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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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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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 ...
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53 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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58 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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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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51 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
66 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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57 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
22 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
68 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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40 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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33 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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32 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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32 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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37 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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46 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 ...
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78 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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24 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 ...
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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 ...
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85 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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55 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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45 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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47 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 $E := M ...
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$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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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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56 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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50 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
84 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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80 views

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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23 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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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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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 ...
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1answer
81 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 ...
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1answer
63 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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Connection between 'canonical projection' and 'implicit function solving' in implicit function theorems

Here are the two versions of the implicit function theorem (surjective/injective) commonly seen, but for these particular statements I took out from Differential Topology by Hirsch (P.214). ...
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Topology well-ordered set $f:\mathbb{Z}_{+}\rightarrow A$

A function $f:\mathbb{Z}_{+}\rightarrow A$ is nonincreasing if $x_{1}<x_{2}$ implies $f\left(x_{2}\right)\leq f\left(x_{1}\right)$. A function is eventually constant if there exists an $N$ such ...
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2answers
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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
45 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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Equivalent vector field with finitely many nondegenerate zeroes

This is an attempt to understand Milnor's proof that it's always possible to compute a vector field's index using a non-degenerate field. The strategy is simple: take $z$ to be an isolated zero of ...
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1answer
37 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
30 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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53 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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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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44 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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82 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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35 views

Question concerning the Lie derivative and the Lie bracket

Let $X,Y$ be vector fields on a differentiable manifold. In a proof I read that for a special chart (namely the chart in which we have $X\equiv e_1$) it holds ...
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1answer
16 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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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 ...
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$H_{n-1}(M;\mathbb{Z})$ 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.
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186 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 ...