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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If $M^n$ is a submanifold of $\mathbb{R}^{n+k}$ such there are L.I. vector fields $v_1,\ldots,v_k$ such that $v_i \perp T_pM$ then $M$ is orientable

If $M^n$ is a submanifold of $\mathbb{R}^{n+k}$ such there are L.I. vector fields $v_1,\ldots,v_k$ such that $v_i \perp T_pM$ for every $p \in M$ then $M$ is orientable. My attempt is: Once $M^n$ ...
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Why are geodesically convex sets diffeomorphic to $\Bbb R^n$?

In the construction of a good cover for a manifold $M^n$, Bott & Tu use the fact that each point in $M$ is contained in a geodesically convex set (after picking a Riemannian metric). They then ...
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are immersions of surface in $\mathbb R^3$ dense in all regular maps?

Let $u\in C^\infty(\Omega,\mathbf R^3)$ with $\Omega$ open set in $\mathbf R^2$. Can we find $u_k\in C^\infty(\Omega,\mathbf R^3)$ with $\mathrm{rank}(Du_k(x))=2$ for all $x\in\Omega$ such that $u_k \...
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Can we prove invariance of dimension directly from the Jordan-Brouwer separation theorem?

Is the following proof correct? Consider spaces $\mathbb{R}^n$ and $\mathbb{R}^m$, where $n<m$, and sphere $S^{n-1}\subset \mathbb{R}^n$. Suppose that we have a homeomorphism $f:\mathbb{R}^m \...
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Hairy dog theorem

I am interested in applications of the hairy ball theorem to non-closed surfaces, such as dogs and cats. As has been pointed out to me in my previous question, it is possible to arrange the two ...
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function from a genus $2$ surface to $S^1$

Let $f\colon \Sigma \rightarrow S^1$ be a map from a genus $2$ surface to $S^1$. If $y\in S^1$ is a regular value of $f$ and $f^{-1}(y)$ is a nonseparating circle of $\Sigma$. How can I prove that $f$...
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Definition of manifold which are subset of euclidean space

According to Guillemin and Pollack "X(which is a subset of R^n)is a k-dimensional manifold if it is locally diffeomorphic to Rk , meaning that each point x possesses a neighborhood V in X which ...
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21 views

Is the set where the exponential map is defined an open subset of $TM$?

Let $M$ be a connected Riemannian manifold. Define $O=\{(p,v) \in TM|\, \,exp_p(v) \text{ is defined} \}$. Is $O$ an open subset of $TM$? I know that for every point in $M$, there is a neighbourhood $...
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Construct smooth mapping $f: B^{n + 1} \to S^n$ with two singularities at which $f$ has degree $+/- 1$.

I'm currently working through a paper by Pjotr Hajlasz who wants to show that For smooth manifolds $M,N$, if $\pi_{[p]}(N) \neq 0$ and $1 \leq p < n = \dim M$, then the smooth mappings $C^\...
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What role does differentiability play in Topology?

My question is stated in the title. As a brief background, I'd like to say I know next to nothing about Topology. The little bit I was exposed to came as an aside in my Multivariate Calculus class; we ...
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Construct a diffeomorphism $\psi: B_1 \to \epsilon\text{-neighborhood of } K$, where $K$ is a subset of a smooth manifold.

I'm currently working through a paper by Brezis on the topology of Sobolev spaces. Right now I am having trouble understanding the following note made by Brezis. Let $M$ be a compact and smooth ...
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Broken line is NOT diffeomorphic to the real line

This is from Bredon's Topology and Geometry, page 71. This comes right after the very definition of differentiable manifold, so I think no use of tangent space or 'differential' is permitted. (Bredon ...
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Describe all smooth surfaces in $\mathbb{R}^3$ with coordinates $(x,y,z)$ such that the pullback of the one-form $\theta:=dy-zdx$ is identically zero.

My question is as the title states: Describe all smooth surfaces in $\mathbb{R}^3$ with coordinates $(x,y,z)$ such that the pullback of the one-form $\theta:=dy-zdx$ is identically zero. Now, ...
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Bounding Geodesic Curvature

Let $\Sigma$ be a smooth surface, let $p,q \in \Sigma$ and let $n_{p}$ be the normal at $p$. Suppose that $d(p,q) < c$ for some constant $c$. That is $p$ and $q$ are pretty close to each other on $\...
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Fixed-point free map of the 2-sphere which has order 4

The antipodal involution of $\mathbb{S}^2$ clearly has no fixed points. However I cannot think up an example of homeomorphism of order 4 which has no fixed points. Could you help me?
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Smooth path lifting on $S^1$ [duplicate]

Given a smooth map $f:S^1\to S^1$, there is a smooth map $g:\Bbb R\to\Bbb R$ such that $f(\cos t,\sin t)=(\cos g(t),\sin g(t))$ and $g(2\pi)=g(0)+2\pi q$ for some $q\in\Bbb Z$. I'm fairly sure one ...
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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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Restriction of a $C^{\infty}$ vector bundle over a regular submanifold.

