Tagged Questions

8
votes
1answer
82 views

Topological manifolds (dimension)

I am taking an introductory course to topology and the professor defined a topological manifold of dimension $n$ if it is hausdorff and if for every point $x$ there exists an open set $U$ around $x$ ...
2
votes
2answers
51 views

Smooth maps on a manifold lie group

$$ \operatorname{GL}_n(\mathbb R) = \{ A \in M_{n\times n} | \det A \ne 0 \} \\ \begin{align} &n = 1, \operatorname{GL}_n(\mathbb R) = \mathbb R - \{0\} \\ &n = 2, \operatorname{GL}_n(\mathbb ...
3
votes
2answers
77 views

Attaching two manifolds along their boundary

I have a question about a proof in John Lee's Introduction to Topological Manifolds. Suppose $M$ and $N$ are two topological $n$-manifolds with nonempty boundary (for reference, the definition I am ...
-1
votes
2answers
99 views

The euclidean space $\Bbb R^n$ is orientable as a manifold.

I know that The euclidean space $\Bbb R^n$ is orientable as a manifold. I think that it is orientable because it has a nowhere vanishing $n$-form. But I am not sure. Please can you explain ...
2
votes
0answers
51 views

Real projective space is Hausdorff

I could not understand the proof of thıs proposition can you help me and give clear explanation.Just can you say how we have (n+1)x2 matrix?? This prove is correct or I need to add something ?? ...
1
vote
1answer
53 views

Real Projective Space

How I can prove this corollary 7.15.I consider that ı can apply the corollary 7.10.But I could not can you help me.
1
vote
1answer
31 views

Locally finite or not

I am tryıng to learn locally finite and can you give an explanation for my green writing please thank you
3
votes
1answer
73 views

What is overlop

I want to ask something that what is overlop.My teacher said that For Ex1, Everything is overlop hence it is not locally finite.For example 2,it doesnt overlop.Actually ı could not understand.please ...
0
votes
2answers
71 views

Topological manifold example

$\theta(x,x^2)=x$ $\Bbb X =${$(x,x^2)| x$ in $\Bbb R$} And V is subset of $\Bbb R$ $dim\Bbb X=1$ My instructor said that this is topological manifold. Why? Please can you explain me? This ...
1
vote
2answers
55 views

An open cover that is not locally finite

I could not understand that why is not locally finite for example 13.4 can you give me explanation please.
2
votes
0answers
74 views

Show that the projection map is Orientation preserving iff n is even

My question is that Orient the unit sphere $S^n$ in $\Bbb R^{n+1}$ as the boundary of the closed unit ball. Let $U$ be the upper hemisphere $U =${$x∈S^n |x^{n+1} >0$}. It is a coordinate chart on ...
1
vote
1answer
109 views

I did all explanation. Can you just teach me how to calculate this interior product?

Orient the unit sphere $S^n$ in $\Bbb R^{n+1}$ as the boundary of the closed unit ball. Show that an orientation form on $S^n$ is $w=\sum _{i=1}^{n+1}(-1)^{i-1}x^i dx^1∧...∧dx^i∧...∧dx^{n+1}$ I ...
2
votes
1answer
53 views

Manifolds with boundary and definition

Can you help for understanding this definitions in a good way.What is the my problem is that I can not image this definition and proposition in my mind and also ı dont understand the reason that ı ...
2
votes
1answer
50 views

Boundary orientation for a cylinder

Please help me.I am think that I can use stokes theorem but ı could not apply.This question is very benefical for me to learn the subject please help me :(
2
votes
0answers
36 views

Orientation-preserving diffeomorphism [duplicate]

Can you help for solving this please. Although I study this subject I could not solve this question please help me ı am willing to learn this question.
1
vote
1answer
34 views

Now I am asking that the topological and manifold boudary for real line I am grateful to explain me more clearly and instructively.

Let M be the subset $[0,1[$ $∪ $ {$2$} of the real line. Find its topological boundary $bd(M)$ and its manifold boundary $∂ M$. I know that while I find the topological boundary, I need to show ...
3
votes
3answers
133 views

Topological boundary vs geometric boundary

Let $M_1=B((0,0),1)=\{(x,y) \mid x^2+y^2<1\}$ $M_2=\{(x,y) \mid x^2+y^2\le1\}$ What are the interior of $M_1$ and $M_2$ ? And What are the boundary of $M_1$ and $M_2$ ? How to find them? ...
4
votes
1answer
72 views

The open Möbius Band is not orientable

Can you explain my green underlying please.I have confused and ı dont understand why by the continuity of the orientation at the points $(0,0)$ and $(1,0)$ are also $e_{1},e_{2}$
16
votes
0answers
137 views

If $S\times\mathbb{R}$ is homeomorphic to $T\times\mathbb{R}$, and $S$ and $T$ are compact, can we conclude that $S$ and $T$ are homeomorphic?

