Articles of low dimensional topology

Seeking a good book or site describing the 3-sphere

would you be able to recommend a good book/chapter or a web site on visualization / structural elements / projection of the 3-sphere. I am trying to locate a good information source on this subject, which will show the math derivations, visualizations, etc. So far I have been able to find a few limited sources. […]

the restriction of a homeomorphism on a subset

Let $X$ be a topological space and $f:X\to X$ be a homeomorphism. Then the induced map $f_1:\pi_1(X,x)\to\pi_1(X,fx)(\cong \pi_1(X,x))$ is an isomorphism (automorphism up to conjugate). In the following we will adapt this viewpoint and denote $f_1:\pi_1(X)\to\pi_1(X)$. I think $f_1$ is kind of special if $f$ can be extended to another ambient space. More precisely I […]

when is the region bounded by a Jordan curve “skinny”?

How can I formalize and prove the following intuition?: Picture a very skinny rectangle, one with base length 1 and sides length $\epsilon$. Or imagine a very flattened ellipse. The interiors of these objects are “skinny” (or we might say $\epsilon$-skinny) in the sense that each point in the interior is very close (or within […]

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 Rolfsen which works with PL knots. I’m really hoping for some grand unifying or almost-unifying theorem. Thank you […]

Prove by elementary methods: the plane cannot be covered by countably many copies of the letter “Y”

As a consideration from the post “Prove by "elementary methods": The plane cannot be covered by finitely-many copies of the letter "Y"”, on the basis of the remark made in previous post by the user Moishe Cohen, is it still possible to apply elementary methods to prove weaker results, namely: The plane cannot be covered […]

A(nother ignorant) question on phantom maps

My last question (Is such a map always null-homotopic?) is quite similar. If you do not care about my motivation for these questions, you can skip to the last line. I asked if some assumptions were sufficient conditions to guarantee the non-existence of “phantoms” (essential maps that are non-essential on every finite sub-complex). Now I […]

Simply Connected Domain of the plane and the Jordan Curve Theorem

While reading Markushevich’s complex analysis book, I realized that his definition of a simply connected domain differs from the one I have seen before. He takes the Jordan Curve Theorem for granted, and denotes the interior of a closed Jordan curve by $I(\gamma)$. Then he defines (pages 70~72 of vol.1) ; A domain $G$ is […]

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

This question already has an answer here: If M is a non-orientable closed connected 3 manifold prove H1(M) is an infinite group. 1 answer

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 $v \in S^2$ we say that the projection, $\pi _v$, from $\mathbb{R}^3$ to the plane orthogonal to $v$ is a regular projection of $\gamma$ […]

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 $H_1=0$? (This question was stimulated by the question Example of a domain in R^3, with trivial […]