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

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9
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4answers
645 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
141 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
votes
1answer
195 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
249 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 ...
4
votes
0answers
92 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?
4
votes
1answer
399 views

How can one prove that manifolds are regular?

First, some clarification of the definition of a manifold that I'm using: A manifold $M$ is a Hausdorff, locally Euclidean and second countable topological space. Now, I am trying to prove that ...
2
votes
1answer
233 views

example of a non differentiable manifold

Take the manifold: The graph of $|x|$ on $(-1,1)$, with the induced topology from $\mathbb{R}^2$. This is a topological manifold, which is homeomorphic to $(-1,1)$ by projection. Is it a ...
1
vote
1answer
158 views

Definition of attaching a cell to a manifold

I know "attaching a handle" to a manifold, but recently I faced "attaching a cell" and I don't know its definition in precise. It seems that the definition is very trivial (!) because my searches did ...
5
votes
4answers
191 views

Is $[0,1]$ a 1-manifold?

Is $[0,1]$ a 1-manifold? I would say no because at either endpoint the open sets containing it aren't homeomorphic to a 1-ball in $\mathbb R^1$.
4
votes
1answer
118 views

Basic property of a tensor

In Jost's Riemannian Geometry and Geometric Analysis (6th ed.) on page 142 there is the following remark concerning the torsion tensor. Remark. It is not difficult to verify that [the torsion ...
7
votes
1answer
521 views

Complement of figure-8 knot

I am reading W. Thurston's famous "3-dimensional Geometry and Topology", but I am stuck at the point where it is said that gluing two tetrahedra in an appropriate way give you the complement of the ...
3
votes
1answer
366 views

Diffeomorphism of tangent spaces

This is my first post on the stackexchange, so I'm sorry if its rambling. I've been working through Lee's Introduction to Smooth Manifolds and I'm having some trouble with one of the exercises. Most ...
1
vote
1answer
96 views

How to show $[\omega]=0$ implies $[\omega^n]=0$?

I'm trying to prove the following: If $(M, \omega)$ is a symplectic manifold and $[\omega]=0$ then $[\omega^n]=0$, where $[\omega]$ is the De Rham cohomology class of $\omega$. Well what I've done ...
0
votes
1answer
200 views

Global stable manifold always an embedded submanifold? Typo or misreading?

I was reading Brin and Stuck's Introdroduction to Dynamical Systems (link to pdf of book can be found by googling "Brin and Stuck's Introdroduction to Dynamical Systems"), and I came across on page ...
0
votes
1answer
75 views

Can anyone check my proof that $H^1(\Sigma-\{p\})=0$ for a compact and orientable surface $\Sigma$?

I have the following problem: Let $\Sigma$ be a compact and orientable surface. Show that $H^1(\Sigma-\{p\})=0$ for every $p\in \Sigma$. Can anyone check my proof and give suggestions? Sketch of ...
2
votes
1answer
304 views

How do I prove that a subset of a manifold is not a submanifold?

I know of ways to prove that a given subset of a smooth manifold is a smooth submanifold, but what if I have some subset which I suspect is not a smooth submanifold? What are some approaches to ...
5
votes
1answer
334 views

Functionally structured spaces and manifolds

The question requires some definitions that I have listed below for your convenience. They can be found in Chapter VI, pages 297-298 of Bredon's Introduction to Compact Transformation Groups. On a ...
2
votes
1answer
276 views

Show that the set $M$ is not an Embedded submanifold

How can I prove that $M=\{(x,y)\in \mathbb{R}^2\ ; y=|x|\}$ is not an embedded smooth submanifold of $\mathbb{R}^2$?
5
votes
1answer
133 views

One-form on quotient manifold

Let $M$ be a smooth manifold with tangent bundle $TM$ and cotangent bundle $TM^*$ and $\psi\in TM^*$ a one-form. We denote the quotient manifold of $M$ by the free and proper $G$-action $\varphi$ as ...
4
votes
3answers
216 views

Is it possible to make the set $M:=\{(x,y)\in\mathbb{R}^2:y=|x|\}$ into a differentiable manifold?