This question is about the content of page 134 of Tu's An Introduction to Manifolds. A $C^{\infty}$ vector bundle or rank r is a triple $(E,M,\pi)$ consisting of manifolds $E$ and $M$ and a ...
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66 views

Preimage of tubular neighborhood

Let $f:M' \to M$ be a map between smooth manifolds. Let $S \subset M$ be a submanifold, and let $T$ be a tubular neighborhood of $S$, ie. $T$ is diffeomorphic to the normal bundle of $S$ in $M$. If $f$...
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Local Properties of Immersions and Submersions

This is from Bredon's Topology and Geometry, pp 83~83. He's definitions of immersion and submersion are the following: Now, in 7.3 Theorem, he explains that (using the implicit function theorem) if ...
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Inequality in chapter “pontryagin construction” from Milnor's topology from the differentiable viewpoint

at the moment I am reading Milnor's book "topology from the differentiable viewpoint". I am stuck on page 49 (chapter "the pontryagin construction", proof of lemma 4). There he claims, that for $0<|...
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Is it true that a compact non-orientable manifold $M^n$ must have $H_{dR}^n(M)=0$?

I have seen this statement assumed to be true several times --- I just can't find a reference, and now I'm starting to suspect there can be catch somewhere.
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A question on the non trivial rank three bundle on $S^2$

We know that number of rank $3$ vector bundles on $S^2$ is just the number of equivalence classes of maps from $S^1$ to $SO(3)$. This implies that there is one and only one non trivial vector bundle ...
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Perturbing a regular submanifold to ensure submersion.

Suppose $M$ is a regular compact submanifold with boundary of dimension $n$ in $\mathbb R^{n+1}\smallsetminus 0$, and assume that for each $v\in S^n$ the ray emanating from $v$ intersects $M$ in at ...
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Why can we always take the zero section of a vector bundle?

$\require{AMScd}$ As I understand it, a rank $k$ vector bundle is a pair of topological spaces with a map between them $$ E\xrightarrow{p}B $$ such that there exists an open cover $(U_\alpha)$ of $B$ ...
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integral constraint induce a manifold on Sobolev space

given the set $$ M:=\{u\in H^2(\Omega):\int_{\Omega}u=m\,\} $$ $m\in \mathbb{R}$, $\Omega $ is a bounded piecewise smooth domain in $\mathbb{R}^n$. also denote by $u(t)$ a map: $u(t):(0,T)\to M$ ...
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Solid tori, meridians, and longitudes

I am working through some of Rolfsen's "Knots and Links" and I have needed to go back and take a more careful look at the first few sections where he carefully discusses curves on solid tori. Let $V$ ...
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Stabilization of embedding?

In D. Freed's lecture notes he mentions "stabilization of embedding" in theorem 4.48. Does anyone know the definition? I can't find it online.
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Hopf fibration homeomorphism injectivity

I was wondering if there is a possibility to proof the homeomorphism between the 1-dimensional complex projective space $\mathbb{C}P^1$ and the 2-sphere $S^2$ via the Hopf fibration (I know already ...
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80 views

Topology of the tangent bundle

Let $M$ be a smooth manifold, and let $\pi:TM\to M$ be its tangent bundle. We define the topology of $TM$ by declaring a subset $V$ of $TM$ to be open if and only if $\psi_\phi(V\cap\pi^{-1}(U))$ is ...
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What is the property I am looking for? [closed]

Warning: This question is not purely mathematics, and requires a bit of "intuition" and "feel" for what I a beginner to topology, must be thinking, in order to be answered satisfactorily. Apologies, ...
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Why is Lemma 6.3 of Milnor's Lectures on the h-cobordism Theorem True?

