For questions on manifolds of dimension $n$, a topological space that near each point resembles $n$-dimensional Euclidean space.

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74 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 $$(D, \partial D) \rightarrow (M, \partial M)$$ ...
2
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1answer
182 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, ...
4
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1answer
64 views

Exterior Derivative Problem

Suppose $\theta$ is a differential $1$-form defined on a manifold and with values in the Lie algebra of a Lie group $G$. On $M\times G$ define the $1$-form $ad(g)\theta$ where $\theta$ is extended ...
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0answers
52 views

Prove equivalence of two conditions to be a smooth $k$-manifold $M^k \subseteq \mathbb{R}^n$

For the first couple classes of differential geometry, we have used the more concrete characterization of a manifold (given in #1 below). I am trying to prove that the following two conditions are ...
4
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1answer
59 views

How to show $f^*E$ is a smooth submanifold…

I'm wondering how to show the following: let $E$, $B_1$ and $B_2$ smooth manifolds. Suppose $\rho:E\rightarrow B_2$ is a smooth vector bundle and $f:B_1\rightarrow B_2$ a smooth map. If we write ...
2
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1answer
41 views

Why $f^{-1} (y)$ is a closed set

Let $M$ be a compact smooth manifold, $f$ is a smooth map between $M$ and $N$. If $y \in N$ is a regular value, then $f^{-1} (y)$ is a closed set. I don't know why $f^{-1} (y)$ is a closed set.
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2answers
458 views

Lie bracket is a connection?

In Road to Reality, section 14.6 on Lie derivative Penrose writes: Now $\epsilon^2 [j,h]$ corresponds to an $O(\epsilon^2)$ gap in the ‘parallelogram’ whose initial sides are $e_j$ and $e_h$ at ...
3
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0answers
106 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 ...
3
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1answer
140 views

What is nonhomogeneous linear mapping?

In Milnor's Topology from the differentiable viewpoint, page 3, he said: One thinks of the nonhomogeneous linear mapping from the tangent hyperplane at $x$ to the tangent hyperplane at $y$ which ...
2
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1answer
105 views

Are there Kirby diagrams for manifolds with boundaries?

There are Kirby diagrams for 3- and 4-manifolds which consist of framed links corresponding to 1- and 2-handles attached to a single 0-handle. Any such diagram will give a unique closed manifold since ...
3
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2answers
262 views

Show that the map on spheres is smooth

For each of the following maps between spheres, compute sufficiently many coordinate representations to prove that it is smooth. $(a):$ $p_{n}:\mathbb{S}^{1}\rightarrow \mathbb{S}^{1}$ is the $n$th ...
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3answers
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Under what conditions the quotient space of a manifold is a manifold?

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 that. ...
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1answer
94 views

Smooth mapping between Manifolds

Given: two Euclidean spaces $\mathcal{P},\mathcal{P'}$ (with their usual smooth structure) and a injective smooth mapping $f:\mathcal{P}\to\mathcal{P'}$ a Riemannian manifold $(\mathcal{M}\subset ...
2
votes
1answer
164 views

Construct a map from unit disk to upper half-plane

I want to construct this map in high-dimensional case. Let $D=\{x \in \mathbb{R}^n:|x|^2<1\}$,and $H=\{u\in\mathbb{R}^n:u^n>0\}$. Well, it is quite clear when $n=2$, but I find it is hard for me ...
7
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1answer
152 views

What is the relationship between Grassmann Manifolds with different dimensions?

I'm an EE student and I'm just beginning to learn about the Grassmann Manifold. As is known that the Grassmann Manifold is a space treating each linear subspace with a specific dimension in the vector ...
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0answers
52 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 ...
15
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1answer
366 views

Can we generalize the regular value theorem even beyond the Ehresmann's theorem?

The formulation is complicated, but the answer may be some clever usage of the partition of unity, because locally the answer is given by the regular value theorem and the whole problem is to glue it ...
7
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1answer
209 views

What's the intuition behind the tangent bundle?

