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.
0
votes
2answers
27 views
Diffeomorphism preserves dimension
I read from Milnor's book $\textit{Topology from the Differentiable Viewpoint}$ this assertion
"If $f$ is a diffeomorphism between opensets $U\subset R^k$ and $V\subset R^l$, then k must equal l, and ...
3
votes
1answer
41 views
Equivalence of Definitions of Principal $G$-bundle
I've finally gotten around to learning about principal $G$-bundles.
In the literature, I've encountered (more than) four different definitions. Since I'm still a beginner, it's unclear to me whether ...
1
vote
1answer
60 views
Prove that the dimension of the tangent space $T_x(X)$ of a k-dimensional manifold is k
In section 2, page 9 of Guillemin and Pollack's book $\textit{Differential Topology}$, he gave a proof that the dimension of the tangent space $T_x(X)$ is equal to the dimension of the manifold $X$. ...
0
votes
1answer
60 views
How many Borel conjectures are there
The following may be referred to as Borel conjecture:
Every strong measure zero set of reals is countable.
On the other hand Wikipedia refers to the following as the Borel conjecture:
Let $M$ and ...
1
vote
0answers
50 views
Kernel of the differential and weak topology
Suppose you have a differentiable map $\Phi : E \rightarrow F$ where $E$ and $F$ are Banach spaces, and a curve $t \mapsto u(t)$ of elements of $E$, with $u(0)=0$ and $\Phi(u(t))$ constant, such that ...
0
votes
0answers
17 views
Density of projection vector fields transverse to an embedded manifold
Say we have some manifold $X$ embedded in $\mathbb{R}^n$. We can then consider a family of vector fields $F_v$ defined by $F_v(p) = \pi_{TpX}(v)$, where $\pi_{TpX}$ denotes the orthogonal projection ...
0
votes
0answers
32 views
On the density of vector fields with only nondegenerate zeroes
Suppose we have a manifold $X$ embedded in $\mathbb{R}^n$. Define the vector field $v_u(p) : X \rightarrow TX$ by taking the point $u \in \mathbb{R}^n$ to its natural (orthogonal) projection onto ...
2
votes
1answer
53 views
question from hatcher basic 3 manifolds
The question is: why should a homologically trivial embedded sphere in a simply connected (not necessarily compact) 3 manifold M bound a compact 3 manifold embedded in M?
I had this problem reading ...
4
votes
0answers
63 views
Soft question: why are there non-smooth manifolds?
Topologists are often very good at explaining the geometric intuition behind certain results and programs of research. For instance, the particular interest in 4 manifolds is often explained by ...
1
vote
1answer
78 views
Showing a diffeomorphism extends to the neighborhood of a submanifold
Does anyone have a proof of problem 14, on page 56 of Guillemin and Pollack? I meant to do it as an exercise (I'm teaching myself the subject) but I'm struggling with the last step. Suggestions?
...
1
vote
0answers
17 views
are closed orbits of Lie group action embedded?
Consider a smooth action $G\curvearrowright M$ of a Lie group on a manifold.
Suppose that an orbit $G\cdot p$ is closed. Is the orbit an embedded submanifold.
In general we know that the orbits are ...
2
votes
1answer
33 views
Uniqueness of “Punctured” Tubular Neighborhoods (?)
Here is a question that has been haunting me for a while: Let $\mathbb{R}^{n-1} \times [0, \infty)$ be the upper half space of $\mathbb{R}^n$ and suppose we have a smooth homeomorphism (not a diffeo) ...
-1
votes
1answer
46 views
Show that $f^{-1}(a)$ is a submanifold of $\mathbb{R}$
Let $f:\mathbb{R}\to\mathbb{R}$ be a real analytic function with infinitely many zeros. Let $a\neq 0$ be a real number. Show that $f^{-1}(a)$ is a submanifold of $\mathbb{R}$.
3
votes
1answer
40 views
Smoothness in Banach space
I need a reference about a definition. Let $n$ be an integer and $G$ be a group of $H^n$(Sobolev) automorphisms of a vector bundle $E$ on some manifold $M$ and $C$ be the space of connections of class ...
