Questions tagged [low-dimensional-topology]

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

Low dimensional topology generally refers to the study of 3 or 4 dimensional topological manifolds (which, as it turns out, is highly related to the study of knot theory: a knot is an embedding of the circle into the 3-sphere, and the property of knots can be completely classified by the topology of the 3-manifold formed from removing the knot from the 3-sphere).

That topologists are interested in low dimensional topology has largely to do with the set of tools available to them. In dimensions 1 and 2, the study of topological manifolds is completely equivalent to the study of Riemannian manifolds, and topological surfaces have long been completely classified. In dimensions 5 and higher, topological manifolds become very pliable: on the one hand this allows for a lot of pretty bad behaviour, on the other one also gets some really powerful tools (h-cobordism theorem, for example). In 3 and 4 dimensions, the study of topological manifolds becomes "just right": the manifolds are floppy enough that (Riemannian/differential) geometry doesn't completely determine topology (existence of exotic 4-manifolds; any 3 (or higher) dimensional smooth manifold admits a negative Ricci curvature metric), but rigid enough that some tools from geometry can be used (Perelman's proof of the Poincare conjecture using Ricci flow, application of Yang-Mills theory to the topology of 4-manifolds).

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Why can't Antoine's necklace fall apart?

Antoine's necklace is an embedding of the Cantor set in $\mathbb{R}^3$ constructed by taking a torus, replacing it with a necklace of smaller interlinked tori lying inside it, replacing each smaller torus with a necklace of interlinked tori lying…
Anon
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Visualization of Lens Spaces

I am trying to visualize lens spaces geometrically. While I am aware of the fact that most manifolds which cannot be embedded in $\mathbb{R}^3$ are hard to visualize because of the obvious limitations, some of them are not too complicated; For…
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Intuition behind Khinchin's constant

Khinchin proved that For almost all reals $r$ with continued fraction representation $[a_o; a_1, a_2, \dots ]$ the sequence $K_n = \left(\prod_{i=1}^{n} a_i\right)^{1/n}$ converges to a constant $K$ (Khinchin's constant) independent of…
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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 this to obtain some quotient space which would be…
Br09
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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 "large" 3-manifold groups, indecomposable under…
user98602
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What is the importance of the Poincaré conjecture?

The Poincaré conjecture is listed as one of the Millennium Prize Problems and has received significant attention from the media a few years ago when Grigori Perelman presented a proof of this conjecture. But why is this interesting at all? What…
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How equivalent are the homeomorphism and diffeomorphism groups of 3-manifolds?

Every topological 3-manifold carries a smooth structure, unique up to diffeomorphism. In addition, "up to homotopy", the maps between them are the same: every continuous map is homotopic to a smooth map, and any homotopy between smooth maps is…
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explicit "exotic" charts

can someone provide explicit charts for non-standard differentiable structures on, for instance $\mathbb{R}^4$ (or some other manifold)?
yoyo
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Is the torus the union of two connected, simply-connected open sets?

Is the torus the union of two connected, simply-connected open sets? A routine computation with the Mayer-Vietoris sequence shows that if so, then their intersection must have exactly three components. Also, exactly one of the components must have…
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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 classifies closed surfaces. I'd like to get the…
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In what kind of space does this object live?

Let me quickly build up some background. One way to build a hypercube is to take cubes, and start gluing them together, face to face, such that each edge is shared by $3$ cubes. You complete the hypercube with $8$ cubes. This involves rotating…
Josh B.
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Embedding compact (boundaryless?) $n$-manifolds in $n$-dimensional real space

I know the embedding theorems that allow you to embed $n$-manifolds into $\mathbb{R}^k$, provided $k$ is chosen large enough. Here I'm interested in the possibility of taking $k=n$ in the case of compact manifolds. From the classification of compact…
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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 $\mathbb{R}^3$ where $\pi_1$ is non-trivial but finite, and…
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To what extent are homeomorphisms just deformations?

Background. It is often said that two spaces are homeomorphic if, roughly speaking, one space can be continuously deformed into the other without any tearing and gluing. It is then emphasized that this is more of a guiding principle than a solid…
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Why is the 3D case so rich?

The Banach--Tarski theorem applies only in the case of three or more dimensions. In 3D, there are five regular solids, two of them being not at all obvious, and the 4D case is also interesting; but the higher-dimensional cases each yield just three…
John Bentin
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