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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Extending metrics

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. Then we have a splitting of the restriction of $TE$ to the ...
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Conditions for a projection of a Knot to be a Knot diagram.

friends. I'm working on a problem, the broad scope of which is to show that given a map $f:S^1\rightarrow \mathbb{R}^3$ be a smooth embedding, and a projection map $\pi_v:S^2\rightarrow P_v$, where ...
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216 views

How to check whether a vector field is Morse-Smale?

Setup and notation: Let $f:M\to \mathbb{R}$ be a Morse-function on the compact $m$-dimensional manifold $M$ and let $X$ be a gradient-like vector field for the function $f$. Denote the unstable ...
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353 views

Tangent space to a product

Can you explain this question explicitly. This is a little bit difficult for me, but I want to learn how to solve. Thank you for help. If $M$ and $N$ are manifolds, let $\pi_1:M\times N\to M$ and ...
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159 views

How does degree theory imply that this mapping $f$ is locally onto?

Let $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ be a smooth vector field ($\mathcal{C}^1$ mapping). Let $0$ be a critical point of $f$, i.e. $H f(0) = 0$. Assume that the index of $f$ at $0$ is ...
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131 views

Why the tangent bundle of a smooth manifold is an oriented manifold?

I need help with the following question. I am not sure how to begin. Any help will be appreciated. Thank you! For any smooth manifold $M,$ the tangent bundle $TM$ is an oriented manifold.
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Discretizing continuous surfaces into semi-regular polygons

I am aware that there have been many works on the problem of discretizing a surface into polygons, however, I wonder if in any work the problem of doing so to get polygons with edges of the same ...
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107 views

Approach topological manifolds with smooth manifolds

Because I'm doing some problems that consider all the manifolds while the situation is really clear when considering only smooth manifolds. Thus my question is can we always appoint a topological ...
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78 views

Suspension of $\mathbb{R}P^2$ Contractible?

Is the suspension of $\mathbb{R}P^2$ Contractible? And if it is, How would you prove it. Thank you!
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147 views

Differential Topology Question on Complex Projective Space

This question seems like it would be very hard to do directly. I wouldn't know where to begin. I was wondering if anyone had a very slick proof of this. The only thing I think is easy is that its ...
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35 views

Matrix Manifolds Question

I am not sure at all how to do the following question. Any help is appreciated. Thank you. Consider $SL_n \mathbb{R}$ as a group and as a topological space with the topology induced from $R^{n^2}$. ...
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Question Concerning the classification of 1-manifolds

I am having trouble proving the following statement used in proving the classification of 1-manifolds. Any help would be great. Thank you. Let $L$ be a subset of $X$ diffeomorphic to an open ...
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277 views

A covering map from a differentiable manifold

Let $p: C \to X$ is a covering map. Suppose that $C$ is a differentiable manifold. Is X - differentiable manifold? More precisely, I am interested in the case where $C$ is Submanifold of Lie algebra, ...
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283 views

Lie Groups induce Lie Algebra homomorphisms

I am having a difficult time showing that if $\phi: G \rightarrow H$ is a Lie group homomorphism, then $d\phi: \mathfrak{g} \rightarrow \mathfrak{h}$ satisfies the property that for any $X, Y \in ...
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161 views

Showing that the exponential map $\mathrm{exp}:\mathfrak{sl}(2,\mathbb{R})\to\mathrm{SL}(2,\mathbb{R})$ is not surjective

I am having a difficult time showing that the exponential map $\mathrm{exp}: \mathfrak{sl}(2, \mathbb{R}) \rightarrow \mathrm{SL}(2, \mathbb{R})$ is not surjective. I have, however, worked out that ...
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295 views

Exponential map and the special orthogonal group

I need to show that the map exp$: \mathfrak{so}(2) \rightarrow SO(2)$ is surjective. I already have that $\mathfrak{so}(2) = \{A \in M(2, \mathbb{R}) \ | \ A^T + A = 0\}$ and the map is given by ...
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73 views

Vector Fields Question 4

I am struggling with the following question: Prove that any left invariant vector field on a Lie group is complete. Any help would be great!
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129 views

Another question on Orientation Preserving Maps

I am stuck on the following question. Sorry about the bad latex skills. Not sure what went wrong. Is the map $f:S^n \rightarrow S^n$ orientation preserving? I constructed an atlas on $S^n$ ...
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265 views

Is manifold mapping degree equal to algebraic degree for polynomials?

