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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What is a regular homotopy?

The definition of regular homotopy from Wikipedia says that two immersion $f,g:M\to N$ are regularly homotopic if they represent points in the same path-component of $\text{Imm}(M,N)$. What does ...
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80 views

Is it possible for a manifold to have a normal vector that is zero everywhere, if so, would this indicate that the manifold is non-orientable?

Basically I've been thinking about defining a non-orientable three-dimensional metric space via defining the normal vector and looking to see if there is two possible vectors for the same point. I'm ...
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59 views

Find the Poincare Dual of a ray in $\mathbb{R}^2-\{0\}$

This is example-exercise 5.16 in Bott and Tu (which I'm independently reading through.) The problem states: Let $M=\mathbb{R}^2-\{0\}$, and $X\subseteq M$ be the closed submanifold ...
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1answer
46 views

What is the definition of a properly-immersed arc?

I am going over a paper in which a theorem includes a condition that certain arcs are "properly-immersed geodesic arcs" in a surface $\Sigma$ with nonempty boundary. I have done a search, but I have ...
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92 views

Morse function on $\mathrm{SO}(n)$

I would like to prove that the following function is a Morse function, $F : \mathrm{SO}(n)\to\mathbb{R}$ $$A=(a_{ij})\mapsto\sum_{i=1}^na_{ii}\lambda_i$$ with $0<\lambda_1<...<\lambda_n$. ...
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1answer
56 views

Euler characteristic of the closed unit ball

I would like to calculate the Euler-Poincaré characteristic of the closed unit ball $B$ by the de Rham cohomology, the Poincaré-Hopf theorem and Morse theory. de Rham cohomology: since $B$ is ...
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122 views

Applying the Frobenius theorem to a decomposable 2-form

So I have the following problem: Suppose $\omega=\phi \wedge \theta$ is a closed decomposable 2-form on $M$ a manifold (decomposable just means it can be written as a wedge of 1-forms). Suppose $p\in ...
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60 views

Proof Transverse Submanifolds

Let $f:M\rightarrow\mathbb{R}^p$ differentiable and $N$ submanifold of $\mathbb{R}^p$. Show that for all $\epsilon>0$ exist $v\in\mathbb{R}^p$ whit $||v||<\epsilon$ such that $f(x)=f(x)+v$ is ...
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73 views

Completeness implies geodesic completeness, a more conceptual way?

We know from Riemannian geometry that for Riemannian manifolds, completeness and geodesic completeness are equivalent, which is usually a consequence of Hopf-Rinow theorem. However, I'm considering a ...
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72 views

Actual Classification re Nielsen-Thurston Theorem (how to)?

according to Nielsen -Thurston Classification: http://en.wikipedia.org/wiki/Nielsen%E2%80%93Thurston_classification If $S$ is compact and orientable surface, then any homeomorphism is isotopic to ...
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56 views

mutually transverse embedded submanifolds, natural bundle surjections, direct sum, isomorphism

Let $N$ be a manifold and let $M_1, \dots, M_n \hookrightarrow N$ be mutually transverse embedded submanifolds, so $M = \cap M_i$ is an embedded submanifold of $N$ with $\text{T}_m(M) = \cap T_m(M_i)$ ...
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132 views

A non-vanishing one form on a manifold of arbitrary dimension

So the problem I have is: Let $\theta$ be a closed 1-form on a compact Manifold M without boundary. Further suppose that $\theta \neq 0$ at each point of M. Prove that $H^{1}_{dR}(M)\neq 0$. The ...
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42 views

A diffeomorphism with $\det(df)>0$ and convexity of matrices

Let $f: \mathbb{R}^{n}\rightarrow \mathbb{R}^n$ be a diffeomorphism such that $\det(df)>0$ at the origin (hence everywhere). Show that there is a continuous map $F: [0,1]\times \mathbb{R}^{n} ...
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1answer
116 views

Laplace-de Rham operator

Consider an operator $\partial = (-1)^k \star^{-1}\,d\star: \Omega^k(\mathbb{R}^n) \to \Omega^{k-1}(\mathbb{R}^n)$. Note that we equivalently can write $\partial = (-1)^{nk + n + 1} \star\,d\star$. ...
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1answer
28 views

The Projection $S^{2}\times S^{3} \rightarrow (S^{2}\times S^{3})/(S^{2} \lor S^{3})$

So a problem I have come across involves constructing some smooth degree 1 maps, and one such map to be constructed is a map $S^{2}\times S^{3} \rightarrow S^{5}$. I've been told that there is such a ...
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137 views

Prove that $TTM$? is a orientable bundle on $TM$

I know that $TM$ is always orientable bundle. But what is $TTM$? How try to prove that $TTM$ is a orientable bundle on $TM$.
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95 views

Properly discontinuous action: equivalent definitions

Let us define a properly discontinuous action of a group $G$ on a topological space $X$ as an action such that every $x \in X$ has a neighborhood $U$ such that $gU \cap U \neq \emptyset$ implies $g = ...
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72 views

Do unit normal vectors of extremal points on a compact hypersurface have opposite directions?