1) Is it possible that be given a suitable smooth atlas to the set $M:=\{(x,y)\in\mathbb{R}^2: y=|x|\}$ so that $M$ to be a differentiable manifold? Why? How? 2) How can I prove that M is not an ...
-1
votes
0answers
32 views

Geometry to the inhabitants of a particular space [closed]

Is it true that inhabitants of a space take up its geometry, and if so, what does this mean? I.e. is geometry always Euclidean to inhabitants, while not necessarily so for external observers? For ...
2
votes
1answer
122 views

Visualize $\mathbb{S}^3/\Gamma$!

I thought the only 3-manifold with positive constant curvature is $\mathbb{S}^3$. But I faced $\mathbb{S}^3/\Gamma$, where $\Gamma$ is a finite subgroup of $SO(4)$ and surprised! My problem is that I ...
0
votes
0answers
31 views

extremal points on a manifold intrinsiclly

I am wondering if there is a geometric object for real analytic manifolds that characterizes extremal points of the manifold intrinsically. For instance, suppose I live in the manifold, can I ...
1
vote
1answer
250 views

How to compute $H^1(\Sigma_g-\{p\})$ using Mayer-Vietoris?

How can I find, using Mayer-Vietoris, $H^1(\Sigma_g-\{p\})$, where $\Sigma_g$ is a genus $g$ surface?
4
votes
1answer
81 views

Geodesic complete subset of a connected manifold

This may be a very silly question but let us consider a connected Riemanian manifold $(M,g)$ and a subset $O\subset M$. Can we have $O$ geodesic complete (in the sense of all geodesics linking two ...
2
votes
0answers
76 views

Vector field from Lamination.

Let $S$ be a smooth closed (i.e. compact without boundary) surface. A geodesic lamination on $S$ is a nonempty closed subset of $S$ which is a disjoint union of geodesics. Suppose $\alpha$ is a ...
2
votes
2answers
49 views

Example of this sort of submanifold

This is from Smooth manifolds and their applications page 7: Let $P=P^r$ be a subset of the smooth manifold $M^k$ of class $m$, defined near each of its points by a system of $k-r$ independent ...
0
votes
1answer
50 views

Basic (multivariable) calculus question

I need some help with basic calculus. I asked a question the other day and got a decent answer but there is one step in the answer I just don't understand. Why is ${\partial y_1 \over \partial x_1} $ ...
2
votes
1answer
85 views

Why can an orientation on $X$ be written as a sum of cycles on this open cover?

I'm reading a proof of the following theorem Let $X$ and $Y$ be compact connected oriented $n$-manifolds, and let $f:X\to Y$ be continuous. Let $y\in Y$, and assume that $f^{-1}(y)$ is finite. ...
5
votes
1answer
198 views

Topologist's Sine Curve not a regular submanifold of $\mathbb{R^2}$?

I am trying to work out the details of following example from page 101 of Tu's An Introduction to Manifolds: Example 9.3. Let $\Gamma$ be the graph of the function $f(x) = \sin(1/x)$ on the ...
5
votes
1answer
160 views

Classification of orientable non-closed surfaces

How does the classification of closed (compact, boundaryless) surfaces imply the classification of all orientable not-necessarily-compact surfaces with boundary? It seems to be that they are all ...
0
votes
1answer
89 views

Invertibility of a function

S is a surface in $\mathbb{R}^{3}$ parameterized by a function $f:S\rightarrow(a,b)^{2}\subset\mathbb{R}^{2}$ $F$ is the function defined by: $F:T^{1}S\rightarrow(a,b)^{2}\times S^{1}$ ($T^{1}S$ is ...
1
vote
0answers
100 views

Tangent bundle of a quotient manifold

I am interested in the tangent bundle of the quotient of a manifold by a proper and free action but I can't find any reference on the net. Does anyone know a book or article where it is described ?
3
votes
1answer
234 views

Visualization of the diffeomorphism!

Basic to all mathematics is the notion-here used quite informally-of a set with structure. For every type of structure there is a notion of equivalence (or isomorphism)-a one-to-one onto ...
9
votes
2answers
440 views

Diffeomorphism of $\mathbb{C}P^1$ and $S^2$.

