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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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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153 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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1answer
84 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
125 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? ...
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85 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
43 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, ...
2
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1answer
154 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 ...
2
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1answer
89 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
42 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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1answer
2k 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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100 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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127 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
163 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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1answer
24 views

Is $f(x)+\sum_{p,i=1,…,m}\lambda_{p,i}x_{p,i}(x)$ globally defined?

Is the map $\Phi: \mathbb{R}^{N\cdot m}\times M\ni (\lambda_{p,i},x) \mapsto f(x)+\sum_{p}\sum_{i=1,...,m}\lambda_{p,i}x_{p,i}(x)\cdot \phi_{p}(x) \in \mathbb{R}$ globally defined, where M is ...
2
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1answer
485 views

Normal Bundle of a Manifold

I was reading "Morse Theory" by J.Milnor and at page number 32 there is remark "It is not difficult that N is an n-dimensional manifold differentiably embedded in $\mathbb{R}^{2n}$ ( N is the total ...
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1answer
88 views

vector bundles and their cross-sections

Let $B$ be a compact Hausdorff space and $C^0(B)$ be the ring of continuous real valued functions on $B$. For any vector bundle $\xi$ over $B$, let $\Gamma(\xi)$ denote the $C^0(B)$-module consisting ...
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140 views

Morse functions are dense in $C^{\infty}(M,\mathbb{R})$ questions.

Hi here is a proof inspired from the reference below. Feel free to get very technical with your comments so that at the end I understand it well. I am more concerned about the questions I added ...
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1answer
89 views

What is the difference between Tangent Bundles and Trivial Vector Bundles.

Tangent bundle: $TM := \bigcup_{p \in M} T_pM$, where $T_pM = \{p\} \times \mathbb{R}^n$. So, $M$ is an $n$ dimensional manifold. Now, letting $V = \mathbb{R}^n$. A trivial vector bundle is $E := M ...
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3answers
978 views

$SO(n)$ is connected

The question really is that simple: Prove that the manifold $SO(n) \subset GL(n, \mathbb{R})$ is connected. it is very easy to see that the elements of $SO(n)$ are in one-to-one correspondence with ...
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64 views

Reference about history of characteristic classes

I'm looking for a good reference about history of characteristic classes. For example, I would like to know how Chern define the Chern class at first in history, or who rewrote the characteristic ...
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80 views

Manifold locally looks like a open set but not as a euclidean space?

I am reading about manifolds from the book by Millman. He says that manifolds locally looks like a open set, but there is no canonical way to make M look like a Euclidean Space, and so we can't define ...
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109 views

References Request (Poincare Duality via Unimodular pairing, de Rham isomorphism)

Chapter 0 of Griffith's Harris contains a substantial amount of topological results that I have not seen elsewhere, namely: Poincare duality as a unimodular intersection pairing on homology. Also ...
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1answer
171 views

Defining a Hyperbolic Metric in a General Surface S

I hope someone can help me or give a ref. I'm trying to understand the general way in which one defines a hyperbolic metric on a given surface $\Sigma$ ( and, if not too complicated, on a manifold of ...
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Does there exist a vector field s.t. all orbits are dense on $\mathbb{R}^2$

Is there a complete vector field such that the all orbits are dense on a contractible manifold? For example, $\mathbb{R}^2$,the interior of a $n$-unit ball.
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1answer
36 views

Let $f:X \to Y$ be a smooth map, $df_x$ is an isomorphism, find parametrizations s.t. $f(x_1,x_2,\ldots,x_k)=(x_1,x_2,\ldots,x_k)$.

The following statement is in page 14 of Guillemin & Pollack Differential Topology: Let $f:X \to Y$ be a smooth map, and suppose that $df_x$ is an isomorphism, show that we can find ...
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45 views

How to compute saddle point index using sourcing flow lines?

Prove $Index_{p}(\bigtriangledown f)$= "dimension of sourcing flow lines from p" ,where p is a critical point. Attempt Near $ p \in Cr(f) $ in some coord. $ f(x) - f(p) = $ $\sum_j x_j^2 - \sum_k ...
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51 views

Proving the existence of a neighborhood and left inverse using calculus in $\mathbb{R}^n$

Suppose that $m < n$, that $U$ is an open set in $\mathbb{R}^m$ and that $f:U \to \mathbb{R}^n $ is a $C^1$ function that has maximal rank (rank $m$) everywhere in $U$. Show that, for each $x \in ...
0
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1answer
246 views

compact open topology on M(X,Y) continuous function space

I'm really confused with the statement. let $X$ be a set endowed with the discrete topology and let $Y$ be any topological space. Show that $M(X,Y)$ with the compact open topology is homeomorphic to ...
3
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1answer
78 views

An immersive map is locally left invertible

Question: Suppose that $m < n$, that $U$ is an open set in $\mathbb R^m$ and that $f : U \rightarrow \mathbb R^n$ is a $C^1$ function that has rank $m$ everywhere in $U$. Show that for every ...
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2answers
307 views

Prerequisites for Freedman's proof of the 4-dimensional Poincaré conjecture

I have a good understanding of differential geometry, enough at least to understand many details of Hamilton & Perelman's approach to the 3-dimensional Poincaré conjecture. I have no such ...
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2answers
60 views

Immersions when the target space isn't a differentiable manifold (but *almost* is)