Problem To be definite, by a hypersurface of $\mathbb R^{n+1}$, I mean a connected submanifold of $\mathbb R^{n+1}$ of dimension $n$ without boundary. Suppose $M$ is a compact smooth orientable ...
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1answer
52 views

Homotopy class of maps to a complex projective space

Let $M$ be a closed oriented smooth 4-manifold. Denote by $[M, \mathbb{C}P^{\infty}]$ homotopy classes of continuous maps from $M$ to $\mathbb{C}P^{\infty}$. I would like to know how to show $$ [M, ...
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38 views

What is the boundary of tubular neighborhood of a compact manifold?

What is the boundary of tubular neighborhood of a compact manifold? for example what is the boundary of tubular neighborhood of a torus $\mathbb{T}^2$ in $\mathbb{R}^3$ or a torus $\mathbb{T}^2$ in ...
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54 views

Two definitions of embedded submanifold

Let $N$ be a smooth manifold. One possible definition (I believe) for an embedded submanifold of $N$ is some $M \subset N$ that is a (smooth) manifold such that the inclusion $i : M \hookrightarrow N$ ...
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1answer
61 views

Reference to finite coverings causing injections on deRham cohomology

So, I've heard that if you have a finite degree covering of a compact connected manifold by another compact connected manifold of dimension $n$ (So $\pi :M \rightarrow N$) gives an injection on the ...
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2answers
64 views

When are the exterior derivative and contraction of forms inverses?

I am trying to get a better feel for both the exterior derivative of a form and the contraction of a form by a vector field $X$. Basically, when are these inverses? If I have a one-form $\omega$ and ...
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74 views

Universal Cover of a Surface with Boundary. What does Cantor set on Boundary Correspond to?

I am trying to understand in more detail the answer to: Universal Cover of a Surface (with Boundary) It is mentioned that the universal cover of a hyperbolic surface $S$ with geodesic boundary is a ...
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1answer
37 views

Geometric Intuition for “Right-Veering” Property of $f$ in MCG(S)?

let $S$ be a compact surface with non-empty boundary, let $\alpha : [0,1] \rightarrow S$ be a Properly-embedded arc (meaning both endpoints of the arc are in $\partial S$) and let $f$ be an element ...
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group operations are smooth in $\text{SL}(n, \mathbb{R})$

I am told the following reason as to why group operations of multiplication and inversion are smooth on $\text{SL}(n, \mathbb{R})$. Multiplication is smooth because the matrix entries of a product ...
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2answers
89 views

Immersion of $\mathbb{R}$ to $\mathbb{R}^2$

I have no idea how to prove that the set $\{(x, |x|): x\in\mathbb{R}\}$ is not the image of an immersion of $\mathbb{R}$ into $\mathbb{R}^2$ For example If $f(t)=(t^3, |t|^3)$ then $f'(0)=(0, 0)$.
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40 views

defining $C^\infty$ structure on finite-dim vector space, homeomeomorphism to tangent bundle, such that independent of choice of bases

If $V$ is a finite dimensional vector space over $\mathbb{R}$, how would I go about defining a $C^\infty$ structure on $V$ and a homeomorphism from $V \times V$ to $TV$ which is independent of bases? ...
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parallelizable manifolds

I know a differentiable manifold $M$ of dimension $n$ is parallelizable if there exist (smooth of course) vector fields $\{X_i\}_{j=1}^n$ which are linearly independent in $T_pM$ at each point $p \in ...
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28 views

something about diffeomorphism

Suppose $A$ and $B$ are both open sets, and there is a diffeomorphism $g$ between them. My book says that the chain rule implies that $Dg$ is non-singular. I don't understand. Can anyone tell my why?
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Is every self-homeomorphism homotopic to a diffeomorphism?

Given a smooth manifold $M$, is every homeomorphism $M \to M$ homotopic to a diffeomorphism? Hirsch's "Differential Topology" has a proof that every $C^1$ diffeomorphism of smooth manifolds is ...
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Manifold Orientability Definition

In Shigeyuki Morita's Geometry of Differential Forms, orientability is defined in the following way: If we can assign an orientation to each point on a manifold $M$ in such a way that the ...
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Is the Whitney embedding theorem tight for all $n$?

Whitney's embedding theorem states that any smooth $n$-manifold $M$ can be smoothly embedded into $\Bbb R^{2n}$. In dimension 1 this is tight: the circle cannot be embedded into $\Bbb R^1$. It is a ...
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1answer
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identity map is not diffeomorphism, $x^3$ is a diffeomorphism [closed]

Consider the real line $\mathbb{R}$ the two following differentiable structures: 1) $(\mathbb{R}, f_1)$ where $f_1(x) = x$. 2) $(\mathbb{R}, f_2)$, where $f_2(x) = x^3$. How do I demonstrate that: ...
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31 views

Index of a non-degenerate critical point of a smooth holomorphic function

I am looking for hints, and ideally for several ways to approach the following problem. Consider a holomorphic function $\mathbf C^n \to \mathbf C$, that is smooth (we view $\mathbf C^n$ as a ...
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1answer
46 views

What can you say about injection, immersion, embedding for the torus?