If $M$ and $N$ are oriented $n$-manifolds and $f: M \to N$ then the degree of $f$ is given by $$ \deg f = \sum_{p \in f^{-1}(q)} sign_p f $$ where $q$ is a regular value and the sign is $+1$ if $f$ is ...
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96 views

Question about Lie Groups

I am having trouble with the following Lie Algebra question. I will appreciate any help greatly. Any Lie group homomorphism $\phi : G \rightarrow H$ is determined by the induced Lie algebra ...
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137 views

Relations between elliptic curves and topological quantum field theory

I heard that there are relations between elliptic curves and topological quantum field theory (TQFT). I googled and found that something called "elliptic genus" might be the key word to relate these ...
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84 views

Smoothing corners of a handle attachment

Say we attach a $\lambda$-handle, $\mathbb{D}^\lambda \times\mathbb{D}^{\mu}$, to a smooth manifold $M, \partial M$ by simply taking the quotient $M \cup_h \mathbb{D}^\lambda \times\mathbb{D}^{\mu}$ ...
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46 views

Topological degree of a map with finite energy

Suppose that $\phi:\mathbb{R}^3 \to S^2$ is of class $\mathscr{C}^1(\mathbb{R}^3\setminus \left\{a\right\}) \cap \mathscr{C}^0(\mathbb{R}^3\setminus \left\{a\right\})$, that is $\phi$ might have a ...
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140 views

On the smooth structure of $\mathbb{R}P^n$ in Milnor's book on characteristic classes.

This question concerns problem 1-B in the book of Milnor and Stasheff part a. They first define the set $F:=\{f:\mathbb{R}P^n\rightarrow\mathbb{R} \mid \text{$f\circ q$ is smooth}\}$ where ...
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smoothness structure on a set

I'm reading Milnor's 'characteristic classes', and in the first chapter he defines smoothness structure on a set M, which is confusing for me, as what follows:(these are not Milnor's words so please ...
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348 views

Prove not a violation of Stokes theorem

The question is as follows: Define the vector field ${\bf F}$ on the complement of the $z$-axis by $${\bf F}(x,y,z)= \frac{-y{\bf i} +x{\bf j}}{x^{2}+y^{2}}.$$ i) Show that ...
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1answer
244 views

Stiefel-Whitney numbers for product bundle

I'm reading Milnor's "characteristic classes" and I want to compute Stiefel-Whitney numbers of $ P^2 \times P^2 $ (product of projective spaces) for one of the problems, I know how Stiefel-Whitney ...
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268 views

Different definitions of handle attachment

This is an extremely technical question about handle attachments.  I am asking why two definitions are equivalent.  My question appears in the second to last paragraph after I've described the two ...
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516 views

Proof that cone not diffeomorphic to plane

What is the simplest way to show that the cone $\{(x,y,z)\in\mathbb R^3\,|\,z=\sqrt{x^2 + y^2}, z\geq 0\}$ is not diffeomorphic to $\mathbb R^2$? After some comments, I realize that this question The ...
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114 views

Does a diffeomorphism between the interiors of two manifolds extend to the manifolds?

Let $M,N$ be manifolds with boundary. Let $f:\mathring{M}\to \mathring{N}$ be a diffeomorphism of their interiors. Does it extend to a diffeomorphism $M\to N$? I suppose we can look at the problem ...
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178 views

Extending a $C^2$-function from a $C^{1,1}$-curve to some neighbourhood

Suppose I have a simple, compact $C^{1,1}$-curve $L$ in $\mathbb{R}^3$ and a $C^2$-function $f$ on it ($C^2$ meaning with two continuous arclength derivatives). Can it be extended to a $C^2$-function ...
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243 views

Torus with positive sectional curvature.

There was this question, whether a torus in dimension n, $T^n$, can carry a riemannian metric with positive sectional curvature. A read a proof, which goes as follows: $T^n$ is complete, because ...
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Is this a trivial Stokes exercise?

I'm working on this exercise from an old Spivak Differential Geometry book which I must be misunderstanding. The question reads: Let $M$ and $N$ be compact $n$-dimensional manifolds with boundary and ...
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246 views

Immersing punctured torus

I am looking for a proof (as elementary as possible) of the fact that punctured n-dimensional torus admits an immersion to $\mathbb{R}^n$. The 2-dimensional case seems to be evident, but I haven't got ...
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Show that $\Gamma(TM) \cong_{\mathbb{R}} \mathsf{Der}_{\mathbb{R}} C^{\infty}(M)$

Let $M$ be a smooth real manifold. I want to show that we have an isomorphism of real vector space $\Gamma(TM)$ of all smooth sections of $TM$ (i.e. of vector fields on $M$) and of real vector space ...
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Lebesgue Measure with given function?

