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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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 ...
1
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52 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 ...
2
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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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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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35 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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92 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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29 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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49 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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37 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
71 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 ...
5
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74 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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1answer
22 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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41 views

Unit square is a smooth 1-manifold

By unit square I mean the boundary of $I\times I$. I drew a unit circle inside the square and projected onto it by the map $(x,y)\to ...
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33 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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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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72 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
61 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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35 views

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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37 views

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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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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41 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
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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 ...
2
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1answer
28 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
34 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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50 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
39 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
61 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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34 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
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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 ...
6
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2answers
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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.
5
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1answer
165 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
39 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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399 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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Book of Pullbacks and Pushouts

what books can I consult for properties of pullback and pushouts in algebraic topology? I need to understand the theory of homotopy in algebraic topology and I started to study pullbacks and push ...
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Gluing tori and surgery related lens spaces

I came up with this question when I was thinking about the lens space obtained by an integer surgery along a Hopf link. Let $T_1, T_1', T_2, T_2'$ be a solid torus $S^1 \times D^2$. We regard the ...
4
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1answer
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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 ...
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23 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 ...
2
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1answer
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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
40 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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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 ...
4
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2answers
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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
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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 ...
3
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1answer
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Expressing $\mathbb{R} P^3$ as a fibre bundle

This question came up in office hours with my differential topology prof and since then I've almost settled on an answer. The question was whether we could write $\mathbb{R} P^3$ as a fiber bundle ...
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1answer
33 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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2answers
45 views

Hessian of Morse function on $S^{n}$ mistake

I am trying to get that $f(x_{0},...,x_{n+1})=x_{n+1}$ has $index_{(0,...,0,1)}=n$ Can you find my mistake or post a partial solution? My attempt I evaluate df using inverse of stereographic ...
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0answers
40 views

Differentiable function on bad sets.

This is exercise (c) on page 6 of ``Elementary differential topology" by Munkres. Find an open subset $U$ of $\mathbb R^2$ and a $C^1$ map $f : A \to \mathbb R$ ($A = \overline U$) such that $Df(x)$ ...
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Functions are smooth iff their product is smooth?

For $k+l=n$, we want to prove that $h:\Bbb{R}^n\to\Bbb{R}^k$ is $C^1$ if and only if $F:\Bbb{R}^n\to\Bbb{R}^k\times\Bbb{R}^l, (x,y)\to (x,h(x,y))$ is $C^1$. Write down and prove a more general ...
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Is there always a smooth variant of a homoeomorphism between smooth manifolds?

Let $M$ and $N$ be smooth homeomorphic manifolds. Let $h:M\rightarrow N$ a homeomorphism. Does there exist $r:M\rightarrow N$ that is still a homeomorphism and additionaly smooth? Can it be chosen ...