Low-dimensional topology generally refers to the study of 3 or 4 dimensional topological manifolds and knot theory.

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A simply-connected closed surface is a sphere

From the Classification Theorem for closed (i.e. compact and boundaryless) surfaces, it follows that $S^2$ is the only closed surface with trivial $\pi _1$. That's easy because the fundamental group ...
3
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2answers
99 views

Compact 3-manifolds with boundary an orientable surface

The usual embedding of the closed orientable surface $M_g$ of genus $g$ in $\mathbb{R}^3$ bounds a compact orientable $3$-manifold which is homotopically equivalent to a bouquet of $g$ circles. If a ...
5
votes
1answer
69 views

Is Whitehead's manifold with a point removed homotopy equivalent to a sphere?

A contractible open subset of $\mathbb{R}^n$ need not be homeomorphic to $\mathbb{R}^n$. The Whitehead manifold is an open subset of $\mathbb{R}^3$ which is contractible but not homeomorphic to ...
3
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1answer
86 views

Classification of Bieberbach groups

Does anybody know if there exists a list of the four dimensional Bieberbach groups presented by generators and relations on the web?. I know there exists the book Crystallographic Groups of ...
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2answers
41 views

Embeddings of surfaces into a 3-manifold

Say we are given a disconnected closed orientable surface $S=S_1\coprod S_2$ with $f=f_1\coprod f_2:S\rightarrow M$ such that the $f_i$ are embeddings and the images are incompressible. Suppose that ...
4
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1answer
313 views

4-Manifolds of which there exist no Kirby diagrams

In 4-Manifold theory one makes often the use of Kirby Diagrams to construct 4-manifolds (compact or non-compact) with specific gauge and topological properties (for example small betti numbers, spin ...
2
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0answers
37 views

Homology, addition of homology classes in construction of Poicare Sphere

I am working through Greenberg and Harper, Lecture notes on Algebraic Topology, and I am having trouble with one exercise. I have spoken with a professor and he encouraged me to ask here or look for ...
3
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2answers
65 views

Non-compact 3-manifold with incompressible boundary

Is there an orientable, irreducible, non-compact, 3-manifold $M$ with $\partial M\cong \Sigma_2$ , genus 2 orientable surface, with $\pi_1(M)\cong \pi_1(\Sigma_2)$ and $M$ not $\Sigma_2\times ...
6
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1answer
318 views

Isotopy and homeomorphism

Let $X$ and $Y$ be topological spaces. Suppose we have an isotopy between maps $f, g: X\to Y$. The question is that is there a homeomorphism $h: Y\to Y$ such that $h\circ f =g$? I am especially ...
2
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0answers
34 views

a diffeomorphism on a 3-manifold

Assume we have an embedded torus $T=S^1\times S^1$ in $3$-manifold $M$. We construct a diffeomorphism $f$ of $M$ as follows: Take a neighborhood $T\times I$ of the torus and set ...
3
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1answer
30 views

Understanding Hempel's proof of uniqueness of cube with handles

In Hempel's 3-Manifolds book, Theorem 2.2 says that if $P$ and $Q$ are two cubes, both with $n$ handles, and both are orientable, then they are homeomorphic. He defines a cube with handles as a ...
2
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1answer
40 views

Representing the $2$-homology classes of a $4$ manifold. Last passage of a Proof

I found a few occurrences of the same proof about representability of elements of $H_2(M,\mathbb{Z})$ for $M$ a closed orientable smooth $4$-manifold. All of them stop at the very end claiming that ...
5
votes
1answer
78 views

Existence of incompressible surface in a non-orientable manifold.

Let $M$ be a compact $P^2$ -irreducible 3-manifold. If $M$ is non-orientable, then there is a compact surface $F$ properly embedded in $M$ such that $F$ is two-sided, non-separating and ...
2
votes
1answer
33 views

figure-8 knot complement

The figure-8 knot seen as a 2-bridge knot with two maxima and two minima of the height function, has a complement in $S^3$ with one 0-handle,two 1-handles, two 2-handles and a 3-handle which cancels ...
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0answers
32 views

Perburb the Monodromy of Lefschetz fibration over a disk

Suppose that $\pi:E \to D$ is a 4-dimensional Lefschetz fibration over a disk,(more general, Lefschetz fibration over a surface with boundary ) and let $\Omega$ be a closed 2-form on $E$ such that it ...
10
votes
3answers
990 views

Blowing Up a Circle To a Hemisphere?

