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.

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Is this map the Gauss map?

Let $M\subset \mathbb{R}^n$ be a closed compact smooth oriantable manifold. Let $E\to M$ be the normal bundle of the embedding and $N\subset \mathbb{R}^n$ a (closed) tubular neighbourhood. $N$ is an ...
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2answers
131 views

If $\operatorname{dim} M > \operatorname{dim} N$, is there an injective smooth map $M\to N$?

Let $m>n$ and suppose $M$ is a smooth $m$-manifold, $N$ is a smooth $n$-manifold. Can there be an injective smooth map $f:M\to N$?
7
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1answer
767 views

How calculate the De Rham cohomology group of $3$-torus: $T^3$?

How do I calculate the De Rham cohomology group of the $3$-torus $T^3$? Here $T^3=S^1 \times S^1 \times S^1 $. Using the Mayer-Vietoris sequence, I can show that $\dim H_3(T^3)=\dim H_0(T^3)=1$. But ...
11
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1answer
394 views

When is a homology class a fundamental class?

Let $X$ be a real connected orientable closed $n$-dimensional compact differentiable manifold. A connected oriented closed $d$-dimensional submanifold $i:M\to X$ (i.e. $M$ is a real connected ...
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2answers
156 views

The most general notion of a directional derivative

Questions: I know you can define a directional derivative on some subset of $\mathbb R^n$, but what can be said about an arbitrary set of points, $S$? What are the most general criteria $S$ must ...
6
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1answer
358 views

When do regular values form an open set?

Let $f:M\to N$ be a $C^\infty$ map between manifolds. When is the set of regular values of $N$ an open set in $N$? There is a case which I sort of figured out: If $\operatorname{dim} M = ...
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91 views

Question about the Structure group of a circle bundle over a Riemann surface

The last two pages of Appendix C in Milnor's Characteristic Classes gives an example of a flat bundle with nonzero Euler class. I have a question about the structure group of this bundle. The ...
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1answer
156 views

question regrading double cone

How to prove that double cone is not a topological manifold.I think I have to do something with the meeting point of the two cones.
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105 views

Can I flip orientation at a point of a non-orientable manifold?

Let $p \in M$ be a point of a non-orientable smooth manifold, $M$. Does there exist a diffeomorphism $f: M \rightarrow M$ with $p \mapsto p$ and such that $df : T_pM \rightarrow T_pM$ is orientation ...
2
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1answer
176 views

Equivalent definition of Tangent Spaces

There are about 4 definitions of tangent spaces 1) using velocities of curve 2) via derivations 3)via cotangent spaces 4) as directional derivatives. I am not getting the intuition about what tangent ...
2
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1answer
167 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} ...
8
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1answer
277 views

Conceptual error in Kosinski's “Differential Manifolds”?

This seems like such an egregious error in such an otherwise solid book that I felt I should ask if anyone has noticed to be sure I'm not misunderstanding something basic. In his section on connect ...
3
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126 views

Reference Request: Poincare-Hopf Index Theorem

When I read Griffiths' Algebraic Curves page 23, it states Poincare-Hopf formula for differential forms that is smooth except finite isolated singularities. I tried to find reference for it, but ...
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1answer
46 views

Proving finiteness of a submanifold

For some reason I am having trouble parsing this bit from Guillemin & Pollack Chapter 2.4: Let $X, Z$ be transversal closed submanifolds in $Y$ (everything is without boundary). Further, let $X$ ...
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216 views

Use of Implicit Function Theorem to provide examples of Manifolds

Suppose we know Definition 1 and Theorem 1, and want to prove Theorem 2 given below. Definition 1: A subset $M$ of $R^n$ is called an k-dimensional manifold (in $R^n$) if for each point $x\in M$ the ...
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1answer
178 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 ...
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36 views

Example of atlas for sequence space

Is it possible to construct and atlas for space $\ell_1$? I.e. give an example of collection $(U_\alpha,\phi_\alpha)$, such that $\cup U_\alpha=\ell_1$ and $\phi_\alpha:U_\alpha\to R^{n}$ is a ...
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68 views

How to use the Prontrjagin-Thom construction to obtain the Gysin map?

