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Questions tagged [differential-topology]

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 preimage is the preimage of the tangent space

Let $M$ and $N$ be smooth manifolds with $S\subseteq N$ a submanifold, and assume a map $f:M\to N$ is smooth and transverse to $S$. Prove that $T_p(f^{-1}(S)) = (df_p)^{-1}(T_{f(p)}S)$ for some $p\in ...
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Differential of Hopf's map

Let $$h : \mathbb{C^2} \rightarrow \mathbb{C \times R} $$ $$h(z_1, z_2) = (2z_1z_2^*, |z_1|^2-|z_2|^2)$$ How do you find the differential of $h$ and show it is onto/surjective? I know that I can ...
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Explicit determination of the open book in $S^3$

I'm familiar with the fundamental concepts of algebraic and differential topology. How can I determine explicitly the topology of a page of the open book in $S^3$ given by for example $$f: \mathbb{C}^...
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Quantum Groups for Generic q and 3d-TQFT. What breaks?

I've just started looking through Quantum Invariants of Knots and 3-Manifolds by V.G Turaev and want to understand what exactly is breaking in the construction of a 3d-TQFT when one considers the ...
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Volume form on a compact manifold is not exact

I am trying to show that a volume form $\mu$ on a compact manifold $M$ is not exact, i.e. show there is no $\alpha \in \Omega^{n-1}(M)$ such that $d\alpha = \mu$. My attempt is the following: Suppose,...
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Proving that the image of an injective, proper immersion is a manifold

I am trying to get through the proof of the statement "if $f: M \to N$ is injective, proper and an immersion, then $f:M \to f(M)$ is a diffeomorphism onto a submanifold". The proof I'm reading says ...
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Excercise 8 chapter 2 section 3 Guillemin and Pollack

I am doing this excercise from Guillemin and Pollack´s ''Differential Topology'': So, the hint pretty much tells you what to do, you will get that for almost any $A$, you have that $df_x + A$ is ...
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Poincaré–Hopf theorem and other topological properties

I am confused about the Poincaré–Hopf theorem. Does it actually implies that $\pi_n(S^n)=Z$? My consideration is as follows. The degree of the map $S^n\rightarrow S^n$ is related to the Euler ...
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Computing $\int_{S^{1} \times S^{1}} d\theta_{1} \wedge d\theta_{2}$

Let $S^{1} \times S^{1}$ be the torus embedded in $\mathbb{R}^{4}$. I want to compute $\int_{S^{1} \times S^{1}} d\theta_{1} \wedge d\theta_{2}$ I believe this should be "essentially" $\int_{0}^{2\...
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How does this example from Spivak that $H_c^n(\mathbb R^n) \ne 0$?

I am not sure how this integral that is being calculated using Stoke's theorem shows that the $n$th de Rham cohomology group with compact supports of $\mathbb R^n$ is not trivial. How does the fact ...
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Is the image of $f: R \to S^1 \times S^1, f(t) = (e^{it}, e^{\sqrt 2it})$ a submanifold?

Let $f: R \to S^1 \times S^1$ be $f(t) = (e^{it}, e^{\sqrt 2it})$,is $f(\mathbb R)$ a submanifold? By easy verification, $f$ is an injective immersion. Then if I want to show that the image is a ...
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What am I doing wrong? Exercise 2, chapter 2, section 3 from Guillemin and Pollack.

I am doing exercise 2, chapter 2, section 3 from Guillemin and Pollack's ''Differential Topology''. Part of the excercise is to prove that given a compact manifold $Y \subset \mathbb{R}^m$, and a pont ...
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Confusion about quotient of the Lie group $\mathbb{S}^1$

I have read that given a Lie group $G$ and a closed subgroup $H$ then $G/H$ is a smooth manifold. I cannot explain though the following example: take as $G = \mathbb{S}^1$ and as $H =\{\pm 1\}$, $H$ ...
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Definition of Differential operator

Definition 2.2, page 19 Let $M$ be a smooth manifold and $E_i \rightarrow M$ be two smooth vector bundles. A PDO $P:\Gamma (M,E_0) \rightarrow \Gamma(M,E_1)$ of order $k$ is a a linear map which ...
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Prove that a function from R to the unit circle is a local diffeomorphism.(2.4.8 G&P)

In order to prove the existence of the function $g$ in the question I want to proof that the following function is a diffeomorphism (I was told a hint that it is a diffeomorphism): $$p(t) = (\cos t, ...
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Defining Bott Class by relative $K$-theory

I am really confused with this construction of Bott Class in Page 127, Example 8.4.12 If $V$ is a complex vector space of dimension $n$, we form the complex $$ 0 \rightarrow \wedge^0 V \...
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Sard's theorem for orientation preserving diffeomorphism of the circle

thanks in advance for helping me. First I'll introduce some definitions: (1) Suppose that $f : \mathbb{S}^{1} \rightarrow \mathbb{S}^{1} = \mathbb{R} / \mathbb{Z}$ is an orientation preserving ...
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A difficulty in understanding the proof of boundary theorem in G&P.

