2
votes
0answers
29 views

Tangent bundles of smooth manifolds

Using the identity $T(M \times N) = T(M) \times T(N)$, it is easy to construct the tangent bundles for various smooth manifolds such as the n-dimensional sphere $S^{n}$. However, I could not figure ...
2
votes
0answers
38 views

Operations on a smooth vector bundle

On a smooth vector bundle, one often defines addition and scalar multiplication to form a vector space. However, doesn't one need to show that these operations are smooth? Is this trivial or is there ...
1
vote
1answer
27 views

Whitney sum of smooth vector bundles

I was reading through Lee's smooth manifolds book, in his chapter on vector bundles. Upon reading about smooth vector bundles and its definition, I was wondering if the whitney sum of two smooth ...
0
votes
0answers
51 views

Klein bottle visualization, parameterization, and isotropic version

Suppose a bright glowing orange mobius strip appeared in space for just an instant and then disappeared, except for its glowing orange edge, which remains suspended motionless in space for a moment, ...
2
votes
2answers
50 views

question about cover maps

Here's a problem I've had a hard time with If $f: M\rightarrow N$ is a cover map and $M$ is a m-manifold, will $N$ also be a m-manifold? A manifold is a space locally Euclidean space that is ...
2
votes
1answer
29 views

Proof that a map from an orientable surface to a non-orientable surface has even degree.

For a smooth map $f:M\to N$ from an orientable closed surface $M$ to a non-orientable closed surface $N$, we define its parity (also called modulo 2 degree, and denoted $\deg_2(f)$) as the parity of ...
2
votes
1answer
61 views

Equivalent topologies on Real projective space $RP^{n}$

This is homework,so no answers please. Prove that the topology on $RP^{n}$ given by the standard smooth structure (lines through the origin in $\mathbb{R}^{n+1}/\{0\}$ and $\tau_{1}$) is equal to ...
0
votes
1answer
43 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 ...
4
votes
1answer
39 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". ...
0
votes
1answer
45 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 ...
4
votes
0answers
102 views

Surgery presentations and gluing

It is well known that every closed oriented 3dimensional manifold can be obtained from a framed link in $S^3$. Let $M$ and $N$ be topological 3-manifolds with boundary such that boundaries $\partial ...
0
votes
0answers
31 views

Doubt about local flatness of low dimensional embeddings

I would like to know if it is possible to have a simple curve $\gamma $ on a surface $S$ such that $\gamma$ is compact and embedded (i.e. with respect to the topology induced from $S$ it is ...
2
votes
1answer
45 views

When gluing maps are isotopic?

Let $M$ and $M'$ be compact orientable connected topological 3-manifolds. (One may need more conditions to answer the question.) Suppose we have two homeomorphisms $f$ and $g$ from the boundary ...
2
votes
1answer
57 views

Boundary connected sum of manifolds

I have two related questions about the boundary connected sum of manifolds with boundaries. Let $T=S^1 \times S^1$ be a torus and let $X=T \times [0, 1]$ be the cylinder over the torus. Let $X'$ be a ...
1
vote
1answer
60 views

Are the topology of a manifold and the topology induced by the metric of a manifold the same?

I am a physics student trying to understand in a rigorous way what a manifold is so please bear with me. Ok, so I am just learning what a topology is and from what I have understood up till now is ...
4
votes
0answers
54 views

Gluing two solid tori along their boundary resulting in a topological manifold

The following question is from a past qualifying exam. Take two solid tori $D^2 \times S^1$, and construct the space $X$ by identifying their boundaries via the map $f \colon \partial D^2 \times S^1 ...
0
votes
0answers
18 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, ...
1
vote
1answer
61 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 ...
1
vote
2answers
54 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 ...
3
votes
1answer
55 views

Graph of a continuous function is a smooth manifold? [duplicate]

Let $f:(a,b)\to \mathbb{R}$ be a continuous function and define $\Gamma(f) = \{(x,f(x)):x\in (a,b)\}$. The two maps $\Psi: \Gamma(f)\to (a,b)$ given by $(x,f(x))\mapsto x$ and $\Phi: (a,b)\to ...
1
vote
2answers
173 views

Definition of “a topological manifold with corners”.

How can we define a topological manifold with corners and its corners? Then, do we use "invariance of domain" to define corners, as we really need this theorem in order to define "boundaries of a ...
3
votes
1answer
42 views

Is the collection of atlases on a set $X$ a set?

Well, the title says it all. I need to know if i can view the collection of all atlases on a given set $X$ as a ordinary set. Is this possible ? All the atlases are only topological atlases, no ...
3
votes
1answer
79 views

Are two spaces obtained from homeomorphic spaces by removing a ball still homeomorphic?

I have a specific example in mind. Consider $S_1,S_2$ two surfaces. Remove two discs to obtain surfaces with boundary $S_1',S_2'.$ If $S_1 \cong S_2,$ does it necessarily follow that $S_1' \cong ...
2
votes
0answers
39 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 ...
1
vote
3answers
58 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 ...
2
votes
0answers
42 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 ...
-2
votes
1answer
107 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 ...
0
votes
1answer
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 ...
0
votes
0answers
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?
0
votes
0answers
70 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 ...
2
votes
1answer
58 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) ...
6
votes
2answers
158 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 $\mathbb{R}^3$ ...
2
votes
1answer
67 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 $V$ ...
0
votes
0answers
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 ...
3
votes
1answer
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 ...
2
votes
2answers
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?
2
votes
2answers
77 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 ...
0
votes
0answers
55 views

Learning about Calabi–Yau manifold

I want to Create a 3d animation of Calabi–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 ...
4
votes
1answer
70 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 ...
3
votes
2answers
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 ...
2
votes
1answer
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 $\mathbb{R}$. ...
1
vote
0answers
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 ...
4
votes
2answers
141 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 ...
1
vote
1answer
39 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, ...
0
votes
1answer
35 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, ...
0
votes
1answer
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 $\Omega_2(t)$. ...
2
votes
1answer
92 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.
0
votes
2answers
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!!
1
vote
1answer
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, ...
7
votes
3answers
77 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. ...