I need help to understand the diagram in Miller's script Vector Fields on Spheres, etc. Chapter 23, p.82 on the bottom of the page. Before, Miller introduces the Prontrjagin-Thom construction: It is ...
2
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0answers
127 views

To what extent is the global angular form well-defined?

I am reading Differential Forms in Algebraic Topology by Bott & Tu. The authors constructed the global angular form $\psi$ for an oriented $k$-sphere bundle $E$ over a smooth manifold $M$. It has ...
3
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1answer
176 views

Transversality is generic

Let $M$ and $N$ be submanifolds of $\mathbb R^n$. I am trying to prove that for almost every $x\in \mathbb R^n$, $M+x$ and $N$ intersect transversely. Intuitively, transversality is a "generic" ...
3
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1answer
110 views

Question on the proof of the Stability Theorem in Guillemin & Pollack

I'm having trouble with the following part in the proof of the stability theorem on Pg 36 of Guillemin and Pollack's Differential Topology: They write that since $X$ is compact, it follows that any ...
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1answer
234 views

Nontrivial h-cobordism

I'm learning the h-cobordism theorem as I want to use it in a talk. I'd like to be able to give an example of an h-cobordism that isn't a cylinder, if possible by drawing a picture. What is the ...
3
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0answers
108 views

Are strongly close maps homotopic?

While reading about various results related to density of smooth functions in the space of continuous functions with strong topology, I've got the impression that it is a general fact that for any ...
4
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1answer
449 views

Problem from “Differential topology” by Guillemin

I am strugling one of the problems of "Differential Topology" by Guillemin: Suppose that $Z$ is an $l$-dimensional submanifold of $X$ and that $z\in Z$. Show that there exsists a local coordinate ...
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1answer
91 views

smooth domain in $\mathbb{C}^2$ and smooth bounday of bounded domain in $\mathbb{C}^2$

How can we define a smooth domain in $\mathbb{C}^2$ and a smooth boundary of bounded domain in $\mathbb{C}^2$? {where $\mathbb{C}^2$ := Cartesian product of complex plane }
14
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1answer
193 views

Are locally homotopic functions homotopic?

Suppose we have two (smooth) functions $f,g:X\to Y$, where $X,Y$ are smooth (second-countable, Hausdorff) manifolds which are locally homotopic (that is, any point in $X$ has a neighbourhood $U$ such ...
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0answers
176 views

Does every non-compact manifold admit an incomplete vector field?

I know that every vector field on a compact manifold is complete. The question of whether every non-compact manifold admits an incomplete vector field seems to follow naturally. I'd hazard a guess ...
2
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1answer
91 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 ...
11
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1answer
419 views

uniqueness of the smooth structure on a manifold obtained by gluing

I've just read a proof that If $M$, $N$ are smooth manifolds with boundary and $f: \partial M\rightarrow \partial N$ is a diffeomorphism then $M \cup_f N$ has a smooth manifold structure such that ...
2
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1answer
187 views

$(n - 1)$-dimensional submanifold of the manifold $\mathbb R^n$

Let $A$ be a symmetric $n \times n$ matrix over $\mathbb{R}$. Let $0 \neq b \in \mathbb{R}$. Show that the surface $M = \{x\in \mathbb{R}^n \mid x^T A x = b\}$ is an $(n - 1)$-dimensional ...
20
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533 views

How does one parameterize the surface formed by a *real paper* Möbius strip?

