# Tagged Questions

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### Complement of a solid genus-2-handlebody in $S^3$

I'm not sure if this is a stupid question or not but is the complement of a solid genus-2-handlebody in $S^3$ also a solid genus-2-handlebody? Thanks!
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### Gluing the ends of a cylinder. Can we get other than a torus?

Let $X=S^1 \times I$ be a cylinder, where $S^1$ is the 1-dimensional circle. If we glue the "bottom" boundary $S^1 \times 0$ and the "top" boundary $S^1\times 1$ by a homeomorphism sending $x\times 0$ ...
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### Is a Möbius Strip in > 4 dimensions impossible?

I seem to remember reading, on a plaque in the math building at Penn State, that Möbius Strips are only possible in 3 and 4 dimensions. In higher dimensional spaces, a Möbius strip will use the extra ...
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### For a Cantor set $\mathcal{C} \subset S^3$ such that $\pi_1(S^3 \setminus \mathcal{C})=0$, prove $S^3 \setminus \mathcal{C}$ can be split by a sphere.

I'm working from the paper Cantor Sets in $S^3$ with Simply Connected Complements by Richard Skora. On page 184 the second sentence states that any Cantor set $\mathcal{C} \subset S^3$ such that ...
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### Questions about the Nature of Chirality (with some focus on dimensionality)

Are all chiralities the same? (Not in the sense of "is the right hand the same as the left hand?" but in the sense of "is the way in which X is chiral the same (or negative of the same) as the way in ...
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### Surgery results in a cylinder

While reading a proof of a theorem about Reshetikhin Turaev topological quantum field theory, I encountered the following problem. Suppose we have several unlinked unknots $K_i$, $i=1, \dots, g$ in ...
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### How to distinguish between knots and links based on knot diagrams/projections

I'm interested in the distinction between knots and links in $\mathbb{R}^3$/$S^3$. In particular, is there an algorithmic way (as in not by sight/intuition) that we can examine the arcs and crossings ...
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### Why surgery produce a new 3-manifold?

I was studying a proof of the fact that any closed orientable 3-manifold is obtained by integer surgery along a link. I read the several proofs but I don't understand well. A proof is as follows. ...
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### Is there a domain in $\mathbb{R}^3$ with finite non-trivial $\pi_1$ but $H_1=0$?

The exterior of the Alexander Horned Sphere has $H_1=0$ but $\pi_1\neq 0$, in fact, $\pi_1$ is infinite. (See Hatcher p.171-172). Is there an example of a domain (connected open set) in $\mathbb{R}^3$ ...
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### Orbit space of S3/S1 is S2

I'm having trouble finishing this homework assignment. I did the first part by showing that the orbits are invariant: every element from the same $(S^1(z_1, z_2) \in S^3/S^1)$ is mapped to the same ...
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### Relation between the braid group and the mapping class group of the plane

According to the following link, page 248, the braid group modulo its center is isomorphic to the mapping class group of the $N$-times punctured plane, i.e. $B_N/Z(B_N)\cong M_N(\mathcal(R)^2)$. Could ...
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### How do you specify a link to a blind combinatorialist?

Regular projections of links look like graphs in the plane. So I'm wondering if it would be possible to specify a link up to isotopy with purely combinatorial data about this graph. If so, what kind ...
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### Annulus Theorem

I'm trying to read Rolfsen's "Knots and Links" and I'm a little discouraged that I can't do one of the first and seemingly more important exercises. The question is Use the Schoenflies theorem ...
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### How is PL knot theory related to smooth knot theory?

I really want to like knot theory but the PL condition seems sort of ugly. I was hoping someone could give me a justification for secretly thinking about smooth knots as I read through a book like ...
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### 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 ...
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### Finding the rank of subgroups of free groups?

How do you find the rank of a subgroup(of finite index) of a free group? I was thinking of looking at the fundamental group of a graph.
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### Representation of $S^{3}$ as the union of two solid tori

Well, I'm trying to prove that you can express the 3-dimensional sphere $S^{3}$ as the union of two solid tori. I tried first use that a solid tori is homeomorphic to $S^{1}$$\times$$D^{2}$ and use ...
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### Prime decomposition of 3-manifolds

Let $H_g$ be a three dimensional handlebody bounded by a genus $g$ surface. Let $M_g$ be a manifold obtained by gluing two copies of $H_g$ via an orientation reversing homeomorphism of the surface of ...
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### For an $n$-dimensional object, how many types of holes are possible?

Update 2012-06-06: At some point I'll attempt to answer my own question by using a dual-fluid model that places the dimensionality and connectivity of "solids" and "holes" on an equal footing. With ...
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### What knot groups are Abelian?

The knot group (the fundamental group of the complement of a knot) of the unknot is $\mathbb{Z}$ and the Hopf link is $\mathbb{Z}^2$, so those are knots (links) with Abelian knot group but are there ...
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### Useful topology on space of smooth structures on $\mathbb R^4$?

Mathoverflow is intimidating, so I thought I'd ask here first (second). If I don't get any useful answers here in a few days, I'll ask there. $Q_0$: Is there any use for a topology on the (continuum ...
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### In/out equivalent to left/right “chirality”

Apologies if this is off-topic, but we're having a problem over on English Language with this question, and I thought you guys might be able to help. Basically it's a matter of topology. We know the ...
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### 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 ...
As we know, an exotic $\mathbb{R}^4$ is a manifold which is homeomorphic, but not diffeomorphic to the standard $(\mathbb{R}^4,id)$, and there are even very explicit descriptions of them (Kirby ...