Well, when we work with a smooth manifold $M$ we can associate with each point $p\in M$ a vector space $T_p M$ of all vectors at $p$ tangent to $M$: this is the space of linear functionals obeying ...
6
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1answer
561 views

Global sections of a tensor product of vector bundles on a smooth manifold

This question is similar to Conditions such that taking global sections of line bundles commutes with tensor product? and Tensor product of invertible sheaves except that I am concerned with ...
3
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0answers
71 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 ...
7
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1answer
506 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 ...
4
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2answers
108 views

Translating french paper into English

I am currently studying a french paper on Einstein manifolds by Berard Bergery and I have doubts that my translation of the following sentence is correct: "De plus, puisque $G$ agit par isometries, ...
3
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1answer
169 views

Translation of french paper into English

I am currently reading a mathematical paper in french and I am not sure how to translate the following sentence: "On suppose que la premiere classe de Chern $c_1(N)$ est $p\alpha$ ou $p$ est un ...
1
vote
1answer
437 views

Understanding the definition and meaning of cotangent space

I would like to understand definition and also meaning of cotangent space. Let's first of all define tangent space: generally we know that equation of tangent line of function $f(x)$ at point $x_0$ ...
3
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0answers
77 views

Does the boundary of a handle decomposition obtain a handle decomposition?

Let $M$ be the $4$-manifold $D^4\cup2\text{-handle}\cup\ldots\cup2\text{-handle}$, where the attachment of the handles is specfied by an oriented framed link $L=L_1\cup\ldots\cup L_n\subseteq S^3$. By ...
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0answers
44 views

Show: $W:=\left\{(x_1,x_2,x_3)\in\mathbb{R}^3:-1<x_i<1, i=1,2,3\right\}$ is a 3-dim. submanifold of $\mathbb{R}^3$

Use two different argumentations to show that $$ W:=\left\{(x_1,x_2,x_3)\in\mathbb{R}^3:-1<x_i<1, i=1,2,3\right\} $$ is a 3-dim. submanifold of $\mathbb{R}^3$. 1) ...
2
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1answer
117 views

Can a tangent vector extend to a vector field?

Let $M$ be a smooth manifold and $p\in M$. I would like to know whether any tangent vector $X_p \in T_pM$ extends to a vector field over $M$. If so is it unique? How can I construct it? Thank you.
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0answers
67 views

warped products

Problem: Consider the following warped product $M^{n+1}=\mathbb{R}\times_{f} \mathbb{P}^{n}$, where $\mathbb{P}$ is a complete n-dimensional Riemannian manifold, $f:\mathbb{R}\rightarrow\mathbb ...
3
votes
1answer
97 views

Natural diffeomorphism between $T\mathbb{S}^n\times \mathbb{R}$ and $\mathbb{S}^n\times\mathbb{R}^{n+1}$

I need to show that there is such a diffeomorphism between these spaces. I've tried looking at the 'faces' of elements on both spaces. It went like this: every element in $T\mathbb{S}^n\times ...
3
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1answer
152 views

Is there a Smooth Real Manifold which is not a Riemannian Manifold?

I am taking a course in Differential Topology right now, but I know of another Subject called "Riemannian Geometry" which studies Riemannian Manifolds. The definition of a real smooth manifold and a ...
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2answers
79 views

show a map of complex projective space is lefschetz

This is a problem from a qualifying exam. Let $A \in GL_{n+1}(\mathbb{C})$. Then $A$ defines a smooth map on $\mathbb{CP}^n$ by $A \cdot [z] = [Az]$ for $[z] \in \mathbb{CP}^n$. We will denote this ...
3
votes
1answer
101 views

On the definition of the exponential map

The exponential map on a manifold $M$ is defined at a point $ p\in T_p(M)$ as $$exp_p:T_p(M)\rightarrow M \\ exp_p(v)=\gamma_v(1) $$ where $\gamma_v$ is the constant speed geodesic with initial ...
1
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1answer
92 views

Group Extension and Classifying Space

If $$ 0 \to H \to G \to G/H \to 0\ $$ is a group extension, under what conditions do we have a fibration of the form $$ BH \to BG \to B(G/H), $$ where $BG$ is a classifying space of $G$? Suppose ...
4
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1answer
111 views

Is there a name for this particular class of topological space?

This is a simple question, but I can't figure out the name for this class of topological space. Say you start with the affine space $\mathbb{R}^n$ for finite n, and equip it with a metric. Now, say ...
1
vote
1answer
157 views

On tangent spaces of Steifel Manifolds

I was trying to read Edelman et al.'s 1998 paper "The Geometry of Algorithms with Orthogonality Constraints" and since I don't have any differential geometry or much linear algebra background I am ...
6
votes
1answer
146 views

Is there any relationship between Cauchy-Riemann equations and vector fields on manifolds?

