Differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.

learn more… | top users | synonyms

4
votes
2answers
426 views

An alternative description of the first Stiefel-Whitney class

I've heard this description of the first Stiefel-Whitney class of a vector bundle but I don't know why this is true. Can anyone help me with this, please? The first Stiefel-Whitney class of a vector ...
4
votes
1answer
72 views

How to make a $C^1$ knot into a $C^\infty$ knot

Suppose I have a $C^1$ imbedding $f: S^1 \rightarrow S^3$. From the point of view of knot theory, what's the "best" way to get a $C^\infty$ curve that "looks like" or is "equivalent to" $f$? For ...
2
votes
1answer
224 views

“Completing” a vector field on a non-compact manifold $M$

Suppose I have a non-compact smooth manifold $M$ and an arbitrary nowhere-vanishing smooth vector field $X$ on $M$ which is not complete. Is there a way to create a smooth vector field $V$ that is ...
1
vote
1answer
276 views

Complete non-vanishing vector field

Let $M$ be a non-compact smooth manifold. Suppose we have a nowhere-vanishing smooth vector field X. Is this vector field complete? I know it is when $M$ is compact. However, I am unsure in the ...
3
votes
1answer
61 views

normal form of an n-form

It is known, that one can convert any function $f(x_1,\dots,x_n)$, defined near $0$, into the function $(y_1,\dots,y_n)\mapsto a+y_1$, by a suitable local change of coordinates, provided $df\neq 0$. ...
2
votes
1answer
62 views

Prove $X =\left \{(x, y) \in \mathbb{R}^3 \times \mathbb{R}^3 \ | \ |x| = 1, |y| = 1, x\cdot y = \frac{1}{2}\right\}$ is a manifold

I am having trouble with the following qualifying exam problem and I would appreciate any help. Thank you. Let $X$ be the set of pairs of unit vectors $(x, y)$ in $\mathbb{R}^3$ such that $x \cdot y ...
1
vote
1answer
77 views

Orientability of $P_{\bf R}T{\bf RP}^{2n}$

I know the following fact : (1) $ {\bf RP}^{2n}$ is non-orientable. (2) $ {\bf RP}^{2n-1}$ is orientable. (3) $P_{\bf R}T{\bf RP}^{2n}$ is orientable. (4) $P_{\bf R}T{\bf RP}^{2n+1}$ ...
0
votes
1answer
59 views

Question about a specific case of the argument principle for maps of circles.

Problem Statement: Let $f:S^1\rightarrow S^1$ be a smooth map of manifolds where $S^1=\frac{[0,1]}{0~1}$, and let $f'(t)\in \mathbb{R}$ be given by the $df_t[1]_t=f'(t)[1]_{f(t)}$ at each $t\in S^1$. ...
3
votes
2answers
141 views

Extending a smooth map

When can I extend a smooth map $f:\mathbb{R^2}-\lbrace 0 \rbrace \to S^1$ to a smooth map $\tilde{f}:\mathbb{R^2} \to S^1$. For instance, consider $g(x,y)=(x,y)/\sqrt{x^2+y^2}$? Am I able to extend ...
3
votes
1answer
267 views

Given a local diffeomorphism $f: N \to M$ with $M$ orientable, then $N$ is orientable.

Given a local diffeomorphism $f: N \to M$ with $M$ orientable. Why is $N$ orientable? My professor wrote this in class without giving a proof and said "you should try to prove this for fun :)". I ...
2
votes
0answers
101 views

“Product” bundle notation.

Let $\newcommand{\Spin}{\operatorname{Spin}}M$ and $M'$ be two manifolds, equipped with a principal $\Spin_n$ and $\Spin_{n'}$ bundle called $P$ and $P'$, respectively. Then there is an induced ...
9
votes
2answers
213 views

What role does differentiability play in Topology?

My question is stated in the title. As a brief background, I'd like to say I know next to nothing about Topology. The little bit I was exposed to came as an aside in my Multivariate Calculus class; we ...
0
votes
0answers
63 views

Compactness of covering space

If we have space $X$ with and $n$ sheeted covering space $Y$ is $Y$ compact iff $X$ is? Torus or sphere, make me believe the answer is yes.
1
vote
1answer
170 views

How to prove that the map is open?

I am trying to prove that $\phi$ is homeomorphism,where $U=\{[1,u,v]|u,v\in \mathbb{R}\}\subset{\mathbb{R}P^{2}}$,and $\phi$:$U\rightarrow\mathbb{R}^{2}$ given by ...
2
votes
1answer
67 views

Given that $X$ is closed and $Y$ is connected, prove that $Y$ is also closed.

