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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Picture behind $SO(3)/SO(2)\simeq S^2$

Is there some kind of intuitive/waving hand argument to explain that $$SO(3)/SO(2) \simeq S^2 \; ?$$
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22 views

define distance in a manifold over the reals

G is a Hausdorff manifold over the reals with a finite atlas: $\exists m$ $G=\bigcup_{1 \leq i \leq n}U_i$, $g_i:U_i \to g_i(U_i) \subseteq \mathbb{R}^m$. Can I somehow define a metric inside G, ...
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18 views

Atlas on a smooth manifold that contains 2 charts

How can I show that if an atlas on a smooth manifold has exactly 2 charts then it is orientable? How do I make sure that the Jacobian of the transition map is positive?
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14 views

Subsets of immersions that are embeddings

Let $X$ be a manifold and let $Y$ be a submanifold, possibly with boundary. I am dealing with a situation where $f:X\to \mathbb{R^3}$ is an immersion, but $f\vert_Y:Y\to \mathbb{R}^3$ is an embedding. ...
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10 views

problem on sheaves

I want to show the following. Suppose $X$ is a smooth manifold and F,G are sheaves of $C^{\infty}_{X}$-modules, then the natural map $Hom(F,G)\to Hom(F(X),G(X))$ is injective. It's easy to see that ...
8
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1answer
118 views

Let $A$, $B$ be subsets of $S^n, n≥2$. Show that if $A$ and $B$ are closed, disjoint, and neither separates $S^n$, then…

Let $A,B$ be subset of $S^n$, $n\geq 2$. Show that if $A$ and $B$ are closed, disjoint, and neither separates $S^n$, then $A\cup B$ does not separate $S^n$. I've thought to do it by contradiction and ...
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1answer
32 views

Intuition regarding the Whitney trick

I read here that a major ingredient in Whitney's strong embedding theorem and later Smale's celebrated h-cobordism theorem is the Whitney trick. Can someone give an intuitive description of the ...
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12 views

Calabi homomorphism of the disk

There is a fact that the homomorphism $Diff_0^{\infty}(\mathbb{D},\partial\mathbb{D},area)\to \mathbb{R}$ is surjective, we can use Calabi homomorphism to prove it, where ...
2
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61 views

Differential of a Sobolev map between manifolds

Let $\Sigma, M$ be compact Riemannian manifolds. By embedding $M$ isometrically into $\mathbb{R}^N$, one can define the Sobolev spaces $W^{k,p}(\Sigma, M)$ by $$W^{k,p}(\Sigma,M) = \{ u \in ...
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18 views

Transversality of Subbundles

It is known that transversality of submanifolds is generic in the sense that two submanifolds could be made transversal by small perturbations. I was wondering if the same is true for subbundles of ...
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27 views

Sequence of critical values has no cluster point (Milnor, Morse Theory)

The following Claim is used in the proof of Theorem 3.5 in John Milnor's "Morse Theory": Claim: Let $f: M \rightarrow \mathbb{R}$ be a differentiable function on a manifold $M$ with no degenerate ...
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1answer
45 views

Symplectic submanifolds in $\mathbb{R}^{4}$

Which symplectic submanifolds can be realized in $\mathbb{R}^{4}$? It easy to show that such submanifolds aren't compact. So, they are spheres with some handles and holes. Which relations between the ...
2
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1answer
23 views

Tangent bundle of $S^1$ is diffeomorphic to the cylinder $S^1\times\Bbb{R}$

How do I construct an explicit diffeomorphism between $TS^1$ and $S^1\times\Bbb{R}$? It will be something like $\phi:TS^1\to S^1\times\Bbb{R}, (x,v)\to(x,...)$. Also we know that for $x=(x_1,x_2)$ ...
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23 views

Real analytic manifold

I found in some lecture notes such a definition of real analytic manifold: Let $X$ be a complex manifold (ringed space, locally isomorpic to...), $i: X \rightarrow X$ - a conjugation ...
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1answer
52 views

Doubt regarding proof from Milnor's notes

I was reading Milnor's 1958 notes in differential topology and came across the following theorem Theorem: Let $U$ be an open set in $\mathbb{R}^n$ and let $f;U \to \mathbb{R}^p$ be differentiable, ...
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1answer
40 views

About “good” covers of manifolds

Let $M$ be compact manifold of dimension $n$. I am wondering if a certain type of covering of M by coordinate charts exists (this is not the usual good covering theorem but seems related to it). I ...
2
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45 views

Are there any results linking hyperbolic operators and Pseudo-Riemaniann geometry?

