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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Derivatives in Topological Vector Spaces and General Spaces

I know that we may have a function $f$ such that all the directional derivative of $f$ in some topological vector space $V$, but the derivative need not exist. Moreover, it's possible for all the ...
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Is every $S^1$ knot orientable?

let $K$ be an $S^1$ knot, i.e., $K$ is a homeomorphic copy of the "standard" $S^1$ living in $\mathbb R^3$ ). I am trying to see if $K$ is orientable. This is what I have: 1) If the map sending $S^1$ ...
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Universe as a finite 3-manifold without boundary

My question is soft and imprecise, as I know very little differential topology. Nevertheless, I hope it makes some $\epsilon>0$ of sense. Assume the Universe is a 3-manifold without boundary, ...
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What is the anticommutator of the interior product and codifferential (adjoint of exterior derivative)?

What is $\eta=i_X \delta + \delta i_X$ acting on differential forms? Here $\delta$ is the usual Hodge adjoint of the exterior derivative and $i_X$ is contraction of a form with the vector field $X$. ...
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Map between Tangent Manifolds Well-Defined?

Let $f: \mathcal{M} \to \mathcal{N}$ be a $\mathscr{C}^{r+1}$ map. We define a map $\mathscr{T}f: \mathscr{T}\mathcal{M} \to \mathscr{T}\mathcal{N}$ as follows: A local representation of the map ...
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Why does the slope of a smooth simple closed curve have winding number one?

$\def\RR{\mathbb{R}}$Let $S^1$ be the circle and let $\gamma : S^1 \to \RR^2$ be a smooth injective map with $\gamma'(t)$ everywhere nonzero. What is the easiest way to show that $t \mapsto ...
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Stokes or homotopy?

The problem states Show that if $X$ is a simply connected manifold, then $\oint_{\gamma}\omega=0$ for all closed 1-forms $\omega$ on X and all closed curves $\gamma$ in $X$. However I have ...
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Embeddings are precisely proper injective immersions.

We call a map $f: X \to Y$ between topological spaces proper if $f^{-1}(K)$ is compact for all compact $K \subset Y$. Where can I find a reference that embeddings are precisely proper injective ...
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Ambient isotopy of based surface knots

Let $S$ be a smooth closed surface of genus $\ell$. Let $p$ be a point of $S$ and $a_i$, $b_i$ with $i=1,\ldots,\ell$ be $2\ell$ curves embedded in $S$ based at $p$ smooth everywhere except perhaps ...
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What can you say about injection, immersion, embedding for the torus?

Define $\varphi_a: \mathbb{R} \to T$ where $T = S^1 \times S^1$ is the torus via$$\varphi_a(x) = (e(x), e(ax)),\text{ }e(x):=e^{2\pi i x}$$and $a > 0$ is some parameter. Determine for which $a$ ...
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Laplace-de Rham operator

Consider an operator $\partial = (-1)^k \star^{-1}\,d\star: \Omega^k(\mathbb{R}^n) \to \Omega^{k-1}(\mathbb{R}^n)$. Note that we equivalently can write $\partial = (-1)^{nk + n + 1} \star\,d\star$. ...
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Embed curves in the plane

The strongest version of Whitney's embedding theorem says that every smooth real $n$-dimensional manifold $M^n$ (Hausdorff and second-countable) can be embedded in $\mathbb{R}^{2n}$. This should mean ...
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Convention of a continued fraction presentation of a lens space

I want to clarify the following two conventions on a surgery description of a lens space. Let $p$ and $q$ are relatively prime integers. Express $$ ...
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Transversality of leaves to the spheres .

Consider a form in the complex plane such that its linear part is $\omega_0=\lambda_1xdy-\lambda_2ydx$ in the Poincare domain: $\lambda_1\lambda_2 \ne 0$ and $\lambda_1/\lambda_2 \notin \mathbb{R}^-$. ...
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45 views

Laplacian and Hodge Laplacian

I am new to the theory of differential forms, but there is one thing that I don't get at all. Imagine that you are on the sphere $\mathbb{S}^2$, then the Laplacian $- \Delta$ is known to be a ...
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a generalization of punctured cylinder

Let $S^1\times \mathbb{R}$ be the infinite cylinder. Pucture it, we have $S^1\times \mathbb{R}-*$. Then $(S^1\times \mathbb{R}-*)\simeq Skeleton^1(S^1\times \mathbb{R})\simeq S^1\vee S^1$. How ...
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cohomology ring of a quotient space

what is the cohomology ring $$ H^*((S^3\times S^3\setminus \{e\})/(a,b)\sim (ab,b^{-1});\mathbb{Z}_2)? $$ Here the unit $e$ and the product $ab$ is of the Lie group $S^3=Sp(1)$.
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Three linearly independent vector fields