Here is a picture of a Möbius strip, made out of some thick green paper: I want to know either an explicit parametrization, or a description of a process to find the shape formed by this strip, as ...
2
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0answers
72 views

4D TQFT construction from a modular tensor category

I know the construction of 3D topological quantum field theory (TQFT) from a modular tensor category. I heard that we can even (mathematically) construct 4D TQFT from a modular tensor category. I ...
5
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1answer
136 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 ...
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1answer
260 views

Co-homology Groups of the Torus

I wanted to explain to me, or give me a reference of how to calculate the cohomology groups of the complex and real, torus $\mathbb{T}^2$. I want to use this as an example in a seminar that I will ...
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1answer
133 views

How to construct a map from $\mathbb{ S}^2=\{(a_1,a_2,a_3) | a_1^2+a_2^2+a_3^2=1\}$ to $\mathbb{ RP}^2$?

How would I construct the map? Once constucted, would I be right in saying that there is no Diffeomorphism to map back? As in $\mathbb{RP}^2$ a closed curve would have to have either $2$ points that ...
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3answers
330 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 ...
4
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1answer
206 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 ...
8
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1answer
234 views

Existence of geodesic on a compact Riemannian manifold

I have a question about the existence of geodesics on a compact Riemannian manifold $M$. Is there an elementary way to prove that in each nontrivial free homotopy class of loops, there is a closed ...
0
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1answer
65 views

2-Manifold an image of the unit disc?

Is every 2 dimensional manifold whose boundary is a cycle, a continuous image of the unit disc? Maybe it happens if the space is good enough? I wanted to prove an equality between two definitions I've ...
3
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0answers
44 views

Linking integral unchanged over continuous deformations

Say we first have two curves, $C_1$ and $C_2$ which are knotted together. Let $C_2'$ be a continuous deformation of $C_2$ such that $C_2$ does not cross $C_1$ as it is deformed into $C_2'$. How ...
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0answers
64 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 ...
19
votes
2answers
870 views

Applications of Pseudodifferential Operators

I am very interested in just about anything that has to do with PDE's, and inevitably pseudodifferential operators comes up. Its obvious that such a novel way of looking at PDE's would be important, ...
3
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1answer
53 views

Are these functions homotopic?

Let $\gamma$ be a smooth, simple, closed curve and let $f : \gamma \to S^1$ assign to each $x \in \gamma$ the unit normal vector there. We can find a diffeomorphism $g: \gamma \to S^1$ and define the ...
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vote
3answers
138 views

A manifold being orientable vs oriented.

I read that an oriented manifold is a manifold with a choice of orientations for each tangent space so that for $p \in M$, there is an open set $U$ and a collection of vector fields $X_1,...,X_n$ so ...
5
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0answers
196 views

Examples of special sphere bundles

I'm interested in examples of sphere bundles which do not arise from vector bundles. I'm not quite clear about the following. So please let me know if anything is false. I believe that a ...
3
votes
1answer
333 views

Splitting of the tangent bundle of a vector bundle

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. I'm interested in a splitting of $i^*(TE)$, the restriction of the ...
2
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1answer
83 views

Question about differential of embedding

For any $C^{\infty}$ manifold $M$, the tangent bundle $TM$ of $M$ is also a $C^{\infty}$ manifold. Hence we can think about the differential $df:TM\rightarrow TN$ of maps $f:M\rightarrow N$ between ...
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1answer
96 views

Regular value of $g \circ f$ is a regular value of $g$

Given smooth maps $f: X \to Y, g: Y \to Z$, where $X, Y, Z$ are boundaryless, compact manifolds of dimension $n$, is the statement in the title true?
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1answer
249 views

De Rham cohomology question

I'm trying to compute a certain DeRham cohomology. Consider $M = S^n-C$, where $C$ is the disjoint union of closed disks $C = \cup_{i=1}^m D_i$, and $m,n \geq 1$. How can we compute the cohomology ...
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1answer
136 views

A “Manifold with Boundary” Question

I'm given the following scenario: Letting $U$ be an open subset of $\mathbb{R}^n$ and $f,g:U\rightarrow \mathbb{R}$, two smooth functions such that $f(\vec x)\lt g(\vec x)$ for all $\vec x\in U$, ...