Suppose $E$ subset $R$ ($R $\is real numbers) where $E$ is Lebesgue measurable, and $f:E\to R$ and defined $g: R\to R$ by \begin{equation*} g(x) = \begin{cases} f(x) & x \in E \\ 0 & x ...
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set of immersions on a manifold is an open set in the set of all mappings to $\mathbb{R}^{2m+1}$

I'm studying a proof of Takens' Delay Embedding Theorem. A key fact used is that the set of immersions is open in the set of all mappings (mappings from an $m$-dimensional manifold M to ...
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67 views

Fundamental group of the following disc

What is the fundamental group of the following space in $\mathbf C^n$? This is the topological space given by $$\{(x_1,\ldots,x_n)\in \mathbf C^n-\{0\} : \vert x_1\vert < 1, \ldots, \vert x_n\vert ...
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80 views

Gluing two copies of $S^3 \times D^4$ along their boundary

I am trying to visualize a seven sphere here: http://www.youtube.com/watch?v=II-maE5HEj0. What is $(u,v)\to(u, u^hvu^j)$ in the attached picture? Are quaternion maps commonly used to glue ...
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187 views

Classification of all 28 Exotic 7-Spheres

In http://en.wikipedia.org/wiki/Exotic_sphere#Explicit_examples_of_exotic_spheres Wikipedia says "As shown by Egbert Brieskorn (1966, 1966b) (see also (Hirzebruch & Mayer 1968)) the intersection ...
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65 views

Convert line integral between different metrics?

If I have $$ \int\limits_0^T \frac{\sqrt{\dot{x}(t)^2+\dot{y}(t)^2}}{\sqrt{2 y(t)}}dt $$ I can convert this problem of finding the solution to the brachistochrone problem to a geometric problem by ...
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108 views

What is the name of the dual of vector fields?

It's all in the title : is there a standard name for the dual vector space of the space of vector fields (on a given manifold) ? I am not speaking about the dual bundle of the tangent bundle, which is ...
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What exactly is a Kähler Manifold?

Please scroll down to the bold subheaded section called Exact questions if you are too bored to read through the whole thing. I am a physics undergrad, trying to carry out some research work on ...
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Non-degenerate smooth functions on a manifold

I am trying to prepare a presentation on "the use of differential geometry in the theory of critical points", but only the case where there is a single variable. (only in dimension 1), and i ask ...
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No Smooth Onto Map from Circle to Torus

My professor was lecturing today and he made this statement which I was unable to verify. (I worded it nicer) There is no map which is both smooth and onto from $S^1$ to $S^1$$\times$ $S^1$. When ...
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Show that we can define a connection on any manifold using partitions of unity

Suppose that $(U,\varphi)$ is a chart on manifold $M$, and $X,V$ are vector fields on manifold $M$, then we can write: $$X=\sum_{i=1}^{i=n}X^{i}\frac{\partial}{\partial x^{i}}$$ on $U$, and define a ...
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smoothness of hopf fibration projection with respect to standard differentiable structure on unit sphere

We know that steographic projection defines a differentiable structure on $S^n$ by sending points on $S^n$ to hyperplane $\{x^{n+1}\}=0$. In fact, stereographic projection $\sigma_P: S^n- P\to R^n $ ...
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314 views

How does Thurston's geometrisation conjecture imply Poincaré's conjecture?

I ran into the geometrisation conjecture a few days ago, and I started wondering how to prove Poincaré's conjecture. Let $M$ be a compact, simply connected, $3$-manifold. Clearly it is irreducible ...
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Orientation on finite dimensional vector spaces over finite fields.

For finite-dimensional $\mathbb R$-vector spaces, we define an orientation to be an equivalence class of ordered bases, where $B_1 \sim B_2$ iff the change of basis matrix $A$ taking $B_{1}$ to ...
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Tangent spaces of compact spaces

In a recent discussion of tangent spaces, it was noted that tangent spaces to a manifold are not compact because by definition they are vector spaces. I was curious as to whether tangent spaces to ...