How can one find three linearly independent vector fields on $S^1\times S^2$? I know that $S^1\times S^2 \cong SO_3( \mathbb{R})$, i.e. the set of orthogonal $3 \times 3$ matrices with determinant ...
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Terminology question : “half smooth, half topological” fibre bundle

First, I know (or I think I know...) the definition of fiber bundle, be it in the smooth or topological category. Here is my situation, which is kind of between the two: I have a smooth manifold $E$, ...
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Why are contact structures studied from a cohomological, rather than homological, perspective?

As far as I know, contact structures are studied from a "cohomology/dual" perspective, meaning mostly from the perspective of contact forms and their respective kernels, instead of from a ...
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63 views

Prove that the Brouwer theorem is false for the open ball $|x|^2 <a$

Prove that the Brouwer theorem is false for the open ball $|x|^2 <a$ Brouwer Theorem: Any smooth map $f$ of the close unit ball $B^n \subset R^n$ tin to it self must have a fixed point. I need to ...
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34 views

Relation between differentials of perturbations of vector fields

Let $A$, $B$ be smooth submanifolds of a smooth manifold $M$ and $X\in C^\infty(TM)$ a vector field such following its flow $\xi^X$ gives a diffeomorphism between $A$ and $B$. Suppose also that ...
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Submanifold of $2\times 2$ Complex Matrices Using Transversality?

Let $M$ be the set of all $2\times 2$ complex matrices with $a_{21}=\bar a_{12}$. $M$ is a smooth manifold diffeomorphic to $\mathbb{R}^6$. Let $W$ be the subset consisting of all matrices in $M$ with ...
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When $f\colon M\to N\times N$ satisfies that $f^{-1}(\Delta)$ is a ball?

Let $M$, $N$ be smooth manifolds of dimension $m+n$ and $n$, respectively. Suppose that $f\colon M\to N\times N$ is a smooth map and $f$ is transversal to the diagonal $\Delta=\{(x,x)\in N\times ...
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67 views

Volume of a paracompact manifold

It is stated, without proof, in Wald (1984) (General Relativity) that given any connected manifold $M$ (which is by definition paracompact), one may define a volume measure $\mu$ such that $\mu[M]$ is ...
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Why any short exact sequence of vector spaces may be seen as a direct sum?

This is actually the first time I have worked with short exact sequences and I have no clue why the following assertion is true: Any short exact sequence of vector spaces $$ 0 \longrightarrow U ...
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Differentiability of a map into $\wedge T^*M$

Let $M$ a differentiable manifold and $p \in M$. Consider $(U;x_1, \dots, x_n)$ a coordinated system around $p$. $\{dx_\phi\}_\phi$ with $\phi \in \cal{P}(\{1, \dots n\})$, $dx_\phi=dx_{i1} \wedge ...
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A question about a part of the proof of the existence of the exterior derivate.

Let $M$ a differentiable manifold, $(U;x_1,\ldots,x_n)$ a coordinated system and $\omega$ a differential form which domain intersects $U$. In $U \cap \operatorname{Dom} \omega$, $\omega$ can be ...
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Charts of $\wedge T^*M$ and $\wedge TM$.

Let $M$ a differentiable manifold and consider $\wedge T^*M$. If $(U, \phi)$ is a chart of $M$ with coordinate functions $(x_1, \dots, x_n)$, then $\{ \frac{\partial }{\partial x_i}\}_i$ and $\{ ...
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Schwartz rule in differentiable manifolds.

Let $M$ a differentiable manifold and $(U,\varphi)$ a chart with coordinate functions $(x_1,...,x_n)$. Let $p \in U$. Given $f:U\longrightarrow \mathbb{R}$, $f \in C^\infty(U)$, it is possible to ...
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Prove that any map $f\colon S^1 \to S^1$ mapping antipodal point to antipodal point has $\deg_2(f)=1$ by a direct computation.