In the case of a $n$-dimensional pseudo-riemannian manifold one has a symmetric bilinear form, $h$, with signature $(1,n)$ and the analogous operator to the Laplace -Beltrami operator,$\Delta$, is the ...
2
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17 views

Tangent space to the graph of a function

$X$ and $Y$ are smooth manifolds and let $f:X\to Y$ be a map. Let $\Gamma$ be the graph of $f$ in $X\times Y$. Prove that $T_{(x,y)}\Gamma$ is the graph of $df(x):T_xX\to T_yY$. $\Gamma$ need not be ...
17
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2answers
372 views

No hypersurface with odd Euler characteristic

Here is a classic problem which I encountered and could not solve: Prove that a simply connected closed smooth manifold has no closed sub-manifold of co-dimension $1$ with odd Euler ...
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1answer
217 views

Restriction of a differentiable map $R^3\rightarrow R^3$ to a regular surface is also differentiable.

This is again an excercise from Do Carmo's book. Prove: if $f:R^3 \rightarrow R^3$ is a linear map and $S \subset R^3$ is a regular surface invariant under $L,$ i.e, $L(S)\subset S$, then the ...
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36 views

Finding the critical points of a map on the torus

Consider the $2$-torus $T$ obtained by revolving about the $z$-axis the circle $(x-2)^2+z^2=1$. I want to find the critical points of the map $f:T\to\Bbb{R}$ defined by $f(x,y,z)=x$. The equation of ...
2
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1answer
29 views

Proving that a regular value of a smooth function isn't in the boundary of the counter-domain

Suppose $X$ is a manifold without boundary and $Y$ is a manifold. Suppose there is a smooth function $f: X \rightarrow Y$ and we are given a $y \in Y$ such that $y$ is a regular value of $f$ and ...
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1answer
23 views

Definition of smooth maps between manifolds

Here is a page from Guillemin-Pollack's differential topology: My question is: At the bottom he defines $df=d\psi\circ dh\circ d\phi^{-1}$. Why doesn't he just define $df=dh$, like here: ...
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24 views

Upper-half space of a manifold with boundary

Suppose that $X$ is a manifold with boundary and suppose that $x$ is a boundary point. Define the upper half space $H_x(X)$ in $T_x(X)$ to be the image of $\mathbb{H^k}$ under $d \phi_0:\mathbb{R^k} ...
3
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1answer
36 views

submanifold of Euclidean space is oriented if and only if normal bundle is an oriented vector bundle.

Let $f:M\longrightarrow \mathbb{R}^{n+k}$ be an immersion of $n$-dimensional manifold $M$ into $\mathbb{R}^n$. Let $\nu(M)$ be the normal bundle of $M$. Prove that $M$ is oriented if and only if ...
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14 views

please tell me what does $C^{0}(M)$ means in quasi states on symplectic manifold $M$

in my project reference paper on Quasi-states on a symplectic Manifold $M$ it is written as $C^{0}(M)$ and $C^{\infty}(M)$ . as i know $C^{\infty}(M)$ is space of all infinitely differentiable ...
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23 views

History of vectorial bundles in articles or papers?

I'm looking for an article or book that gives a thorough and interesting history of bundles and vectorial bundles in algebraic topology. I'm looking for it for my own learning, please help its ...
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24 views

Tangent space of the tangent bundle 2

This is a continuation of the problem given in Tangent space of the tangent bundle which I repeat: Let $M$ be a differential $k-$manifold in $\mathbb R^n$, $g:TM\rightarrow\mathbb R$ given by ...
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1answer
39 views

Regular values and manifolds with boundary

Question: Let $X^m$ and $Y^n$ differentiable manifolds. $f:X\rightarrow Y$ a differentiable map. Show that if $\partial X=\emptyset$, $y\in Reg(f)$ and $f^{-1}(y)\neq\emptyset$, then $y\not\in\partial ...
2
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1answer
40 views

Definition for Euler characteristic without CW-complexes

It is possible to have a definition of the Euler characteristic without using CW-complexes? (I'm referring to the definition given by Wikipedia : ...
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2answers
328 views

Composition of smooth maps

From section 1, problem 3 of Differential Topology by Guillemin and Pollack: Let $X \subset R^N, Y \subset R^M, Z \subset R^L$ be arbitrary subsets, and let $f : X \to Y, g : Y \to Z$ be smooth ...
2
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1answer
40 views

measure on non-oriented Riemannian manifold

Let $M$ be a non-oriented Riemannian manifold of dimension $m$. Nash embedding theorem implies that there exists an isometric embedding $\phi: M\longrightarrow \mathbb{R}^n$ for $n$ sufficiently ...
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what does diffeomorphism mean for differential map?

Does isomorphic differential map mean $det(DF)\neq 0$ ? Is $de(DF)\neq 0$ in the following example? For y regular value of $F:M^{n+1}\to N$, we have $dim(F^{-1}(y))=1$ i.e. circles and arcs ...
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My mistake on proving “$deg(f,y)=0$ if f can be extented”.