I am having trouble with the following qualifying exam problem. Suppose $f: X \rightarrow Y$ is a smooth immersion between smooth manifolds of the same dimension. Given that $X$ is closed and $Y$ ...
-7
votes
1answer
1k views

Vector field on an odd sphere [closed]

Let $x^1,y^1,\ldots,x^n,y^n$ be the standard coordinates on $\mathbb{R}^{2n}$. The unit sphere $S^{2n-1}$ in $\mathbb{R}^{2n}$ is defined by the equation $\sum_{i=1}^n(x^i)^2+(y^i)^2=1$. Show that ...
0
votes
1answer
77 views

Finding the kernel of Pushforward of $f:\mathbb R^n\rightarrow \mathbb R^k$

Let $U$ be an open subset of $\mathbb R^n$, $f:U\rightarrow\mathbb R^k$ a smooth map such that its pushforward is onto, for each $x\in U$, i.e. $$f_{*x}:T_xU\rightarrow T_{f(x)}\mathbb R^k$$ is ...
6
votes
1answer
268 views

Uniqueness of Smoothed Corners

Let $M^m$ be a smooth manifold with boundary. We may form a new topological manifold by "adjoining a handle to $M$", I.e. by choosing a smooth map $h : \partial \mathbb{D}^{\mu} \times ...
3
votes
1answer
74 views

Let $M$ and $N$ be smooth manifolds and $f: M\rightarrow N$ a diffeomorphism. Prove that the map $df:TM \rightarrow TN$ is a homeomorphism.

I am going through qualifying exam questions and I am stuck on this problem. I don't think it should be too difficult, but I am having a lot of difficulty. I am not even sure how to start. Some ...
1
vote
1answer
189 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
257 views

A surjective map which is not a submersion

Is there an example of a smooth map between smooth manifolds which is surjective, but not a submersion? I feel there can't be one, but don't know of a proof. Nor do I know of a counter-example. ...
6
votes
2answers
198 views

Topological space M with partition of unity--->M paracompact. John Lee Problems

Suppose $M$ is a topological space with the property that for every open cover $X$ of $M$, there exists a partition of unity subordinate to $X$. Show that $M$ is paracompact.
1
vote
1answer
151 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 ...
1
vote
1answer
258 views

How to show that open interval is manifold but closed one is not

Prove that we can define manifold's structure for $1.$ An open interval $A=(0,1) $ $2.B=\{(x,y)\in R^2 | ||(x,y)||<1\}$ And that we can't define manifold's structure for $3.$ An closed interval ...
2
votes
2answers
188 views

Finding the degree of a map

I am having trouble computing the degree of a certain map using the fact that $f: N \rightarrow M$ where $M$ and $N$ are both $n$-dimensional manifolds induces a homomorphism between the nth ...
3
votes
1answer
114 views

Find an closed 1-form on $\mathbb{R}^2 \backslash (0,0)$ that is not exact.

I need help with the following problem. I am not sure how start and I would be very appreciative if someone could help me with this (I believe easy?) example. Find an closed 1-form on ...
5
votes
2answers
187 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 $$\partial\partial M = \emptyset?$$
6
votes
1answer
529 views

de Rham comologies of the $n$-torus

I'm attempting to calculate the de Rham cohomologies of the $n$-torus: $n \choose k$. I'd like to use a Mayer-Vietoris sequence relating $H^kT^{n}$ to $H^kT^{n-1}$ and $H^{k-1}T^{n-1}$ so I can use ...
7
votes
2answers
177 views

Example of a diffeomorphism of class $C^{k}$ which is not $C^{k+1}$

Can anyone give me an example of a map $f:\mathbb{R}\to\mathbb{R}$, which is a diffeomorphism of class $C^{k}$ but it is not a diffeomorphism of class $C^{k+1}$?
9
votes
0answers
202 views

Reality check: $\mathcal{O}_{\mathbb{P}_\mathbb{R}^1}(1)$ and $\mathcal{O}_{\mathbb{P}_\mathbb{R}^1}(-1)$ are both Möbius strips?

This question follows up this previous question, which has an accepted answer that I am having trouble believing. (Edit: the answer has now been fixed; thanks Georges!) I have been working on giving ...
0
votes
0answers
77 views

Existence of Solution: Embedding from 2D Euclidean space to a circle

Given a real matrix $X$ with $n$ rows and 2 columns, can the matrix be transformed to a real matrix $Y$ such that all the points formed by the rows of $Y$ lie on a circle (2d) and their inter-point ...
1
vote
0answers
135 views

An example of a differentiable manifold class $C^k$ but not class $C^{k +1} $

I'm looking for an example of a differentiable manifold of class $C^k$ but not class $C^{k +1}.$ I found an exercise in Hirsh's book, which suggests that the graph of $f (x) = |x|^{\lambda}$, where ...
4
votes
0answers
102 views

Extending metrics

Let $\pi:E\to M$ be a rank $k$ vector bundle over the compact manifold $M$ and let $i:M\hookrightarrow E$ denote the zero-section. Then we have a splitting of the restriction of $TE$ to the ...
1
vote
0answers
119 views

Conditions for a projection of a Knot to be a Knot diagram.

friends. I'm working on a problem, the broad scope of which is to show that given a map $f:S^1\rightarrow \mathbb{R}^3$ be a smooth embedding, and a projection map $\pi_v:S^2\rightarrow P_v$, where ...
5
votes
1answer
207 views

How to check whether a vector field is Morse-Smale?

