a Heegaard Splitting.
\begin{proof}Look at the dual $1$-skeleton of a triangulation of the
manifold.\end{proof}

\sep{Decomposition of Manifolds and the Connected Sum}

\thmtit{Prime decomposition of manifolds}
Every closed, orientable, nontrivial \thrm can be written as
$M_1 \# M_2 \# \cdots \# M_n$ where each $M_i$ is prime. The
decomposition is unique up to order and homeomorphism.
\begin{proof}Hard. See Hempel ch.3.\end{proof}

\sep{The Fundamental Group of a Three-Manifold}

\thmtit{Fundamental group and manifold decomposition}
If $\pi_1(M) \cong G_1 * G_2$ (a free product of groups), where $M$ is
a compact \thrm with every component of $\del M$ incompressible, then $M$ can be written as the connected sum $M=M_1 \#�14359�M_2$, where $\pi_1(M_i) \cong G_i$.
\begin{proof}Based on a topological proof of Grushko's theorem (an
algebraic theorem on the “splitting” of free groups). See Hempel
ch.7.\end{proof}

\thmtit{Manifolds with free groups}
The only prime, compact \thrms with nontrivial free fundamental
groups are sphere bundles over $S^1$ and handlebodies.

\thmtit{Subgroups of fundamental group}
For every finitely presented subgroup $H$ of $\pi_1(M)$, for a \thrmm,
there exists a compact \thrm $N$ embedded in $M$ with fundamental
group including as $H$ into $\pi_1(M)$.
\begin{proof}Similar to that for the loop theorem, using
a tower construction of submanifolds.\end{proof}

\sep{Existence Theorems for Incompressible Surfaces}

\describe{We study surface embeddings for the same reason we
study the fundamental group. The embedding of one manifold in another
(of higher dimension) is an “inductive” way of studying manifolds,
and extremely useful.
Incompressible surfaces are those which give us the most direct
information about a \thrm, because their fundamental group
includes naturally into the manifold's fundamental group. Compressible
surfaces have more complicated groups which provide no extra
information.}

\thmtit{Incompressible surfaces and homology}
A compact orientable \thrm with infinite first homology group $H_1$ contains an orientable nonseparating properly embedded surface (hence
is either $S^2\cross S^1$ or contains an incompressible surface (and
so is Haken).

\thmtit{Incompressible surfaces in compact manifolds}
A compact, orientable \thrmm, whose boundary contains a
surface of positive genus, contains a properly embedded orientable
incompressible surface whose boundary is nontrivial in the homology
group $H_1(\del M)$.

\thmtit{Incompressible spheres and Heegaard splittings}
If a closed, orientable \thrmm contains an incompressible sphere, then it
contains some incompressible sphere intersecting a given Heegaard
splitting of the manifold in a simple closed curve.

\sep{Seifert Fiber Space Theorems}

\thmtit{$\mathbb{Z}$]] normal subgroup}
Any compact orientable irreducible \thrmm whose fundamental group
contains an infinite cyclic normal subgroup $\mathbb{Z}$ is an SFS.

\thmtit{$\mathbb{Z}\cross\mathbb{Z}$]] subgroup}
Any closed orientable irreducible \thrmm whose fundamental group
contains a $\mathbb{Z}\cross\mathbb{Z}$ subgroup either has an embedded incompressible
torus or is an SFS.

\thmtit{Homeomorphism classes}
If a compact \thrmm has two SFS structures not related by a
fiber-preserving homeomorphism, then either (a) $M$ is covered by
$S^3$ or $S^2\cross\mathbb{R}$, (b) $M$ is covered by $S^1\cross S^1 \cross�14396�S^1$, (c) $M$ is a solid torus $S^1 \cross D^2$, or (d) $M$ is an
interval-bundle over the torus $T^2$ or Klein bottle $K^2$.

\thmtit{Incompressible surfaces}
A closed incompressible surface in an SFS is isotopic to either a
vertical surface (the union of regular fibers) or a horizontal surface
(transverse to the fibers).

\thmtit{SFS covering}
Any compact orientable irreducible \thrm which has infinite
fundamental group and is covered by an SFS is itself an SFS.
(Meeks/Scott)

\thmtit{Non-irreducible SFS's}
The only non-irreducible Seifert fiber spaces (compact, connected) are
$S^1\cross S^2$, $S^1 \tilde{\cross} S^2$, and $\mathbb{R} P^3 \# \mathbb{R} P^3$.

\thmtit{Essential surfaces in SFS's}
Every $2$-sided horizontal surface in a compact irreducible SFS is essential. Every $2$-sided vertical surface is essential, with two exceptions: (i) a regular neighborhood of a singular fiber (a torus), and (ii) an annulus cutting off from the manifold a solid torus with the product fibering.

\sep{Non-separating surfaces}

\thmtit{Non-separating spheres}
The only prime orientable \thrm with a non-separating sphere is $S^2\cross S^1$.

\thmtit{Non-separating surfaces}
An orientable prime \thrm containing a properly embedded orientable non-separating surface is either Haken or $S^2\cross S^1$. \begin{proof}Compress the surface until an incompressible surface is obtained.\end{proof}

\sep{Topological Rigidity}

\thmtit{Topological rigidity of \thrms}
Two homotopy equivalent closed orientable prime \thrms with infinite
first homology groups are homeomorphic.

\thmtit{Haken \thrms}
Two homotopy equivalent manifolds, one Haken and one prime, are homeomorphic.

\sep{Classification Theorems}

\thmtit{Lens spaces}
The lens spaces $L(p,q)$ are classified by (i) $L(1,q)=S^3$ and (ii) $L(p,q)=L(p,q')$ if either $q \equiv \pm q' \mod p$ or $qq' \equiv \pm 1 \mod p.$ \begin{proof}Ask when a homeomorphism on the boundary of a torus can be extended to a homeomorphism on the whole torus.\end{proof}

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# References

\begin{enumerate}

1. {\bf J. Hempel,} {\it 3-Manifolds}, 1976. Chapters 1-7.
2. {\bf W. Thurston,} {\it The Geometry and Topology of

Three-Manifolds}, 1978-1999. Not yet used.

1. {\bf W. Jaco,} {\it Lectures on Three-Manifold Topology},

1980. Chapter 2. Parts of chapters 1,3.

1. {\bf M. Lackenby,} {\it Three-Dimensional Manifolds},

1999. Sections 1-6, other parts.

1. {\bf M. Lackenby and others,} {\it Lectures on Seifert Fiber

Spaces}, personal notes, 2001. Parts.
#{\bf A. Hatcher,} {\it Notes on Basic $3$-Manifold Topology},
2000? Chapter 1.
\end{enumerate}

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page revision: 7, last edited: 19 Sep 2009 20:35