The theorem and its proof is given below: But I could not understand the last line in the proof in particular: Why $F^{-1}(Z)$ is a compact one dimensional manifold with boundary? And why this leads ...
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A difficulty in understanding a case in the intersection theory mod 2(p.80 Guillemin and Pollack)

The following is written just before the boundary theorem in Guillemin & Pollack : But I see that if Z is not transversal to X this is not true,why the book did not consider this case? why the ...
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1answer
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Existence of a Tubular neighborhood of a hypersurface

Suppose $H$ is a co-dimension 1 embedded submanifold of $M$. Let $X$ be a vector field on $M$ such that $\forall x \in H$, $T_xM = T_xH \oplus X_x$. Now I want to show that there exixts an open set $U$...
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If $(U,\varphi)$ is a coordinate chart around $p \in M$, where $M$ smooth manifold, then how does $\varphi$ induce coordinates on $T_p M$?

I am studying differential topology and I have some trouble understanding how coordinates are induced on the tangent space at any point. Let $M$ be an $n$-dimensional smooth manifold, and let $p \in ...
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1answer
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Euler characteristic of matrix manifolds

I'm reading through examples of computing Euler characteristic of manifolds. I know how to compute it for generic manifolds like sphere and torus. But what about matrix manifolds? I'd like to know how ...
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A diifficulty in understanding a sentence in a paragraph in Guillemin and Pollack p.77

The paragraph is given below: But I have a difficulty in understanding the sentence starting in the forth line by "If we furthur ...." until its end, could anyone explain it for me please? thanks!
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1answer
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A vector field corresponding to the complement of the tangent bundle

Let $M$ be a $m$ dimensional orientable manifold, and $N$ a $m-1$ dimensional orientable submanifold in $M$, then we know at each point $x \in N$, $T_{x}M = T_x N \oplus$its complement. I need to ...
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1answer
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de Rham cohomology of doubly punctured torus

Let $T^2=S^1\times S^1$. I'd like to know all de Rham cohomology groups of $M=T^2-\{a,b\}$ but I couldn't find a result. So I want to compute it and I'm thinking of using Mayer Vietoris sequence. I ...
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1answer
27 views

Exterior derivative is linear PDO of order $1$

I am using this definition of PDO on page 19. I want to verify my understanding by the example of exetior derivative. The exterior derivative $d:\Omega^p(M) \rightarrow \Omega^{p+1}(M)$ can ...
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Holomorphism vs Smoothness

I want to understand the relation between Holomorphism and Smoothness. I want to elaborate the question as there are some underlying intricacies involved in the definitions: Smooth: A function is ...
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1answer
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A difficulty in understanding a part of a paragraph in P.41 in Guillemin & Pollack (2)

The paragraph is given below: But I do not understand: 1-In the forth line why we can not have the case $df_{x} =$ constant other than 0, could anyone explain this for me please? 2-In the sixth ...
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+300

Is every loop in a 3-manifold homotopic to some loop on its boundary?

Consider a solid region of Euclidean 3-space, or more precisely, a compact, connected 3-dimensional submanifold $U \subset E^3$ bounded by a smooth oriented surface $\Sigma = \partial U$. Very ...
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Which Nonorientable 3 manifolds have torsion in $H_{1}$?

In thinking about closed, nonorientable 3 manifolds I've been interested in finding nonorientable loops which represent torsion classes in homology. More precisely, for a closed nonorientable 3 ...
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1answer
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Extending to a local frame that agrees with given orientation

Suppose that $(e_1, \ldots, e_k)$ is an oriented basis for $T_pM$ where $M$ is an oriented Riemannian manifold. In general, we know that we can extend to a smooth local frame $(X_1, \ldots, X_k)$ on $...
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Euler class as obstruction to have a never vanishing cross section

We know that (see Hatcher's vector bundles and K-theory Prop. 3.22) the Euler class of an orientable vector bundle or rank $r$, $E\to M$ is the first obstruction to the existence of a never vanishing ...
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1answer
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$yz\,dx-2xz\,dy+(xy-y^3z)\,dz=0$

My attempt at the question What I don't know is do we integrate partially w.r.t. z to find the value of Φ (z) here? $$ \require{begingroup} \begingroup \newcommand{\dd}{\;\mathrm{d}}$$ ...
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1answer
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Transformation for Integrals over Manifolds

Most of modern books on integration theory, when constructing the Lebesgue integral, do not introduce manifolds prior. The transformation for Lebesgue integrals can then be stated as follows: Let $\...
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1answer
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existence of a non-negative smooth function on a neighborhood of point on a boundary of a smooth manifold.