Well, suppose we have $f : \mathbb{C} \to \mathbb{C}$ analytic, then if $f = u + iv$ the functions $u,v : \mathbb{C} \to \mathbb{R}$ satisfy the Cauchy-Riemann equations: $D_1u=D_2v$ and $D_2u=-D_1v$. ...
2
votes
1answer
130 views

Precise definition of isotropic curve of a conformal structures on a manifold?

Could you please provide me with the precise definition of isotropic curves of a conformal structure on a manifold $M$? If there is such a definition, then can I say the following: if $c$ is an ...
2
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0answers
332 views

local parametrization of regular surface

I am doing excercises of Do Carmo's dg of curves and surfaces Chapter 2.2 and need some help with the following excercise: Show that the set $S=\{(x,y,z)\in R^3;z=x^2-y^2\}$ a regular surface and ...
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0answers
122 views

4-manifold: $0$-handle $\cup$ $2$-handles along a framed link in $S^3$ (intersection form = linking matrix = presentation matrix of $H_1(\partial M)$)

Let $L$ be a framed link in $S^3$, consisting of framed knots $L_1,\ldots,L_m$. Let $A=[a_{ij}]\in\mathbb{Z}^{m\times m}$ be its linking matrix, with $a_{ii}=$ framing of $L_i$ and $a_{ij}=$ linking ...
1
vote
1answer
407 views

Constant Rank Theorem and Submanifolds

I'm related to my previous question here. The problem is: I am given a $C^r$ manifold $M$ and a connected subset $A$ of $M$, and a $C^r$retraction $f:M\rightarrow A$ such that ...
3
votes
2answers
141 views

whether gluing the faces of tetrahedron in pairs would get a manifold?

I think gluing the faces of tetrahedron in pairs would get a manifold. Because gluing in pairs will make the result of gluing without boundary and there is not Y structure in the result. But it is ...
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0answers
159 views

Is this enough to show that this map has constant rank?

My question is related to this question I am given a $C^r$ manifold $M$ and a connected subset $A$ of $M$, and a $C^r$retraction $f:M\rightarrow A$ such that $f\vert_A=id:A\rightarrow A$. Then $A$ is ...
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0answers
73 views

On the proof that the inverse value set of a regular value is a submanifold

I have a doubt on the proof of the following, well-known theorem: Let $f:M^m\rightarrow N^n$ ($m\geq n$) be a $C$ map, $r\geq 1$.If $y\in f(M)$ is a regular value, then $f^{-1}(y)$ is a $C$ ...
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2answers
168 views

Example of a manifold?

Why is this picture an example of a $1$-dimensional manifold? My thought process is: the circle must have a point removed from it because otherwise it would be self-intersecting, and self-intersection ...
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0answers
49 views

JSJ-decomposition of a non-hyperbolic 3-manifold

Suppose $M$ is a $3$-manifold. Then you can split it over spheres. This is the "prime decomposition" and is unique. You can then split the components of this decomposition along tori. If you leave the ...
3
votes
3answers
485 views

Why do we need Hausdorff-ness in definition of topological manifold?

Suppose $M^n$ is a topological manifold, then $M^n$ locally looks like $\mathbb{R}^n$. $M^n$ is locally Hausdorff, since $\mathbb{R}^n$ is Hausdorff and Hausdorff-ness is a topological invariant. ...
2
votes
1answer
139 views

Typo or additional hypotheses needed in problem 5, p.71 of Bredon's Topology and Geometry?

The problen can be foun in p.71 of Topology and Geometry. I state it below for convenience. Problem: Consider the half open real line $X=[0,\infty)$. Define a functional structure $F_{1}$ by taking ...
6
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1answer
191 views

Isotopy preserving inverse image $f_t^{-1}(V)$ of a homotopy

During a lecture I was given a bunch of easy propositions entitled as "observations" by the lecturer. But one of them seems more difficult and I have absolutely no idea how to "observe" it... I'd be ...
3
votes
1answer
226 views

A functional structure on the graph of the absolute value function

Let $X$ be the subspace of $\mathbb{R}^2$ consisting of the graph of the absolute value function. That is, $X=\{(x,|x|) : x\in\mathbb{R})\}$. We define a functional structure on $X$ by restricting ...
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0answers
91 views

For what kinds of manifolds $\dim T_pM=\dim M$ holds?

Does the truth that $\dim T_pM=\dim M$ hold only for differentiable manifolds or for all topological ones?