Without using the Borsuk-Ulam theorem. Prove that any map $f\colon S^1 \to S^1$ mapping antipodal point to antipodal point has $\deg_2(f)=1$ by a direct computation. I know that $f$ map anitpodal ...
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Partial derivatives on Manifolds

Let $F : A \times B \to C$ be a map of smooth manifolds. Define the following maps ("partial derivatives"): $E_1 F: TA \times B \to TC$ $E_1 F(a,v,b) = D_a F(-,b) v $ where $v \in T_a A$ $E_2 F: A ...
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Explicit Calculation of the Euler class for the 2-Sphere using transition functions

I have been trying to learn about characteristic classes for months now, and every time I try a simple example something goes wrong. Any insights would be greatly appreciated. I am trying to follow ...
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Integral Curve of the vector field

If we have a 2-sphere with coordinates $x=r \cos \theta \cos \phi$, $y= r \sin \theta \sin \phi$ and $z=r \sin \theta$ and the vector field $X= (-r\sin \theta \cos\phi, r \cos \theta \sin \phi, r \cos ...
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Fundamental group of cusp of a negatively curved manifold

Let $M$ be a complete, noncompact Riemannian manifold with finite volume and whose sectional curvatures vary within the interval $[a,b]$, $-1\leq a<b<0$. It is known that such manifold has ends ...
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Making a gradient-like vector field a gradient vector field via choosing a Riemannian metric.

Let $\xi$ be a vector field on manifold $M^n$ which is a gradient-like vector field for a some Morse function $f$. Prove that there exists a Riemannian metric on $M$ such that $\xi$ is a gradient ...
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Locally exact vs globally exact

Why the volume form in Sphere is locally exact but not globally exact? here the integral is integral $$\int_{S^n}w$$ with $$w = \frac{1}{r} \sum_{i=1}^{n+1} (-1)^{i-1} x_i dx_1 \cdots\hat{dx_i},\cdots ...
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119 views

Are strongly close maps homotopic?

While reading about various results related to density of smooth functions in the space of continuous functions with strong topology, I've got the impression that it is a general fact that for any ...
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41 views

Proof a $(2n-1)$-compact manifold

I have no idea how prove that $$\{(z_0,\ldots,z_n)\in\mathbb{C}^{n+1} \quad| \quad z_0^d+z_1^2\ldots+z_n^2=0, \quad |z_0|^2+|z_1|^2\ldots+|z_n|^2=2\}$$ is a $(2n-1)$-compact manifold. How give the ...
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last step in proof of existence of coordinate vector field

This is problem 7 on page 172 of Spivak's Differential Geometry pt. 1. Given a smooth manifold $M$ and a smooth vector field $X$ on $M$, Check that if the coordinate system $x$ is $x = \chi^{-1}$ ...
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Covering groups of compact connected Lie groups is compact.

Hi: I have a question as follows: Is it true that the covering group of a compact, connected Lie group is also compact? Thanks very much!
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Is the smooth map with constant rank dense?

Let $M$ and $N$ be two Riemannian manifold. Assume that $f:M\rightarrow N$ is a smooth map and $\dim M < \dim N$. Can we find a smooth map $g$ $C^{k}$-close to $f$ which is immersive at each point ...
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How to prove there exists a solution? Guillemin Pollack

Prove there exists a complex number $z$ such that $$ z^7+\cos(|z|^2)(1+93z^4)=0. $$ (For heaven's sake don't try to compute it!)
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What is a regular homotopy?

The definition of regular homotopy from Wikipedia says that two immersion $f,g:M\to N$ are regularly homotopic if they represent points in the same path-component of $\text{Imm}(M,N)$. What does ...
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Almost complex structures on a manifold and its exotic copy

Consider a compact simply-connected smooth $4$-manifold $M$ and its exotic copy $M'$. Identify the underlying topological manifolds. Dold-Whitney theorem guaranties ($\mathbb R$-linear) isomorphism of ...
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Is it possible for a manifold to have a normal vector that is zero everywhere, if so, would this indicate that the manifold is non-orientable?

Basically I've been thinking about defining a non-orientable three-dimensional metric space via defining the normal vector and looking to see if there is two possible vectors for the same point. I'm ...
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Find the Poincare Dual of a ray in $\mathbb{R}^2-\{0\}$

This is example-exercise 5.16 in Bott and Tu (which I'm independently reading through.) The problem states: Let $M=\mathbb{R}^2-\{0\}$, and $X\subseteq M$ be the closed submanifold ...
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parallelizable manifolds

I know a differentiable manifold $M$ of dimension $n$ is parallelizable if there exist (smooth of course) vector fields $\{X_i\}_{j=1}^n$ which are linearly independent in $T_pM$ at each point $p \in ...
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identity map is not diffeomorphism, $x^3$ is a diffeomorphism [closed]

Consider the real line $\mathbb{R}$ the two following differentiable structures: 1) $(\mathbb{R}, f_1)$ where $f_1(x) = x$. 2) $(\mathbb{R}, f_2)$, where $f_2(x) = x^3$. How do I demonstrate that: ...