In an exercise I am asked to find a (smooth) submersion of $S^3$ onto the sphere $S^2$. So far I have a submersion of $S^3$ onto $\mathbb{C}P^1$. Are $\mathbb{C}P^1$ and $S^2$ diffeomorphic? If so, ...
1
vote
3answers
194 views

Extending a map of manifolds continuously

Let $M$ and $N$ be manifolds, and $A \subset M$ compact. Let $f:A \rightarrow N$ be a continuous mapping. Show there exists an open neighborhood $U$ containing $A$ and continuous extension $g:U ...
1
vote
1answer
42 views

Reference request for studying on space forms

I would like to study on Space form, But I dont know what book or notes are suitable for beginning basically. Can someone help me? Thanks.
2
votes
2answers
56 views

Function and its exterior derivative

Is there an example of a function $f:M\to \mathbb R$, where $M$ is a differentiable manifold, such that $f$ is constant on the hypersurface $\Sigma$ and its exterior derivative $df\neq 0$ on ...
3
votes
1answer
111 views

Relative homology groups

The local homology of a manifold $X$ at a point $x$ is defined as the relative homology $H_n(X, X-{x};\ \mathbb Z)$. It holds true for relative homology that under certain conditions $H(X,A) = ...
1
vote
0answers
74 views

On the canonical neighborhoods

Suppose $M$ is a 3-dimensional manifold, John W. Morgan and Frederick Tsz-Ho Fong in their "Ricci Flow and Geometrization of 3-Manifolds" book as a definition of canonical neighborhoods have ...
1
vote
1answer
28 views

If $M=U\cup V$, $U$ and $V$ have finite dimensional cohomology then $U\cap V$ has finite dimensional cohomology..

I need some help with the following: Let $M$ be a differentiable manifold such that $$\textrm{dim}(H^k(M))<\infty$$ for every $k=0, \ldots, n$ where $H^k(M)$ is the $k$-th De Rham cohomology group ...
5
votes
1answer
120 views

Integrating 2-form

In $\mathbb{R}^3$ I consider the compact 2-dimensional manifold $$ M=\left\{(x,y,z)\in\mathbb{R}^2: z=xy\right\} $$ which is orientated by the (global) map ...
0
votes
2answers
89 views

Concept map for manifolds

What exactly is manifold? What concepts do I need to learn in order to take on manifolds and concepts related to it?
1
vote
1answer
127 views

Find orientation of manifold

In $\mathbb{R}^3$ consider the the 2-dim manifold $$ M:=\left\{(x,y,z)\in\mathbb{R}^3:z=xy\right\}. $$ Let the orientation of $M$ be in such a manner that in the Point $(0,0,0)$ the vector ...
3
votes
1answer
214 views

A question about Moyal product

In Geometric quantization theory the Moyal product is one of main tools. We know Moyal product for the smooth functions $f$ and $g$ on $ℝ^{2n}$ takes the form $f\star g = fg + \sum_{n=1}^{\infty} ...
1
vote
0answers
55 views

Given an abstract open book obtaining a 3-manifold

Let $\Sigma_{\phi}$ be the mapping torus of $\phi$, i.e., $\Sigma \times [0,1] / \sim$ where $(\phi(x),0) \sim (x,1)$ for all $x \in \Sigma$. Also let $$M_{\phi} = \Sigma_{\phi} \cup_{\psi} ...
2
votes
1answer
285 views

Definition clarification on orientation on a manifold.

I have been trying to self-learn differential geometry. I think I may have misunderstood/missed out on something along the way. It is said that for $X$ an $n$-form, $M$ a differentiable manifold, ...
4
votes
1answer
266 views

Two definitions of smooth manifolds

In Milnor/Stasheff they give the definition of smooth manifold as follows (page 4): A subset $M \subset \mathbb R^A$ is a smooth manifold of dimension $n \ge 0$ if, for each $x \in M$ there exists a ...
3
votes
3answers
305 views

Coordinates (in vectors space and on manifold)

I recently realized that I was confused by the coordinates of a vector and the coordinates on a manifold. It probably happened because I think of manifolds as subsets of $\mathbb R^n$ and most of the ...
5
votes
2answers
149 views

What is $S^3/\Gamma$?

Let G is a group and H is a subgroup of G. I know $G/H$ is the quotient space but I have no idea about what $S^3/\Gamma$ is, where $S^3$ is the sphere and $\Gamma$ is a finite subgroup of $SO(4)$. In ...