Can you blow up a circle drawn in the Euclidean plane to a hemisphere in Euclidean 3-space ? I am not concerned about preserving surface area or angles. I am interested in preserving an arrangement ...
5
votes
2answers
60 views

End of 3-manifold

Let $M$ be an irreducible, orientable, open 3-manifold with finitely generated fundamental group, this gives us a Scott's Core $C\hookrightarrow M$ so that the inclusion is an homotopy equivalence and ...
2
votes
1answer
48 views

Reeb orbit and open books

Weinstein conjecture is about existence of a closed orbit of the Reeb vector field on every contact manifold. On the other hand, we know every contact 3-manifold admits a compatible open book, which ...
4
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1answer
77 views

limit set of Kleinian groups with closed manifolds as quotient

I'm trying to convince myself that if $M\cong\mathbb{H}^3/G$ is a closed hyperbolic 3-manifold then the limit set $\Lambda(G)$ equals the whole Riemann sphere $S_\infty^2$. My idea of the proof goes ...
5
votes
1answer
113 views

Characterization of open sets in $R^3$ homeomorphic to $R^3$.

Background: By the Riemann mapping theorem, for any non-empty, simply connected open subset $U \subset \mathbb{C}$, $U \neq \mathbb{C}$ there exists a biholomorphic map (in particular a ...
5
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3answers
93 views

One-sided submanifolds in Hempel's 3-Manifolds

Early on in Hempel's book 3-Manifolds, he discusses two-sided submanifolds: if $N$ is a manifold of dimension $n$, and $M$ is a submanifold of dimension $(n-1)$, then $M$ is two-sided if there is an ...
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0answers
64 views

Exterior of a torus knot is Seifert-fibered.

I am trying to show that if $K\subset S^3$ is a $(p,q)$ torus knot, then the knot exterior $X_K=S^3\setminus N(K)$ is Seifert-fibered space, where $N(K)$ is a tubular neighborhood of $K$ in $S^3$. ...
2
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1answer
69 views

Restriction of ${\rm spin}^c$ structures

Suppose I have an oriented 4-manifold $X$ with boundary $\partial X$ an rational homology 3-sphere. If the restriction map $${\rm Spin}^c(X) \rightarrow {\rm Spin}^c(\partial X) $$ is surjective then ...
3
votes
1answer
45 views

planarity of trivalent graphs with a cyclic ordering on the edges in each vertex

Let $G$ be an (undirected) trivalent graph. For each vertex $v$ of $G$ we choose a cyclic ordering on the edges coming into $v$ (so if vertex $A$ has neighbors $B, C$ and $D$ we decide whether the ...
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0answers
30 views

Homeomorphic surfaces embedded in 4-space

A surface-knot is a closed connected surface embedded in the Euclidean 4-space $\mathbb{R}^4$. We consider the projection of the surface-knot into $\mathbb{R}^3$ with the singularity set contains of ...
2
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1answer
60 views

Regular neighbourhoods of non-orintable surfaces in $S^4$

Suppose that $F \subset S^4$ is a non-orientable surface. Let $N \subset S^4$ be a regular neighbourhood of $F$. Clearly, the boundary of $N$ is a circle bundle over $F$. Which is its Euler number? My ...
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0answers
39 views

Plane curves admitting several ways to seal it up by the disc

Let $\gamma:S^1\to\mathbb R^2$ be an immersed closed curve with only isolated transversal self-intersections. Say that $\gamma$ is sealed by the disc if there is an immersion $f:D^2\to\mathbb R^2$ ...
5
votes
1answer
54 views

Components of the space of immersions 2-manifold into $\mathbb R^3$

Let $M$ be a $2$-sphere with $g$ handles. Consider the space of maps $M\to \mathbb R^3$, which are immersions [i.e. smooth maps with nondegenerate differential in each point $x\in M$], with ...
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0answers
56 views

double decker set in a surface-knot

Surface-knot is an embedded surface in $\Bbb{R}^4$. Project the surface in $\Bbb{R}^3$ gives the surface diagram with set of singularity points consists of double points, triple points and branch ...
13
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3answers
317 views

What's true in $\mathbb{R}^4$, false in $\mathbb{R}^3$ and uninteresting in $\mathbb{R}^5$?

What are some interesting and easy-to-understand (for non-differential geometers) facts about subobjects of $\mathbb{R}^4$ that are not only false in $\mathbb{R}^3$, but also specific to the structure ...
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0answers
37 views

Reducible Heegaard splittings can be written as connected sums

Recall that a Heegaard splitting is a decomposition of a three manifold into a triple $(V,W,S)$ where V and W are solid handlebodies meeting along their common boundary S. A Heegaard splitting is ...
6
votes
1answer
122 views

Is $(\#^k \Bbb{RP}^2) \times I$ an $\mathbb{RP}^2$-irreducible 3-manifold?

