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 […]

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 […]

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 […]

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

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 $\gamma$ […]

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 […]

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 […]

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 […]

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

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 […]

Intereting Posts

Where is axiom of regularity actually used?
Minimizing $L_\infty$ norm using gradient descent?
Divisor on curve of genus $2$
an example of a continuous function whose Fourier series diverges at a dense set of points
How can I use math to fill out my NCAA tournament bracket?
An example of a bounded pseudo Cauchy sequence that diverges?
Irreducible cyclotomic polynomial
Showing that if $\lim_{x\to\infty}xf(x)=l,$ then $\lim_{x\to\infty}f(x)=0.$
Proof of the independence of the sample mean and sample variance
Derive an algorithm for computing the number of restricted passwords for the general case?
What is known about nice automorphisms of the Mandelbrot set?
Connection between rank and matrix product
Transforming from one spherical coordinate system to another
Eigenvalue of an Euler product type operator?
Is my proof valid? Integration of logarithmic function.