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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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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260 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 ...
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116 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
43 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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211 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
168 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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66 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 ...
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126 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
167 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
102 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 ...
9
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232 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 ...
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106 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
421 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
90 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
190 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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164 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
88 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 ...
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380 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
182 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 ...
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495 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
68 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
247 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
302 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
193 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
222 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
62 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 ...
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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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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
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2answers
793 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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3answers
135 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 ...
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190 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
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1answer
298 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
79 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
93 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?
0
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1answer
236 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
135 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 $\vec x\in U$, ...
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138 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 ...
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1answer
110 views

Diffeomorphism between a triangle and a square?

Is there always a diffeomorphism between $(0,1)^2$ and any given (not degenerate) triangle?
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1answer
526 views

Problem 3-38 in Spivak´s Calculus on Manifolds

This is not homework. Problem 3-38 reads: Let $A_{n}$ be a closed set contained in $(n,n+1)$. Suppose that $f:\mathbb{R}\rightarrow \mathbb{R}$ satisfies $\int_{A_{n}}f=(-1)^{n}/n$ and $f(x)=0$ for ...
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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 ...
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200 views

Do we need additional assumptions for problem 3-37 (b) in Spivak´s calculus on Manifolds?

Problem 3-37 (b) reads: Let $A_{n}=[1-1/2^{n},1-1/2^{n+1}]$. Suppose that $f:(0,1)\rightarrow \mathbb{R}$ satisfies $\int_{A_{n}}f=(-1)^{n}/n$ and $f(x)=0$ for any $x\notin$ any $A_{n}$. Show that ...
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267 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 ...
3
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1answer
98 views

Suspension Operation on the Pontryagin-Thom Construction

I have a feeling that this is well-known: View the Pontryagin-Thom construction as the bijective correspondence between $[M,S^r]$ and the set of (appropriate equivalence classes of) framed ...
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3answers
135 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.
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357 views

Solving problem 3-29 in Spivak´s Calculus on Manifolds without using change of variables

Problem 3-29 (p. 61) in the section treating Fubini´s theorem reads: Use Fubini´s theorem to derive an expression for the volume of a set of $\mathbb{R}^{3}$ obtained by revolving a Jordan-measurable ...
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2answers
265 views

is the fixed set of a smooth involution a submanifold?

Let $f:X\rightarrow X$ be a smooth map of a smooth manifold with $f^2=\operatorname{id}$. Is the subset $\{x\in X\mid f(x)=x\}$ a smooth submanifold? I tried to find an argument with the implicit ...