Consider $S$ a surface homeomorphic to a connected sum of $n$ projective planes, $n \geq 2$. Can there be a two sided projective plane embedded in $[-\epsilon,\epsilon]\times S$?
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0answers
37 views

Platonic Hopf. Given the vertices of a tetrahedron circumscribed by unit sphere, find the stereographic projection of inverse Hopf fibers

I am trying to find the equations in $\mathbb{R}^3$ for the fibers of the four vertices of the tetrahedron circumscribed by the unit sphere. I want to find $s\circ h^{-1}$, where $s$ is the ...
5
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1answer
225 views

Cancelling 3-handles in Kirby diagrams

Recently been trying to understand the proofs of Gompf and Akbulut that certain 4-manifolds are $S^4$ (these 2 papers: Gompfs paper in Topology Vol. 30 Issue. 1, Akbulut). In which they use a clever 2 ...
2
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1answer
82 views

Kirby-like diagrams for $M^n$ when $n > 4$

Are there any attempts on constructing Kirby-like diagrams for representing manifolds $M^n$ with $n > 4$. What are the references on that ? I think you run out of dimension in which you can draw ...
2
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1answer
122 views

How can a Kirby diagram fail to determine a handle decomposition?

I've read that a handle decomposition for 4-manifold determines a unique smooth structure, and I've also read that every 4-manifold admits a Kirby diagram. So when does a Kirby diagram fail to ...
3
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1answer
49 views

Reference for low-dimensional topology

I have read topology and algebraic topology by Munkres and I want to start low-dimensional topology. What is a good reference for stating low-dimensional topology?
17
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2answers
282 views

What closed 3-manifolds have fundamental group $\Bbb Z$?

For certain small groups, it is easy (and desirable) to classify closed (and orientable if necessary) 3-manifolds with that group as their fundamental group. (Essentially due to Waldhausen is that for ...
5
votes
1answer
145 views

Showing every knot has a regular projection using differential topology

Can we use differential topology to prove that every smooth knot has a regular projection? Here is some background: Let $\gamma : S^1 \rightarrow \mathbb{R}^3$ be a smooth unit-speed imbedding. For ...
2
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1answer
58 views

The composition of a tricolorable knot with another knot is always tricolorable

Prove that the composition of a tricolorable knot and another knot (except the unknot, whether tricolorable or not) is tricolorable. I understand that the composition of two tricolorable knots ...
3
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1answer
62 views

Does the Dirac belt trick work in higher dimensions?

If the Dirac belt is in 4-space, is it still true that when the belt is initially given a 360 degree twist then it cannot be untwisted? I assume this is so because SO(n) is not simply connected, but ...
7
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1answer
137 views

If $M$ is a nonorientable $3$-manifold, why is $H_1(M, \mathbb{Z})$ infinite? [duplicate]

Let $M$ be a compact connected $3$-manifold with boundary $\partial M$. If $M$ is nonorientable and $\partial M$ is empty, then how do I see that $H_1(M, \mathbb{Z})$ is infinite?
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3answers
171 views

The shape of a room that's bigger on the inside

I have an advanced facility with many floors containing experimental technology. One of these floors goes on forever. What shape is it? More specifically: within the room, you can move as far as you ...
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votes
2answers
87 views

$3$-manifold has same homology groups as a $3$-sphere. [closed]

Let $M$ be a compact connected $3$-manifold with boundary $\partial M$. If $M$ is orientable, $\partial M$ is empty, and $H_1(M; \mathbb{Z}) = 0$, does it follow that $M$ has the same homology groups ...
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0answers
38 views

Example of a doubly degenerate Kleinian group which does not come from a mapping torus

Doubly degenerate Kleinian groups are discrete subgroups of $PSL(2,\mathbb{C})$ whose limit set is all of $S^2$, the boundary of $\mathbb{H}^3$. A standard example of such a group is given as ...
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0answers
161 views

Prerequisites for studying Perelman's proof of the Geometrization Conjecture

I want to set a course toward understanding Perelman's proof of the Geometrization Conjecture. I realize this will be a lengthy undertaking, but hopefully only on the order of one to two years. I am ...
6
votes
2answers
222 views

Homotopically trivial $2$-sphere on $3$-manifold

Let $S^2$ be an embedded sphere in a $3$-manifold $M^3$ such that $[S^2]$ is trivial in $\pi_2(M)$. Can we find an embedded disk $D^3$ in $M$ such that $\partial D^3=S^2$?
3
votes
3answers
281 views

Knot Group and the Unknot

Hi I am stuck in trying to show that given a knot $K$ such that the knot group $\pi_1(K)=\mathbb Z$ then $K\simeq U$. I tried to use the fact that the infinite cyclic cover is the universal cover but ...
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0answers
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The Developing Map

I am looking for resources to learn about the developing map, but the only source I know is Thurston's book. Could anyone direct me to other sources? Lecture notes, articles, books, etc? Thanks in ...
3
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1answer
99 views

Existence of a vector field with one singularity on a surface

In a paper I'm reading, it is stated that "It is known that on a compact, connected, oriented two dimensional manifold, there exists a vector field with only one singularity". Where can I find a ...