If $S \times \mathbb{R}$ is homeomorphic to $T \times \mathbb{R}$ and $S$ and $T$ are compact, connected manifolds (according to an earlier question if one of them is compact the other one needs to be ...
3
votes
1answer
52 views

Classifying Vector Bundles

Given a manifold $M$, is there a way of classifying up to isomorphism all possible vector bundles over $M$ of a given rank? Some other questions on this site deal with specific cases, which all seem ...
0
votes
2answers
52 views

Transition formula for 1-forms

I try to solve this question but ı could not.ı am working for my exam.please help me.
0
votes
1answer
33 views

prove that $supp(π^{∗} f) = (supp f)×N.$ Please can you check my answer? Also more explanation please.

My question is that Let $f \colon M \to R$ be a $C^{\infty}$ function on a manifold $M$. If $N$ is another manifold and $π \colon M \times N \to M$ is the projection onto the first factor, prove ...
1
vote
2answers
28 views

Manifold Boundary versus Topological Boundary.

Let $M$ a $n$-manifold whit boundary, i.e., for each $x\in M$, there exist $U_x\subseteq M$ open in the topology of $M$ such that $U_x$ is homeomorphic to $\mathbb{R}^n$ or homeomorphic to ...
0
votes
1answer
27 views

Lie bracket in local coordinates

Can you help for solving this.I have an manıfold exam and ı am working but ı have a problem about lie bracket. And ı am putting what ı did..
2
votes
1answer
59 views

Orthogonal group is a regular submanifold of $GL(n,\Bbb R)$

I want to show that $O(n)$ is a regular submanifold of $GL(n,\Bbb R)$. I think that I can use constant rank theorem but how? I am putting the picture that what I did. Please help me I want to learn. ...
0
votes
1answer
93 views

Qualifying Exam Question on Manifolds

I am practicing qualifying exam problems and I am having trouble with the following question. Any help is greatly appreciated. Let $P$ be a polygon with an even number of sides. Suppose that the ...
1
vote
1answer
42 views

Show that 2 sets are not homeomorphic

Prove that a closed interval $A=[0,1]$ and $B=\{(x,y)∈R^2 \mid ||(x,y)||≤1\}$ are not manifold I'm struck with this problem.Can anyone explain how and what property should i use to show that for any ...
8
votes
2answers
144 views

How to deal with Homeomorphisms?

I have one doubt that may be too general, I don't know, so sorry if this is not a good place to ask it. I've also seem many other people with the same problem that I have, so I think that if this ...
5
votes
2answers
157 views

Why is $\partial\partial M=\varnothing$?

Why is the border of the border of an oriented differentiable $n$-dimensional Manifold $M$ empty, that is $$\partial\partial M = \emptyset?$$
0
votes
0answers
10 views

fibered solid tori matched by a fiber preserving homeomorphism

how do I proof that two seifert fibered solid tori $V$ and $V'$ (not ordinary fibered) with the same fiber parameters matched together by a fiber preserving homeomorphism do not become a solid tori. ...
3
votes
1answer
59 views

Approach topological manifolds with smooth manifolds

Because I'm doing some problems that consider all the manifolds while the situation is really clear when considering only smooth manifolds. Thus my question is can we always appoint a topological ...
1
vote
0answers
38 views

Smoothing corners of a handle attachment

Say we attach a $\lambda$-handle, $\mathbb{D}^\lambda \times\mathbb{D}^{\mu}$, to a smooth manifold $M, \partial M$ by simply taking the quotient $M \cup_h \mathbb{D}^\lambda \times\mathbb{D}^{\mu}$ ...
1
vote
0answers
50 views

Differentiable Manifold Hausdorff, second countable

Why do we generally require that a differentiable manifold be Hausdorff and second countable? Is this universally accepted in the definition? My Professor only required the Hausdorff condition for ...
4
votes
2answers
60 views

Closed ball not a manifold.