Milnor's statement is: "Let $M^r$ and $N^s$ be sub-manifolds of $V^{r+s}$ which are all smooth, compact, oriented and without boundary. If $p$ is a point of $M^r$ contained in an $r$-cell $U$,...
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Existence of closed level sets on a surface for some field

Consider an infinite 3D space with only 2 things in it: wind and a solid object. Wind evidently blows around this solid object over its rigid surface. Bascially we are trying to set up a pure field. ...
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Explicit Dehn twist for $S^n\times S^n$

Fix $n$ odd and let $M=S^n\times S^n$. The diffeomorphism group Diff($M$) acts on the homology group $H_n(M)\simeq \mathbb Z^2$ inducing a surjection $d: \text{Diff}(M) \rightarrow \text{SL}(2,\mathbb ...
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Classifying continuous maps from closed 2-manifolds to various closed manifolds

I believe my question should be simple. The question is more physically oriented and originated from one of Witten's papers, "On Holomorphic Factorization of WZW and Coset Models", where he considered ...
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Showing deformation retract : $B^3 \setminus T^2$, $R^3 \setminus S^1$, $S^2 \bigvee S^1$

Here what i want to show is $B^3 \setminus T^2$, $R^3 \setminus S^1$, $S^2 \bigvee S^1$, $i.e$, three spaces are deformation retract to each other. Can you give me some hints or concept(?) ...
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Regarding Problem 5 on p. 71 of Bredon's Topology and Geometry

Someone else asked this question here long ago. Sadly, no answer was provided. I though the natural approach would be trying to show that $x\rightarrow x^{2}$ is an isomorphism of functionally ...
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Nonexistence of a continuous injection $f:S^2 \rightarrow \mathbb{R^2}$

What is the "easiest" way to show that there is no continuous injection $f:S^2 \rightarrow \mathbb{R^2}$? Sure the Borsuk-Ulam theorem implies that result, but this may be a "difficult" way.
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Linking Number $0$ When One Knot is a Boundary

I am given two smooth knots $K$ and $L$ embedded in $\mathbb{R}^3$. I am also given that $K$ is the boundary of an oriented, compact surface disjoint from $L$, call it $S$. I must prove that the ...
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Transition maps on Grassmanian $Gr(2,5)$

I need to provide charts and transition maps on Grassmanian $(2,5)$. (All $2$-dimensional subspaces in $5$-dimensional space). I know how the charts look, used definition from this document: http://...
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Derivation of metric from product topology?

Suppose you have two topological spaces, g11 and g22, which are "components" of a more general topology. For example, suppose that a metric has components g11, g12, g21, and g22. And suppose you want ...
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Why doesn't this argument show the Möbius bundle is trivial?

I wrote the following argument to prove that $S^1$ is parallelizable, that is, to show that the tangent bundle is trivial. It looks fairly reasonable to me. Let $\tau=2\pi$. We define a map $\...
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Normal sections

Let $S$ be a surface and let $p \in S$ with normal $n_{p}$. Let $q \in S$ nearby $p$, say within the injectivity radius of the exponential map at $p$. Consider the plane $\Pi = \operatorname{span}\{...
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Milnor's definition of smooth manifold

In Milnor's book "Topology from a differential viewpoint" on page one he defines a smooth manifold to be a subset $M \subset \mathbb R^n$ which is locally diffeomorphic to some open subset of $\mathbb ...
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How to build smooth variations along arbitrary paths at the endpoint?

Let $M$ be a smooth manifold, $\gamma:[0,a] \to M$ a smooth path. Assume we are given another smooth path $\phi:(-\epsilon,\epsilon) \to M$ that starts from an endpoint of $\gamma$, i.e $\phi(0)=\...
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Differential of the multiplication and inverse maps on a Lie group

I'm trying to solve the following two problems: Let $G$ be a Lie group with multiplication map $\mu\colon G\times G\to G$, and let $\ell_a\colon G\to G$ and $r_b\colon G\to G$ be left and right ...
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curl-free vector field on 3-torus

Let $U$ be an open and simply-connected subset of $ \mathbb{R}^3$. Then for every curl-free vector field $v \: \colon U \to \mathbb{R}^3$ there is a potential $\phi \in C^{\infty}(U; \mathbb{R})$ such ...
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Reference request for Thom's Transversality Theorem.

I am trying to read the book Introduction to the h-principle by Eliashberg and Mishachev. I am unable to understand the proof of Thom's Transversality theorem in the book. So if anyone can give any ...
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How to prove that the Lefschetz number is invariante under homotopy?

How to prove that the Lefschetz number is invariante under homotopy? We define the Lefschetz number as the number of $f : M \to M$ as the number of intersection of the map $g(x) = (x,f(x))$ with the ...
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Is true that a closed $k-$form on $T^n$, the torus, is exact if, and only if, the integral of $\omega$ over every compact submanifold is zero?

Is true that a closed $k-$form on $T^n$, the torus, is exact if, and only if, the integral of $\omega$ over every compact and oriented submanifold is zero?