Statement: Show $deg(f,y)=0$, when $f:\partial M^{n}\to N ^{n}$, y is a regular value, $\exists$ extension $F:M^{n+1}\to N$ and M and N are compact smooth mflds. The outline: 1) $F^{-1}(y)$ is a ...
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1answer
38 views

Tangent bundle of a manifold [duplicate]

can anyone help me with this problem: Show that for a manifold $M$, the tangent bundle $TM$ also has the structure of a manifold. If $M$ is an n-manifold, what is the dimension of $TM$? for the 1st ...
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1answer
35 views

Construction of Hodge decomposition

We know Hodge decomposition splits any $k$-form into three $L^2$ components. And I see some proofs, none of them provide an explicit constructive method. Is there any general method to construct one? ...
9
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1answer
721 views

How to apply Stokes' Theorem for manifolds with boundary

Original motivation: How can I apply Stokes' Theorem to the annulus $1 < r < 2$ in $\mathbb{R}^2$? Concerns: Since the annulus is a manifold without boundary, it would seem that Stokes' ...
2
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48 views

handle moves: proof

In several 4-manifold textbooks, when handle moves (creation, cancellation, sliding) are discussed, they are explained using very helpful drawings. However, I would like to know if there is a ...
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1answer
35 views

Contractibility of the space of sections of a fiber bundle

Let $\pi: E \to M$ a fiber bundle and $\Gamma(M,E)$ the space of smooth sections of the bundle with topology induced by the Whitney topology on $C^{\infty}(M,E)$. Assume that each fiber is ...
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10 views

Proof that I can always get a height function that is Morse.

So a height function $h(x_{1},...,x_{m})=x_{k}$ for mfld $M^{m}\subset \mathbb{R}^m$. I proved that Morse functions are dense in $C^{\infty}(M,\mathbb{R})$. So I can approximate h by Morse functions, ...
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1answer
28 views

What is the local trivialization $(\pi^{-1}(U),\Phi)$ associated with a coordinate chart $(U,\varphi)$?

In this set of notes on Vector Bundles: http://www.math.toronto.edu/mgualt/MAT1300/week10.pdf (example 3.13), they say that given a coordinate chart $(U,\varphi)$, there is an associated local ...
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29 views

Example: Maps of Constant Rank - Composition not of Constant Rank

In Lee's Introduction to Smooth Manifolds, there is chapter on Submersions and Immersions, that is maps which derivative is of full rank everywhere. Despite their permanence properties, maps of ...
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1answer
20 views

What is a coordinate function $x^i$ of a manifold, given a chart $(U,x)$?

I am trying to understand the notes here: http://unapologetic.wordpress.com/2011/04/13/cotangent-vectors-differentials-and-the-cotangent-bundle/. Specifically, this sentence: If we have local ...
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2answers
158 views

Structure of a $ C^{\infty} $-manifold

I was studying differentiable manifolds (an introduction) and found the following example, but I am confused. Example The function \begin{align} f: &\mathbb{R}^{3} \to \mathbb{R}, \\ f: ...
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1answer
32 views

What exactly are the basis $\{ \frac{\partial}{\partial x_i}\mid_p \}$ of the tangent space of a manifold?

From http://en.wikipedia.org/wiki/Tangent_space#Definition_via_derivations, I understand that if $\gamma: (-1,1) \to M$ is a curve (and $M$ a manifold), with tangent vector $\gamma'(0)$, then the ...
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40 views

English edition of Vol 9 of Dieudonné's Foundations of Modern Analysis?

I have found the first 8 volumes of Dieudonné's Foundations of Modern Analysis in English translation, but I'm having difficulty locating volume 9. I have searched the catalogues of numerous libraries ...
2
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1answer
58 views

Notation and hierarchy of cartesian spaces, euclidean spaces, riemannian spaces and manifolds

I am confused by some definitions. Forgive the looseness of my language. A Cartesian space is basically a space of points that can be represented by n-tuples ( and other things, but I won't go into ...
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682 views

Is every Compact $n$-Manifold a Compactification of $\mathbb{R}^n$?

I read the result that every compact $n$-manifold is a compactification of $\mathbb{R}^n$. Now, for surfaces, this seems clear: we take an n-gon, whose interior (i.e., everything in the n-gon except ...
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21 views

surjective differential at tangent vector is zero

The context is proving $deg(f,y)=0$ where $f:\partial M^{n}\to N^{n}$. page 2 at http://www.math.polytechnique.fr/~gravejat/SemiElev/Poincare-Hopf.pdf. Also,page 2 at ...
5
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1answer
353 views

Mapping Degree of a Smooth Map from a Compact Manifold without Boundary

There is a comment in Milnor's "Topology from a Differentiable Viewpoint," that I don't quite understand: Let $f$ be a smooth map from $M$ to $N$, where $M$ is compact without boundary, and $N$ is ...