Basically what I’m looking for is a topological group that is also a non-orientable, n-dimensional manifold

Why each sphere admits an orientation reversing diffeomorphism onto itself? (For even dimensional ones can we take conjugation map?) And why complex projective spaces do not admit? Is there a geometric way to see this without using characteristic classes?

Let me ask if the following is possible: Let $L_1$ be some curve segment in the $\mathbb{R}^3$ space which has the length $1$. Let $L_2$ be some curve segment in the $\mathbb{R}^3$ space which you can not define the length. Then some $\mathbb{R}^3$ ambient isotopy takes $L_1$ to $L_2$. In other words, can an $\mathbb{R}^3$ […]

Please consider the following links with four components My question is if such two links are isomorphic. The corresponding Jones polynomials are respectively It is observed that the ratio of the Jones polynomials is $q^{9⁄2}$. It is to say the only difference between the two Jones polynomials is a simple monomial. According with such fact […]

Suppose $A$ and $B$ are disjoint subsets of the plane, both closed, nonempty, and connected. Define $E(A, B)$ as the set of points in the plane equidistant from $A$ and $B$. For example, if $A$ is a point and $B$ is a straight line, $E$ is a parabola. (1) I think that $E$ is always […]

We counteract this answer by adding the rigidity assumption: Is there still a counterexample? Let $(X_{n},d_{n})_{n \in \mathbb{N}}$ be a sequence of complete geodesic metric spaces such that: $X_{n}$ is a regular$^1$ CW-complex of constant local dimension$^3$ $n$, it is of finite type$^4$, boundaryless$^2$, unbounded, uniform$^5$, and it is the $n$-skeleton of $X_{n+1}$, which is […]

Identify the topological type obtained by gluing sides of the hexagon as shown in the picture below Clearly the boundary is encoded by the word $abcb^{-1}a^{-1}c$ I do not understand how the surface is glued together – could you help me, please? Edit: This answer is a Klein bottle (see answer below) Similar problem: see […]

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 […]

Can we embed $S^{p} \times S^q$ in $S^d$ with all the nice properties, what are the allowed values of $p$ and $q$ for $d=2,3,4$ where $p+q \leq d$? =For $d=2$= I suppose that we cannot embed $S^{1} \times S^1$ in $S^2$. =For $d=3$= Can we embed $S^{2} \times S^1$ in $S^3$? =For $d=4$= Can we […]

I know that there is no complete surface embedded in $\mathbb{R}^3$ of constant curvature -$k$ for any $k$. But you can clearly embed the hyperbolic plane (curvature -1) into hyperbolic 3-space (curvature -1). You can also embed Euclidean planes (curvature 0) as horoballs into hyperbolic 3-space (curvature -1) But can you embed surfaces of curvature […]

Intereting Posts

Why is $(1+\frac{3}{n})^{-1}=(1-\frac{3}{n}+\frac{9}{n^2}+o(\frac{1}{n^2}))$ and how to get around the Taylor expansion?
Are there statements that are undecidable but not provably undecidable
What is the probability on rolling $2n$ dice that the sum of the first $n$ equals the sum of the last $n$?
Proving the Möbius formula for cyclotomic polynomials
How does one approximate $\cos(58^\circ)$ to four decimal places accuracy using Taylor's theorem?
Isomorphism of Vector spaces over $\mathbb{Q}$
Least Upper Bound Property $\implies$ Complete
Fractional Calculus: Motivation and Foundations.
Prove $f(x) = 0 $for all $x \in [0, \infty)$ when $|f'(x)| \leq |f(x)|$
Moebius band not homeomorphic to Cylinder.
When does $e^{f(x)}$ have an antiderivative?
Technique for finding the nth term
Possible road-maps for proving $\lim_{x\to 0}\frac{\sin x}{x}=1$ in a non-circular way
Do polynomials in two variables always factor in linear terms?
if $A^2 \in M_{3}(\mathbb{R})$ is diagonalizable then so is $A$