Define $\varphi_a: \mathbb{R} \to T$ where $T = S^1 \times S^1$ is the torus via$$\varphi_a(x) = (e(x), e(ax)),\text{ }e(x):=e^{2\pi i x}$$and $a > 0$ is some parameter. Determine for which $a$ ...
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26 views

Possibility of the cellular decomposition of a manifold

I am working to find if it's possible to find a cellular decomposition of $S^2\times S^1$ as following: $e^0\cup e^1\cup e_1^2\cup e_2^2\cup e^3$. I cannot find such a decomposition. And I try to ...
3
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1answer
87 views

Non homeomorphism

I want to show that the sphere $S^2$ and the torus $T^2$ are not homeomorphic, using the notion of intersection modulo $2$. I have to show that any two loops on the sphere $S^2$ have an even number of ...
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1answer
49 views

Smooth function on intersection is the difference of two smooth functions

I am trying to understand a proof from Loring W. Tu's An introduction to Manifolds. In order to prove Proposition 26.2, The author must show that if $\{U, V\}$ is a open cover of a manifold $M$ and ...
2
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1answer
43 views

definition of differential does not depend on the choice of the curve, differential is a linear map

Let $\varphi: M \to N$ be a differentiable map. How do I show that the definition of the differential $d\varphi_p: T_pM \to T_{\varphi(p)}N$ of $\varphi$ at $p$ does not depend on the choice of the ...
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1answer
62 views

Every immersion is locally an embedding?

Let $\varphi: M \to N$ be an immersion and let $p$ be a point in $M$. How would I go about showing that there exists a neighborhood $V \subset M$ of $p$ such that the restriction $\varphi|V$ of ...
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1answer
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Let ($\mathbb{R}^n$,*) be a division algebra over $\mathbb{R}$, $n>1$, $\Rightarrow n$ is even

I want to prove: Let ($\mathbb{R}^n$,*) be a division algebra over $\mathbb{R}$, $n>1$, $\Rightarrow n$ is even. I'm stuck. My thoughts are: I want to use the hairy ball theorem and I want to ...
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27 views

Alexander Method for non-orientable Surface

From Farb and Margalit,Primer on Mapping Class Group, p.62, we know For example for torus with four puncture we can choose following claret red curves. Curves are choosen such that their complement ...
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1answer
35 views

no point where assumptions of Inverse Function Theorem hold?

Let $\varphi: \mathbb{R}^3 \to \mathbb{R}$, $\psi: \mathbb{R}^3 \to \mathbb{R}$ are continuously differentiable. Define $f: \mathbb{R}^3 \to \mathbb{R}^3$ by$$f({\bf x}) = (\varphi({\bf x}), \psi({\bf ...
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1answer
130 views

Differential forms, number of zeros on disk

Let $f$ be a holomorphic function on $U \subset \mathbb{C}$ and $f'(z) \neq 0$, $z \in U$. Then how do I show that all zeros of $f$ are simple and positive, and that if I have a disk $D \subset U$ ...
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0answers
63 views

Pullback of a 1 form on the circle

Q: Let $M$ be a smooth compact manifold, and suppose there is a smooth map $F:M \rightarrow S^{1}$ whose derivative is non-zero at every point. Prove that the de Rham cohomology space $H^{1}(M)$ is ...
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1answer
70 views

Eembedding of product $\mathbb{S}^2\times\mathbb{S}^3$ into $\mathbb{R}^6$

It is easy to see that $\mathbb{S}^n$ can be embedded in $\mathbb{R}^{n+1}$ and therefore $\mathbb{S}^2\times\mathbb{S}^3$ can be embedded in $\mathbb{R}^7$. The question is how to prove that ...
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1answer
91 views

How does the Möbius group act on circlines?

This is a continuation of my earlier, rather vague question. I am interested in studying the action of the Möbius group $PGL(2,\mathbb{C})$, on the circlines in the extended complex plane $\mathbb{C} ...
3
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1answer
60 views

How do you work with the space of circles on the sphere considered as the projective line?

I'm trying to prove some things about the action of the Möbius group on the "circlines" in the extended complex plane, ie. circles on $\mathbb{C}P^1$. I find that while I have a good grip on Möbius ...
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30 views

Prove that in the half plane $\{x>0\}$, ω is the differential of a function.

Define a 1-form $ω$ on the punctured plane $R^2-\{0\}$ by $ω(x,y)=\frac{-y}{x^2+y^2}dx+\frac{x}{x^2+y^2 }dy $ a) Calculate $∫_Cω$ for any circle C of radius r around the origin b) Prove that in ...