0
votes
1answer
17 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 ...
-1
votes
0answers
41 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
32 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 ...
1
vote
1answer
34 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
109 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 $\mathbb{R}^3$ ...
1
vote
0answers
25 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
28 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
33 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
48 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
45 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
31 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
40 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
68 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
36 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
34 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 ...
3
votes
2answers
92 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
36 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
27 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
18 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
38 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
73 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
26 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
65 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. ...
1
vote
1answer
39 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 ...
0
votes
1answer
16 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.
0
votes
0answers
41 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 ...
0
votes
0answers
46 views

Is a manifold over $\mathbb{R}$ normal?

We have manifold $G$ over the reals with its finite atlas ($g_i:U_i \to g_i(U_i) \subseteq \mathbb{R}, G=\bigcup U_i$). The atlas induces a topology in the normal way ($A \subseteq G$ is open iff ...
0
votes
2answers
71 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 ...
2
votes
1answer
44 views

Topological property of some manifold

I am provided with a smooth map $g : N \rightarrow N^\prime $ between differentiable manifolds. $N$ is assumed to be compact and connected. Moreover, the differential $dg_x: T_x N \rightarrow ...
0
votes
1answer
37 views

Assumptions required for an implicitely defined surface/manifold to have a specified dimension

What are some normal assumptions made on implicitly defined manifolds? More specifically, by implicitly defined manifold, I mean the definition of a surface such as $g(x,y)=x^2+y^2-1=0$ for the ...
1
vote
1answer
48 views

Using transition maps as a comparison tool between charts on a manifold.

In the wikipedia article http://en.wikipedia.org/wiki/Chart_%28topology%29#Transition_maps we read A transition map provides a way of comparing two charts of an atlas. To make this comparison, we ...
3
votes
1answer
51 views

smooth lie group action

Let $\theta:G\times M\to M$ be a smooth left action of a Lie group $G$ on the manifold $M$. Suppose $G$ is compact and $M$ is Hausdorff. Let $K$ be a compact set in $M$. Is it true that $G_K:=\{g\in ...
3
votes
1answer
37 views

$S^1\times S^1$ diffeomoprhic to torus of revolution.

I am searching for an diffeomorphism between $$S^1\times S^1$$ and the torus of revolution $$\{(x,y,z)\in\mathbb{R}^3|z^2+(\sqrt{x^2+y^2}-a)^2=r^2\}.$$ I know it's true, I am just looking for an ...
0
votes
1answer
74 views

Intuition of a Submanifold

Could someone explain the intuition behind a submanifold. When, for example, is it appropriate to work with immersed submanifolds vs embedded submanifolds? Why is it important for a submanifold to be ...
2
votes
0answers
62 views

How can I visualize what open sets “look” like in unfamiliar topological spaces?

The question is extremely general, but I do have a specific case I'd like to look at, and I'm hoping that some combination of specific pointers and general advice will help me out. Consider the ...
3
votes
2answers
198 views

The boundary of an $n$-manifold is an $n-1$-manifold

The following problem is from the book "Introduction to topological manifolds". Suppose $M$ is an $n$-dimensional manifold with boundary. Show that the boundary of $M$ is an $(n-1)$-dimensional ...
0
votes
0answers
20 views

Prove that if $A$ and $B$ are two $k$-frames that there is an invertible $n\times n$ matrix $M$ such that $M\cdot A=B$.

Ok, so a $k$-frame (in case this isn't common terminology) is an $n\times k$ matrix with rank $k$. I realize that an invertible $n\times n$ matrix, when multiplied on the left by a $k$-frame, will ...
1
vote
1answer
60 views

example of a topological space such that there exists a sequence that escapes to infinity but has convergent subsequence

Find an example of a topological space such that there exists a sequence that escapes to infinity but has a convergent subsequence This actually is from exercise 2.15 of introduction to smooth ...
0
votes
1answer
39 views

Locally Finite Cover

Give a hint to the following problem: Let $M$ be a second-countable manifold, $N\subset M$ closed subset, $\Omega\supset N$ its open neighbourhood. Then $M$ has an locally finite cover $\{U_i\}_{i\in ...
5
votes
4answers
172 views

Algebraic varieties in $\mathbb{C}^n$ cannot have interior points

I know that the zero-set of a non-zero polynomial in $\mathbb{C}[x_1,...,x_n]$ can not have interior points, but I'm trying to find a proof that doesn't require a knowledge of complex analysis like ...
4
votes
1answer
91 views

Defining a quotient manifold with gluing

I'm trying to find conditions on the gluing map between two manifolds so that the quotient space will be a smooth manifold, and the inclusion map will be a diffeomorphism. Specifically, Suppose ...
2
votes
2answers
69 views

Equivalent definitions of manifolds

From Lee's Introduction to Smooth Manifolds, p.3: Question Concerning the exercise; what if there is a point $x$ in our manifold $M$ such that it has a neighborhood $N$ that is homeomorphic to ...
1
vote
0answers
36 views

Does there exists a simple imbedding theorem for general topological $n$-manifolds?

I am interested in finding some paper or book where i can find how to build an imbedding $e\colon M^n \hookrightarrow \mathbb{R}^q$ of an arbitrary Hausdorff topological $n$-dimensional manifold $M$ ...
2
votes
1answer
50 views

Intuition on the Loop Theorem

Probably the simplest statement of the Loop Theorem in 3-manifolds is as follows: Let $M$ be a 3-manifold and let $D$ be a 2-disk. If there is a map $$(D, \partial D) \rightarrow (M, \partial M)$$ ...
1
vote
1answer
118 views

Atlas on sphere $S^n$

Prove that there does not exist an atlas of the sphere $S^n\subset R^{n+1}$ with exactly one chart. Update Solution: Suppose there is an atlas with only one chart to cover $S^n$. By definition, ...
2
votes
0answers
72 views

Understanding the topology of Casson and Kinky Handles

I am trying to understand kinky handles (and later on: Casson Handles itself) by means of Kirby Calculus. From Akbulut, I have learned roughly: 1-Handles can be drawn as unknots (with dots on it, to ...
6
votes
2answers
395 views

Under what conditions the quotient space of a manifold is a manifold?

Well, there are many operations we can do with topological spaces that when we apply on topological manifolds gives us back topological manifolds. The disjoint union and the product are examples of ...
0
votes
0answers
46 views

Showing something is homeomorphic to $S^2$.

Suppose $X,Y$ are compact surface such that $X\#Y \approx X$ for any compact $X$. Show that $Y$ is topologically equivalent to the sphere. I was thinking for a while about this. It seems pretty ...
2
votes
0answers
57 views

A connected sum and wild cells

Can we find such a connected sum of two spheres (in any dimension) that is not homeomorphic to the sphere? $\def\R{\mathbb R}$ It seems that there should be examples like that, because there are lots ...
5
votes
1answer
320 views

Is a connected sum of manifolds uniquely defined?

It is a standard excercise in differential geometry to prove that a connected sum $M\#N$ of two smooth manifolds $M,N$ of the same dimension is uniquely defined (under some assumptions regarding ...