# Tagged Questions

28 views

### Gluing oriented manifold along boundaries

Let $M_1$ and $M_2$ be oriented manifolds with boundaries. Suppose they have homeomorphic boundaries. I want to glue $M_1$ and $M_2$ along the boundaries via some homeomorphism. To ensure that the ...
33 views

### Converse to the Jordan-Brouwer separation theorem

By the Jordan curve theorem, if $C \subset S^2$ is (the image of) a simple closed curve, then $S^2 \setminus C$ has precisely two connected components. This statement admits the following "converse". ...
40 views

### Compactness of the Grassmannian $G(k,n)$

Related to this question, suppose we define $G(k,n)$ to be the set of $n\times k$ matricies with rank $k$, equipped with the quotient topology of $\mathbb{R}^{nk}$ by the equivalence relaiton A\sim ...
95 views

17 views

### Definition of Non-characteristic Manifold

What is the definition of non-characteristic manifold in the following context. "Since $f (x, y)=0$ is assumed to be a non-characteristic singularity manifold, we have $f_{x}\neq 0$." Thanks, ...
60 views

### Deleting a subset from a topological manifold

Consider an $n$-topological manifold $M$. We remove a subset $A$ from $M$. Are there cases where $M-A$ is no longer a topological manifold. In case we suppose that $M-A$ is still a manifold, what ...
47 views

### Some questions about the proof of the General Linear Group being a manifold.

I understand the idea behind proving that GL(n,$\mathbb{R}$) is a smooth manifold by first using the fact that it is isomorphic to $\mathbb{R}^{n^{2}}$ and using the continuity of the determinant ...
41 views

35 views

### Topologies of flag manifolds

I'm currently reading an article discussing flag manifolds and the action of $\mathrm{PSL}(n,\mathbb{C})$ on them. A flag (in my view at least) is a nested sequence $(y^1,\ldots,y^{n-1})$ of subspaces ...
57 views

### What is a “control point”?

I'm trying to figure out a good definition of control point for use in wikipedia (see https://en.wikipedia.org/wiki/Control_point_(mathematics) ) There seems to be a bias towards ascribing a ...
39 views

### How do we understand and visualize the meridians of a 3-sphere?

The equator of a n-sphere is (n-1)-sphere. Then the equator of a 3-sphere is a 2-sphere. Does this mean the meridian of a 3-sphere is a hemisphere (half 2-sphere)? In contrast, Wikipedia describes the ...
96 views

### Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere. [closed]

Trying to give a proof of the following statement: Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere. The statement seems rather intuitive for the case of three ...
30 views

### Is there a smooth map from the square to the deltoid?

Is there a $C^\infty$ map between a unit square in $\mathbb R^2$ and a deltoid like this one The deltoid is obtained by varying the angles $\theta_1$, $\theta_2$ in the equations \begin{align} x_2 ...
49 views

### Is $(\mathbb{R},+)$ a smooth manifold?

I feel like $(\mathbb{R},+)$ is, but I'm not really sure. How would I know whether or not it is?
64 views

### Why is the vertex called non-manifold vertex?

I am working on triangle meshes in one 3D reconstruction project for a while. I know what one manifold vertex looks like and how to detect them. But I hope to understand the definition of non-manifold ...
57 views

### Does the connected sum depend on direction of gluing?

The connected sum of two surfaces (2-manifolds) is defined by removing a disk from each and gluing the cut edges: (Image adapted from Wikipedia) Does the resultant surface (up to homeomorphism) ...
145 views

### Closed, orientable surface whose genus is very hard to find intuitively

I'm introducing the Classification Theorem on closed and orientable surfaces in a talk on (intuitive) topology, and to motivate it I'd like an example of an embedding of a surface in $\mathbb{R}^3$ ...
43 views

### Proof of Euler Characteristic for Sphere

Theorem 1. All cell decompositions of a sphere $S$ have Euler characteristic 2. This is well-known, but I had this idea for an intuitive proof: for any cell decomposition $\Gamma$ with $V$ ...
32 views

### Antipodal map and parallel transport on $S^3$

I'm reading Thurston and Levy's "Three-dimensional geometry and topology" where they have an informal explanation of what life in $S^3$ would look like. For the following detail that I am concerned ...
38 views

### An open cover $\{U_\alpha\}$ of $X$ is locally finite iff each $U\alpha$ intersects $U_\beta$ for only finitely many $\beta$

I am trying to prove: Lee, Smooth Manifolds, Exercise 2.9. Show that an open cover $\{U_\alpha\}$ of $X$ is locally finite if and only if each $U\alpha$ intersects $U_\beta$ for only finitely many ...
54 views

### Is the boundary of an open subset of $\mathbb{R}^n$ always a topological manifold?

Let $U \subseteq \mathbb{R}^n$ be open in the usual topology. Is its boundary, $\partial U$, necessarily a topological manifold?
61 views