I've come across this situation in a number of places but it's most glaring in the lecture notes I'm currently reading. (PCMI lectures on the geometry of outer space). We have a map from a circle ...
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1answer
88 views

coordinate system of a sphere

I am looking for a coordinate system for the sphere that has constant Lamé parameters. In fact, the Lamé coefficients of the usual spherical coordinate system are: $L_1 = R$ $L_2 = R\sin(\phi)$ As ...
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1answer
40 views

surface curvature

I would like to proof the existence or the non-existence of a finite surface which has 2 different radius of curvature $R_1$ and $R_2$ that are: constant on the whole surface finite different each ...
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1answer
85 views

Extending pullback of a vector field

Consider the vector field $\frac{\partial}{\partial x_1}$ on $\mathbb{R^2}$. Let $\psi_N : S^2 \setminus\{N\} \to \mathbb{R^2} $ and $\psi_S : S^2 \setminus\{S\} \to \mathbb{R^2} $ be the ...
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64 views

Proving $d(f^*e_i')=0$

Let $f:M\to N$ be a differentiable map between two manifolds. $e_i$ is a basis vector of $N$ with respect to some chart and $e_i'$ its dual (i.e. $e_i'(e_j)=\delta_{ij}$). How do I prove the ...
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1answer
103 views

Morse height function for general compact manifold

Can you give me the form of the height function for any compact manifold embedded in the reals? Maybe the projection of the parametrization onto a basis vector ex. For the n-sphere is ...
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1answer
178 views

(Morse) non-degenerate iff transverse to the zero section

So Morse $f:M\to \mathbb{R}$ has nondegenerate critical point p iff $df|_{p}\pitchfork 0$-section. Attempt nondegenarate p iff Hessian has full rank at p iff ...
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1answer
20 views

If two maps' derivatives have unit length, then the derivative of the product is $\pm 1$

Let $M$ be a space and $I$ the unit interval. Definition A map $f : I \to M$ is a parametrization by arc-length if $f$ maps $I$ diffeomorphically onto an open subset of $M$, and if the "velocity ...
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416 views

When does a SES of vector bundles split?

Given a short exact sequence of smooth vector bundles, $$0\to A \to B \to C \to 0$$ on a manifold $M$, it is an easy exercise that sequence splits. One approach is to pick a Riemannian metric on ...
3
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2answers
117 views

$H_{n-1}(M;\mathbb{Z})$ is a free abelian group

need help with this problem: show that if $M$ is closed connected oriented n-manifold then $H_{n-1}(M;\mathbb{Z})$ is a free abelian group. thanx.
5
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1answer
476 views

Construction of lens spaces

I have a question about the surgery construction of lens spaces. Let $T=S^1 \times D$ be a solid torus. Let $T'$ be another torus. We fix a meridian $m$ and longitude $l$ of the torus. Then the lens ...
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1answer
70 views

Sufficient conditions for smooth pushout

We restrict ourselves to the category of smooth manifolds and smooth maps. Suppose we have a pair of smooth maps $f:A \to X$ and $g:A \to Y$. A pushout is a pair of smooth maps $p:X \to Z$ and $q:Y ...
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454 views

Why care about group actions?

Let X be a space (topological space, manifold, etc) and let the group G act continuously on X. What extra (homotopical, homological, cohomological, diffeomorphical etc) data can extracted from X when ...
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229 views

Turn a torus inside out

Let $T=D^2 \times S^1$ be a solid torus, where $D^2$ is a 2-dimensional disk and $S^1$ is a circle. Suppose we have another solid torus $T'$ and we have a homeomorphism $f$ sending a meridian ...
2
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1answer
52 views

Cancelling Handle Attachments

Let $(W, \partial W)$ be an $n$-dimensional manifold with boundary. Suppose that $(W', \partial W')$ is obtained from $(W, \partial W)$ by attaching a $k$-handle via an embedding $\phi: S^{k-1}\times ...
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1answer
555 views

Conformal map iff holomorphic

It seems like if $U$ is an open subset of the complex plane, $\mathbb{C}$, then a function $$f: U \rightarrow \mathbb{C}$$ is conformal if and only if it is holomorphic and its derivative is ...
2
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1answer
77 views

Is this a local diffeomorphism?

I want to find a local diffeomorphism $\Bbb{R}^2\to\Bbb{R}^2$ that is not a diffeomorphism onto its image. This is what I thought: $f(x,y)=(\sin 2\pi x, \cos 2\pi y)$. Does that work? Seems ok to me.
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107 views

I don't understand a paragraph about tangent space

I don't understand how author associate the smooth manifolds and linear subspace. TM is a linear subspace,what 's the mean of T?A set of vector? And find the definition on Wikipedia. I still ...
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2answers
292 views

Any N dimensional manifold as a boundary of some N+1 dimensional manifold?

Is this statement true: Question: Can any N dimensional manifold be realized as a boundary of some N+1 dimensional manifold? If so/not, how to prove/disprove it? I read a TQFT paper from Edward ...
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1answer
54 views

Proof of Lemma in “Differentiable Viewpoint”

On page 11 of Milnor's Differential Topology book there is Lemma 1. In the proof of Lemma 1 it says, to define, $ F:M\to N\times \mathbb{R}^{m-n}$ by $F(\xi) = (f(\xi),L(\xi))$. The derivative ...