Articles of geometric topology

Is there an example of a non-orientable group manifold?

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

Orientation reversing diffeomorphism

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?

Two ambient isotopic curve segments, one has the length and the other does not

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 $\mathbb{R}^3$ […]

Are isomorphic the following two links?

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 […]

line equidistant from two sets in the plane

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 […]

Is such an infinite dimensional metric space, weakly contractible?

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

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 […]

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 […]

Embed $S^{p} \times S^q$ in $S^d$?

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 […]

Embedding manifolds of constant curvature in manifolds of other curvatures

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 […]