### Help in showing that $f:M\rightarrow \mathbb{R}$ has atleast two critical points

Here is the question: Let M be a compact smooth manifold, $f:M\rightarrow\mathbb{R}$ be a smooth non-constant function. Show that $f$ has at least two critical points. I am trying to show by ...
52 views

### Learning about Calabiâ€“Yau manifold

I want to Create a 3d animation of Calabiâ€“Yau manifold. I tried to learn it but it deals with some very advanced math. What math knowledge have to know before trying to understand the calabi-yau ...
63 views

### Collar neighbourhoods for topological manifolds.

The well-known collar neighbourhood theorem states: Let $M$ be a smooth manifold with compact boundary $\partial M$, then there exists a neighbourhood of $\partial M$, which is diffeomorphic to ...
75 views

### complement of a finite subset of a path-connected space is path-connected

Given a smooth connected manifold $X$ of dim $\geq 2$, I need to show that $X\setminus Y$ is connected, for some $Y\subseteq X$ finite. The claim is intuitively obvious to me, but is not finding the ...
39 views

### Why would $[0,1) \times \eta$ (with lexicographic order topology) not be a manifold for $\eta > \omega_1$?

From Wikipedia's entry on the long line: And if we tried to glue together more than $\omega_1$ copies of $[0,1)$, the resulting space would no longer be locally homeomorphic to $\mathbb{R}$. ...
44 views

### Removing the boundary from a manifold with boundary

Let $M$ be a manifold with boundary $\partial M$. Can we say that $M'=M-\partial M$ is a closed manifold. I think it is correct, for example take $D$ to be the disk this a manifold with boundary the ...
132 views

### Any N dimensional manifold as a boundary of some N+1 dimensional manifold?

Is this statement true: Question: Can any N dimensional manifold be realized as a boundary of some N+1 dimensional manifold? If so/not, how to prove/disprove it? I read a TQFT paper from Edward ...
38 views

### Extension of funcion

I think its right but Im not sure. I have topological space (exactly manifold - second countable, Hausdorff, local Euclidean topological space) M, dim M=m. Let $A \subset M$ is closed set, dim A=n, ...
31 views

### Relative Compactness $\Rightarrow$ Compactness

I try to figure out: $(\overline{A}^U\text{ compact in }U )\Rightarrow( \overline{A}^X\text{ compact in }X)$ ...while $U\in\mathcal{T}$ It's clear for the case: $\overline{A}^X\subseteq U$ But else, ...
19 views

### About the boundary of a set of the form $Q_i = \bigcup_{t \in (0,T)}\Omega_i(t) \times \{t\}$

Let $\Omega$ be a bounded (open) domain. For every $t \in [0,T]$, let $\Omega_1(t), \Omega_2(t)$ be open subsets of $\Omega$, with $S(t)$ the interface separating $\Omega_1(t)$ and $\Omega_2(t)$. ...
87 views

### Orientability of Grassmannians

How can I understand when $Gr(n,k)$ is orientable and when not? I found that answer is yes if and only if $n \vdots 2$, but I do not know how to prove it.
80 views

### A subset $A$ of a manifold $X$ that is a manifold but not a submanifold of $X$

Let $X$ be a manifold and $A$ a subset of $X$. Is it possible for $A$ to be a manifold without being a submanifold of $X$. Thank you for your help!!
28 views

### Involutive Properties of Space-structures on Smooth Manifolds

I am currently reading Quantum Invariants of Knots and 3-Manifolds by Turaev, and I am having a hard time understanding a statement made on page 120. He is explaining the property of space-structures, ...
75 views

### Embeddings (how to prove them exactly)

For which of the following sets is the statement: '$A$ can be embedded in $B$' true? I can try to decide this intuitively but don't know if I'm right, and surely don't know how to formally prove it. ...
51 views

### Manifold is $2$nd countable iff it has a countable atlas

I am trying to prove that if a smooth manifold has a countable atlas than it has a countable basis. If I have $(U_n, \varphi_n)$ a countable atlas, how can I find a basis of the topology? If I take ...
19 views

### Atlases and Transition Maps

What is the difference between open sets and open balls? Definitions of atlases for manifolds do not seem to specify any difference.
53 views

### Topological Manifolds & Covers

This problem is from John Lee's "Introduction to Smooth Manifolds" 1-4. Let M be a topological manifold, and let U be an open cover of M . (a) Assuming that each set in U intersects only finitely ...
114 views

### Is a closed compact 2-Manifold that is embedded in euclidean 3-space always orientable?

I am sorry if this is a trivial question but I am a little confused right now so please bear with me. Since non-orientable compact 2-manifolds without boundary cannot be embedded in three-dimensional ...