My book on differential geometry claims that a closed ball in $\Bbb R^n$ can never be a differentiable manifold because of the boundary points. The book doesn't really give an explanation for why this ...
3
votes
2answers
57 views

Homeomorphism between $\mathbb{R}_+ \times \mathbb{R}_+$ and $\mathbb{R}_+ \times \mathbb{R}$

I am trying to find a (smooth, if possible) homeomorphism between $\mathbb{R}_+ \times \mathbb{R}_+$ and $\mathbb{R}_+ \times \mathbb{R}$. i have come with some ideas, but the resulting functions are ...
2
votes
1answer
71 views

prove that the sphere with a hair in $IR^{3}$ is not locally Euclidean at q. Hence it cannot be a topological manifold.

A fundamental theorem of topology, the theorem on invariance of dimension, states that if two nonempty open sets $U ⊂ R_{n}$ and $V ⊂ R_{m}$ are homeomorphic, then n = m. prove that the sphere with a ...
2
votes
1answer
50 views

Fundamental group of the following disc

What is the fundamental group of the following space in $\mathbf C^n$? This is the topological space given by $$\{(x_1,\ldots,x_n)\in \mathbf C^n-\{0\} : \vert x_1\vert < 1, \ldots, \vert x_n\vert ...
2
votes
0answers
63 views

Pullbacks as manifolds versus ones as topological spaces

Let $Y_1\overset{f}{\longrightarrow}X\overset{f_2}{\longleftarrow} Y_2$ be smooth maps with a common target. Suppose that we have a pullback $Z$ of the diagram in (Mfd). Questions: Suppose that we ...
2
votes
0answers
133 views

Topological proof that this set is a topological manifold

let $S \subseteq \mathbb R^3 \times \mathbb R^3$ be the set of pairs $(x,y)$ where x,y are orthogonal unit vectors in $\mathbb R^3$. i am trying to show that this is a topological manifold without ...
5
votes
1answer
108 views

$\mathbb{C}\mathbb{P}^1$ is homeomorphic to $S^2$

I just read on wikipedia that the the complex projective line is homeomorphic to the riemann sphere. How do I prove this? But, before that I have an extremely silly doubt that has been eating me. In ...
1
vote
2answers
113 views

Help understanding manifolds and topological spaces

I'm used to think about linear algebra with matrices and vectors, I don't have particular problems with geometry either, I'm having hard times understanding what is the meaning of a manifold and a ...
5
votes
0answers
57 views

Homological definition of orientation at a boundary point?

For a topological manifold $M^m$, an orientation at a point $x \in M$ can be defined as a choice of generator for $H_m (M, M-x)$. For a topological manifold with boundary this definition still makes ...
1
vote
1answer
42 views

Is the configuration space a manifold? a CW complex?

The ordered configuration space of $n$ points in a topological space $X$ is defined as $F(X,n)=\{(x_1,\ldots,x_n)\in X^{n} | x_i\neq x_j \mbox{ for } i\neq j\}$ and the unordered configuration space ...
2
votes
0answers
13 views

How to build the space BTOP

Can anybody explain how is the procedure for building the space BTOP, which classifies microbundles of topological manifold ? Is there any good (and easy to read) references on this subject ? Thanks ...
2
votes
1answer
68 views

How to finish this proof (or sketch)?

I'm trying to prove that a manifold $M$, that is connected, is pathwise connected. I know the standard proof of this theorem: just use that the set of points that can be joined to a point $x \in M$ is ...
0
votes
1answer
39 views

Relatively compact subsets of a manifold.

So I'm going through Otto Forster's "Lectures on Riemann Surfaces", and I need another hint (shame). This is in the "Cohomology Groups" sections, as part of a problem to show that for $X$ a compact ...
1
vote
2answers
70 views

What is the topological classification of connected 1-manifolds? [duplicate]

Possible Duplicate: The only 1-manifolds are $\mathbb R$ and $S^1$ Any manifold is homeomorphic to the disjoint sum of its connected components. Therefore, the full classification of ...
4
votes
3answers
266 views

Topology on Klein bottle?

I was trying to show that the Klein bottle was second countable. My try was to use that it has the subspace topology of $\mathbb R^3$. Then I noticed that it is not imbeddable into $\mathbb R^3$. ...
2
votes
1answer
85 views

Existence of bump functions which are positive on a prescribed set

Let $U \subset \mathbb{R}^n$ be an open subset of Euclidean space. I feel like there should be a smooth function $f: \mathbb{R}^n \rightarrow \mathbb{R}$ with $f|_U > 0$ and ...
2
votes
1answer
58 views

Why is this not a proof of Invariance of Domain?

We know that if $f:K \to X$ is continuous and injective, $K$ is compact, and $X$ is Hausdorff, then $f$ is a homeomorphism $K \cong f(K)$. So suppose $f:U \to \mathbb{R}^n$ is continuous and ...

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