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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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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39 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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32 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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1answer
69 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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2answers
124 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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40 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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14 views

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
32 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
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
32 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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33 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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2answers
75 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 ...
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1answer
62 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.
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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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30 views

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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48 views

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 ...
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1answer
89 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 ...
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22 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
76 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 ...
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1answer
39 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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1answer
66 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 ...
3
votes
2answers
92 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
39 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 ...
3
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1answer
50 views

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
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 ...
2
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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 ...
2
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0answers
39 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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20 views

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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41 views

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 ...
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25 views

Is the $\varepsilon$-neighbourhood theorem used in proving Homotopic transverse extension?

In Guillemin & Pollack page 71 I can't see where "For compact mfld Y,the map $\pi:Y^{\varepsilon}\to Y$ is a submersion" is used to show: "If for $f:M\to N$, closed subset $C\subset M$, closed ...
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2answers
50 views

Why $df_{0}\simeq \mathrm{identity}$

Milnor lemma 2 pg 34 "Any orientation preserving diffeomorphism f on $R^m$ is smoothly homotopic to the identity" So he proves that $f\simeq df_0$ ,which he says is clearly homotopic to the identity. ...
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1answer
53 views

The boundary of this set is smooth?

Let $\Omega_1 \supset \Omega_2 \supset....$ a decreasing sequence of bounded, convex and smooth sets. My intuition says that the set $int(\overline{\bigcap_i \Omega_i})$ (where int denotes the ...
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31 views

Intersection form and poincaré duality

Let $ M $ be a $2n$-dimensional compact connected oriented smooth manifold and let $A$, $B$ be two $n$-dimensional submanifolds that intersect transversally. Denote by $A \cdot B$ the sum of the ...
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1answer
36 views

Extension of funcion

I think its right but Im not sure. I have topological space (exactly manifold - second countable, Hausdorff, local Euclidean topological space) M, dim M=m. Let $A \subset M$ is closed set, dim A=n, ...
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3answers
149 views

Applications of Galois theory for topology

Is there any applications of Galois theory in topology? I already have learned Galois theory, and applied it in algebra. Can I get solution of some big problem about topology using Galois theory? ...
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2answers
51 views

If a connection on a principal $G$-bundle restricts to an $H$-subbundle, must its holonomy lie in $H$?

Let $P \to M$ be a principal $G$-bundle, equipped with a principal connection $D$. Let $Q \subset P$ be a principal subbundle with fiber $H$, where $H \leq G$ is a (let's say closed and connected) ...
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2answers
147 views

Why 'closed differential forms' are called 'closed'?

As is well known a differential form $ \omega $ is called closed differential form if it satisfies $ \mbox{d} \omega = 0 \, \, (\ast) \, $ where $ \mbox {d} $ is the exterior derivative. I think the ...
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Cohomology of covering space

Let $B$ be a base space and $E$ be a covering space of $B$ what is the relation between $H^2(B,\mathbb{Z})$ and $H^2(E,\mathbb{Z})$.?
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The first proof for Poincare lemma in history

How can I get a reference about the first proof of Poincare lemma in history? I already know some methods of proof, but I do want to know the original approach. Thanks for your help!