Tagged Questions
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 ...
2
votes
1answer
32 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
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 ...
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.
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 ...
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
44 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
70 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 ...
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 ...
2
votes
1answer
27 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 $R^{n^2}$. ...
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 $M \cup_h \mathbb{D}^\lambda \times\mathbb{D}^{\mu}$ ...
0
votes
3answers
46 views
Lebesgue Measure with given function?
Suppose $E$ subset $R$ ($R $\is real numbers) where $E$ is Lebesgue measurable, and $f:E\to R$ and defined $g: R\to R$ by \begin{equation*}
g(x) =
\begin{cases}
f(x) & x \in E \\
0 & x ...
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 $$\{(x_1,\ldots,x_n)\in \mathbf C^n-\{0\} : \vert x_1\vert < 1, \ldots, \vert x_n\vert ...
8
votes
2answers
106 views
No Smooth Onto Map from Circle to Torus
My professor was lecturing today and he made this statement which I was unable to verify.
(I worded it nicer)
There is no map which is both smooth and onto from $S^1$ to $S^1$$\times$ $S^1$.
When ...
1
vote
2answers
112 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 ...
2
votes
0answers
33 views
Topological inequivalent manifolds obtaining by removing a surface from a manifold
Are there any general techniques for classifying the inequivalent topologies that can be obtained by removing a 2-surface S from a 4-manifold M? I am particularly interest in the case where both M and ...
1
vote
0answers
39 views
Continuity in the Strong(Whitney) Topology
Let $P,M$ and $N$ are smooth manifolds and let $F:P\times M \to N$ be a smooth map. We know its associated map $\tilde F:P \to C^\infty(M,N)$ given by $p \mapsto F_p(m)$ is continuous if and only if ...
0
votes
0answers
35 views
Topology on set of maps between manifolds
So the professor in class gave the following two theorems without proving ( or maybe I missed it).
Theorem 1. Let $M$,$N$ be two smooth manifolds. $L$ a submanifold of $N$. And $M$ is compact. Then ...
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 ...
4
votes
1answer
51 views
Approximate continuous mapping by smooth mappings on manifold(Bott, Tu book)
So I was looking at the proof given in Bott, Tu "Differential Forms in Algebraic Topology" of how to approximate continuous mapping by smooth mappings between manifolds. It is Proposition 17.8 on Page ...
1
vote
1answer
69 views
How is PL knot theory related to smooth knot theory?
I really want to like knot theory but the PL condition seems sort of ugly. I was hoping someone could give me a justification for secretly thinking about smooth knots as I read through a book like ...
3
votes
0answers
33 views
Is a solution to some partially differential equation homeomorphic (or diffeomorphic) to a solution of an equation with a different covariance group?
Consider some solution $\psi(x,t)$ to the linear Klein-Gordon equation: $-\partial^2_t \psi + \nabla^2 \psi = m^2 \psi$. Up to homeomorphism, can $\psi$ serve as a solution to some other equation ...
2
votes
1answer
93 views
Metric tensor of complex numbers & Hamiltonian Mechanics
The Euclidean $\mathbb{R}^2$ geometric space can be mapped onto $\mathbb{C}$.
In other words I see it like this
$$\vec{v} = x\vec{x}+y\vec{y} = x\vec{1}+y\vec{i}= \begin{bmatrix}x \\y\end{bmatrix} ...
1
vote
1answer
77 views
equivalence of different definitions of isotopy
Here are two supposedly equivalent definitions of a smooth isotopy (M and N are smooth manifolds):
A smooth level preserving imbedding $M \times I \rightarrow N \times I$
A smooth map $ F: M\times I ...
2
votes
1answer
53 views
3-Ball. 3−manifold in $\mathbb{R}^3$
The $3-$ball ${B_R}^3 = \{(u,v,w) \in \mathbb{R}^3 | u^2+v^2+w^2 \le R^2\}$ is a $3-$manifold in $\mathbb{R}^3$;
orient it naturally and give
$${S_R}^2 = \partial {B_R}^3 = \{ (u,v,w)\in \mathbb{R}^3 ...
5
votes
1answer
120 views
Proving that $O(n,m)$ is simply connected.
My question is the following:
Under which conditions on given integers $n\le m$ is $$O(n,m) = \{A \in \mathbb R^{m\times n} : A^TA = \mathbf 1\}$$ simply connected? Does anyone know a reference for ...
3
votes
2answers
134 views
About covering maps and sections!
If $q: E\rightarrow X$ is a covering map that has a section $(i.e. f: X\rightarrow E, q\circ f=Id_X)$ does that imply that $E$ is a 1-fold cover?
1
vote
1answer
141 views
Infinitely sheeted covering spaces!
I was wondering what the fundamental group of an infinitely sheeted covering space of say some surface might be?
