Articles of low dimensional topology

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 […]

Exotic Manifolds from the inside

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 diagrams, etc). Being descriptions from the “outside” (forgive the less exact tone) here, I wonder if there are “inside” descriptions, i.e.: When you are sitting inside […]

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 interested in the case when $Y$ is a surface and $f, g$ are embeddings. Is this true or […]

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 $\partial D^3=S^2$?

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 structure, etc.). This raises a couple of questiona: 1.Is any compact or non-compact 4-manifold obtainable as a (finite or infinite) handle diagram ? 2.What are the […]