0
votes
2answers
43 views
Application of the transversality theorem
I am trying to do this question in Bredon's Topology and geometry about using the transversality theorem to show that the intersection of two manifolds is a manifold.
Now it goes as follows:
Let ...
-1
votes
0answers
66 views
explanation of examples related to Boundary Orientation.
I found the example from my textbook. I understood similar example related to boundary orientation on $∂ H^n$ But I could not understand these two example which I posted. Please can you explain me ...
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 ...
1
vote
2answers
46 views
Parametrization of $n$-spheres
This comes from Guillemin and Pollack's book Differential Topology. The book claims that one cannot parametrize a unit circle by a single map. I thought we could (by a single angle $\theta$).
I ...
1
vote
2answers
26 views
Question about index of critical points.
I don't really understand what index of a critical point is and I am trying to do a very simple example. I was wondering if someone could help me figure out what the index of the critical point ...
7
votes
0answers
64 views
Let $A$, $B$ be subsets of $S^n$, n≥2. Show that if $A$ and $B$ are closed, disjoint, and neither separates $S^n$, then…
Let $A$,$B$ be subset of $S^n$, $n\geq 2$. Show that if $A$ and $B$ are closed, disjoint, and neither separates $S^n$, then $A\cup B$ does not separate $S^n$.
I've thought to do it by contradiction ...
4
votes
2answers
122 views
Does Differential Topology or Differential Geometry play a larger role in Chaos Theory?
I'm an undergraduate on somewhat of a time constraint in school. I have room in my remaining schedule for a semester of either Differential Geometry or Differential Topology.
I understand the ...
2
votes
1answer
79 views
When is a topological space a manifold?
I'm looking for someone to point me in the direction of papers or books which discuss when a topological space (perhaps with the conditions locally compact, Hausdorff) is a topological manifold, ...
3
votes
2answers
125 views
Relationship between trace of a linear map and the number of points it fixes.
Problem Statement: Let $\Phi_A:T^2\rightarrow T^2$ be a smooth mapping into the torus induced by a linear map $A\in SL_2(\mathbb{Z})$ under the quotient relation that identifies 0 and 1. Assume that A ...
1
vote
1answer
42 views
Extending curves
I have the following situation, $N$ is a $k$-manifold, $X$ a compact $(k+1)$-manifold and $F:X \to N$ a smooth map. Let $y$ be a regular value of both $F$ and $F|_{\partial X}$, hence $F^{-1}(y)$ is a ...
0
votes
0answers
21 views
How can I align the angle between points with the magnetic heading as the points move?
I have 3 robots which must track a point. The distance between all the robots and the point is known so a triangle can be formed between any 2 robots and the point.
If I find the angles in the ...
1
vote
1answer
36 views
Why is $\theta \not \in C^{\infty}(S^1)$?
Why is $\theta \not \in C^{\infty}(S^1)$? I know that since $\int_{S^1} d\theta = 2\pi$ then $d\theta$ is not exact. Thus since $d(\theta)=d\theta$, $\theta$ must not be $C^{\infty}$ but it seems ...
3
votes
2answers
94 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
48 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
67 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
89 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
40 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
42 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
40 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
31 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 $t\in S^1$. ...
2
votes
2answers
52 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 ...
0
votes
0answers
37 views
1-manifolds classification
There is a classification of one-dimensional manifolds: any connected compact one-dimensional manifold is diffeomorphic to a circle $S^1$, noncompact is a line, i.e. $\mathbb{R}$. I understand it on a ...
3
votes
1answer
42 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
50 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 ...
5
votes
2answers
101 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
36 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
76 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
47 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 $Y$ ...
0
votes
1answer
419 views
Vector field on an odd sphere
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
50 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 ...
5
votes
1answer
132 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
43 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 ...
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 ...
0
votes
1answer
53 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. ...
5
votes
2answers
67 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
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 ...