Setup and notation: Let $f:M\to \mathbb{R}$ be a Morse-function on the compact $m$-dimensional manifold $M$ and let $X$ be a gradient-like vector field for the function $f$. Denote the unstable ...
2
votes
1answer
333 views

Tangent space to a product

Can you explain this question explicitly. This is a little bit difficult for me, but I want to learn how to solve. Thank you for help. If $M$ and $N$ are manifolds, let $\pi_1:M\times N\to M$ and ...
2
votes
2answers
156 views

How does degree theory imply that this mapping $f$ is locally onto?

Let $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ be a smooth vector field ($\mathcal{C}^1$ mapping). Let $0$ be a critical point of $f$, i.e. $H f(0) = 0$. Assume that the index of $f$ at $0$ is ...
1
vote
1answer
128 views

Why the tangent bundle of a smooth manifold is an oriented manifold?

I need help with the following question. I am not sure how to begin. Any help will be appreciated. Thank you! For any smooth manifold $M,$ the tangent bundle $TM$ is an oriented manifold.
2
votes
2answers
112 views

Discretizing continuous surfaces into semi-regular polygons

I am aware that there have been many works on the problem of discretizing a surface into polygons, however, I wonder if in any work the problem of doing so to get polygons with edges of the same ...
3
votes
1answer
105 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
1answer
74 views

Suspension of $\mathbb{R}P^2$ Contractible?

Is the suspension of $\mathbb{R}P^2$ Contractible? And if it is, How would you prove it. Thank you!
1
vote
1answer
126 views

Differential Topology Question on Complex Projective Space

This question seems like it would be very hard to do directly. I wouldn't know where to begin. I was wondering if anyone had a very slick proof of this. The only thing I think is easy is that its ...
2
votes
1answer
35 views

Matrix Manifolds Question

I am not sure at all how to do the following question. Any help is appreciated. Thank you. Consider $SL_n \mathbb{R}$ as a group and as a topological space with the topology induced from $R^{n^2}$. ...
2
votes
0answers
56 views

Question Concerning the classification of 1-manifolds

I am having trouble proving the following statement used in proving the classification of 1-manifolds. Any help would be great. Thank you. Let $L$ be a subset of $X$ diffeomorphic to an open ...
1
vote
2answers
261 views

A covering map from a differentiable manifold

Let $p: C \to X$ is a covering map. Suppose that $C$ is a differentiable manifold. Is X - differentiable manifold? More precisely, I am interested in the case where $C$ is Submanifold of Lie algebra, ...
1
vote
1answer
274 views

Lie Groups induce Lie Algebra homomorphisms

I am having a difficult time showing that if $\phi: G \rightarrow H$ is a Lie group homomorphism, then $d\phi: \mathfrak{g} \rightarrow \mathfrak{h}$ satisfies the property that for any $X, Y \in ...
2
votes
2answers
160 views

Showing that the exponential map $\mathrm{exp}:\mathfrak{sl}(2,\mathbb{R})\to\mathrm{SL}(2,\mathbb{R})$ is not surjective

I am having a difficult time showing that the exponential map $\mathrm{exp}: \mathfrak{sl}(2, \mathbb{R}) \rightarrow \mathrm{SL}(2, \mathbb{R})$ is not surjective. I have, however, worked out that ...
1
vote
1answer
286 views

Exponential map and the special orthogonal group

I need to show that the map exp$: \mathfrak{so}(2) \rightarrow SO(2)$ is surjective. I already have that $\mathfrak{so}(2) = \{A \in M(2, \mathbb{R}) \ | \ A^T + A = 0\}$ and the map is given by ...
1
vote
1answer
73 views

Vector Fields Question 4

I am struggling with the following question: Prove that any left invariant vector field on a Lie group is complete. Any help would be great!
2
votes
1answer
128 views

Another question on Orientation Preserving Maps

I am stuck on the following question. Sorry about the bad latex skills. Not sure what went wrong. Is the map $f:S^n \rightarrow S^n$ orientation preserving? I constructed an atlas on $S^n$ ...