Let $X$ be an n-dimensional manifold with boundary and let $x \in \partial X$. Show that there exists a smooth non-negative function $f$ on some open neighborhood $U$ of $x$, such that $f(z)=0$ iff $z ...
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1answer
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The radial plane is not locally compact

I'm trying to prove that the radial plane is not locally compact. I assume on the contrary that is locally compact and take any arbitrary point in it (say 0). Now 0 has a compact neighborhood $K$. $\...
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Show the set of mixed strategy - utility profiles such that each player is indifferent between all of her strategies is a manifold.

I am looking for help on how to solve this game theory/manifolds question. My thoughts: Informally, E is the set of mixed strategy - utility profiles such that each player is indifferent between ...
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1answer
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Optimal way to prove smooth homotopy between polynomials

I am trying to prove that given $p: \hat{\mathbb{C}} \to \hat{\mathbb{C}}$ a polynomial given by $p(z)=a_nz^n+...+a_0$ then, $p$ is smoothly homotopic to the polynomial $q(z)=a_nz^n$. I am using the ...
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Positivity of Currents

I already asked about this a couple of weeks ago but had introduced some rather annoying notation. I decided to reformulate the question in a more compact format. Suppose $\psi$ is the complex ...
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1answer
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A difficulty in understanding a part of a paragraph in Guillemin & Pollack p.60

I do not understand the highlighted part of the paragraph given below: Could anyone explain it for me please?
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1answer
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Definition of connected sum and orientation problem

I am reading Kosinski's book. To define the connected sum of $M_1^n$and $M_2^n$ (oriented and closed manifolds) we choose two embeddings of the disk $h_i:\mathbb{D}^n\to M_i$ such that $h_1$ preserves ...
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Cohomology of n-sphere minus k discs

If $M=S_n \backslash K$, where $K$ is the union of $k\geq1$ disjoint disks $D_i$, how would you compute the de Rham cohomology of $M$?
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Construct function with given conditions in topological space

I have to give an example (just construct not very formally really) of a function $f \in H^1(\Omega)$ that: 1. $\Omega = (0,1)\times(0,1), \Gamma_1=\{(x,0);0\leq x\leq 1\}\cup\{(y,0);0\leq y\leq 1\}, \...
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1answer
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What does the Jacobian matrix of the projection mapping for Normal bundle look like? (2.3.14 G&P)

I want to solve this question: I feel like the previous question is similar to the one given in this link: Natural projection of tangent bundle is submersion Am I correct? but what does the ...
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1answer
55 views

Non-vanishing section on compact manifolds

Now if we have a compact smooth manifold M and a rank k vector bundle on it. Then I want to find a non-vanishing smooth section on M if $k>dim M$. But I have met some difficulties: The main idea is ...
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1answer
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Smooth no-where vanishing form

Does there exist any no-where vanishing smooth $1$-form on $S^2$. I , think there is such one. For example, consider the smooth $1$-form $\omega=dx+dy+xdz$ on $\Bbb R^3$ consider the pull-back of $\...
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Computing de Rham Cohomology

I'm stuck on the following problem. Let $X=S^{n}\setminus A$, where $A$ is the union of $k\geq 1$ disks $D_{k}$. Use the Mayer-Vietoris sequence to compute the de Rham cohomology $H_{\mathrm{dR}}^{...
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Problem understanding Milnor Proof: (Theorem 1, Vector Fields chapter)

I am readng Milnor's ''Topology From a Differentiable Viewpoint'' and in the chapter about vector fields, page 38, there is a Theorem that states that: Given any vecotr field $v$ on $M \subset \...
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foliations with diagonal leaves

The torus has a foliation by Villarceau circles which are "diagonal" in the sense that the projection from the torus onto either factor, when restricted to a circular leaf of the foliation, is a ...
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Doubts about a definition of the variational derivative

I'm reading some lecture notes on Lagrangian mechanics. The author defines the variational derivative of a function of curves roughly as follows: Let $Q$ be a manifold and $\Gamma_{a,b}$ be the ...