I'm thinking it should be an infinite cyclic group, but this is more intuitively, i ...
1
vote
1answer
68 views
Morse index and Euler characteristic
I found the following problem and I couldn't solve it.
Let $X$ be a compact manifold and $f$ a Morse function (all of its critical points are non degenerate) on $X$. Prove that the sum of the Morse ...
1
vote
0answers
49 views
How to prove this isotopy exist?
Let $M$ be a topological manifold and $N$ is a subset of $M$. Let $f_t$ be an isotopy from $id $ to $f$ which is a homeomorphism on $M$. Suppose $f(N)=N$ and there is an isotopy $g_t$ on $N$ such that ...
3
votes
0answers
89 views
Covering spaces!
If one wanted to find all connected covering spaces of a product of two spaces, say $S^1\times RP(3)$, how would you go about it?
I'm thinking finding the fundamental group of $S^1\times RP(3)$, and ...
1
vote
0answers
30 views
Other definitions of singularity
Many definitions of a singularity of a manifold $X^n$ are concentrating on the defining equations of it and the vanishing of the (partial) derivatives. My questions:
What if $X^n$ isn't algebraic ...
5
votes
0answers
161 views
How to prove a manifold is diffeomorphic to Euclidean space?
Problem is this: suppose a manifold
$$M=\bigcup_{n\in\mathbb{N}} U_n,$$
where each $U_n$ is diffeomorphic to Euclidean space, and $U_n$ is contained in $U_{n+1}$. Then please show that $M$ is ...
2
votes
3answers
89 views
Finding the rank of subgroups of free groups?
How do you find the rank of a subgroup(of finite index) of a free group?
I was thinking of looking at the fundamental group of a graph.
1
vote
1answer
32 views
Transversality of a mapping
The question I have is:
Show that the mapping $g:R^2 \rightarrow R^3 $ given by :
$y_1 = x_1 (x_1 ^2 -x_2 ^2 +1), y_2=x_2, y_3=x_1 ^2$
is transversal to all lines $y_2 = \textrm{constant}$ in the ...
2
votes
0answers
125 views
Fundamental Group!
Say you have two surfaces of genus 2, say $X$ and $Y$ and you want to attach them via homotopy attaching maps $f$ along their waist curves. Then what will the fundamental group of the adjunction space ...
0
votes
0answers
28 views
How to relate $S^1$ (SO(2)) to a stretched version of $S^1$ ($SO(2) \subset SO(3)$)?
I want to decompose the quaternion parameterization of $SO\left(3\right)$ into the subspace of $SO\left(2\right)$ defined by choosing the z-axis as the rotation axis, and find a relationship between ...
2
votes
1answer
84 views
Reference for topology and fiber bundle
I am looking for an introductory book that explains the relations of topology and bundles.
I know a basic topology and algebraic topology. But I don't know much about bundles. I want a book that
...
0
votes
0answers
56 views
Most elementar definition of embedding and imbedding
I found some comments about embeddings but they are talking about manifolds or surfaces. So, I'd like to know what is the difference between embedding and imbedding, but not with many structures, I ...
0
votes
1answer
58 views
Line with a knot in $\mathbb R^3$ isotopic to standard embedding $\mathbb R\subset \mathbb R^3$?
I have come across the following exercise in Kosinski's 'Differential Manifolds':
Exercise: Consider an imbedding $\mathbb R\to \mathbb R^3$ where the image is "the line with a knot":
...
6
votes
4answers
169 views
How do you imagine the shape of a manifold $S^2 \times S^1$?
In 3-dimensional manifold theory, I have encountered the manifold $S^2 \times S^1$ many times.
(The following story can be applied not only this manifold but also for any 3-dimensional manifold.)
But ...
4
votes
1answer
98 views
Prime decomposition of 3-manifolds
Let $H_g$ be a three dimensional handlebody bounded by a genus $g$ surface.
Let $M_g$ be a manifold obtained by gluing two copies of $H_g$ via an orientation reversing homeomorphism of the surface of ...
3
votes
1answer
65 views
Gluing axiom of a TQFT
In the book, Lectures on tensor categories and modular functors by Bakalov and Kirillov they construct a TQFT.
When they come to prove the gluing axiom, they just mention that "...This statement is ...
3
votes
2answers
223 views
Homotopy Question Help?
Let $X$ be a topological space and suppose $X_1$ and $X_2$ are spaces obtained by attaching an n-cell to $X$ via homotopic attaching maps. Show that $X_1$ and $X_2$ are homotopy equivalent.
Proof:
...
2
votes
2answers
67 views
Is there any notion for a certain type of embedding of a smooth curve in a 2-d euclidean space?
There is a smooth 1-manifold (a smooth curve of infinite arc length) embedded in a 2 dimensional euclidean space. This curve (of infinite arc length) is such that, there is one and only one point $P$ ...
