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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What does it take for a smooth homeomorphism to be a diffeomorphism?

I have an open subset $A$ of $\mathbb{R}^k$ and a subset $B$ of $\mathbb{R}^n$, $n>k$, that are homeomorphic and $f:A\longrightarrow B$ is a smooth homeomorphism between two sets. I'm wondering if ...
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30 views

Let $\deg$ be the topological degree. Then $\deg(fg) = \deg(f)\deg(g)$, with $f, g : M \to N$

Recall that the topological degree is defined as: Let $f : M \to N$ a $C^k$ function and $y$ be a regular value of $f$. Then we define: $$\deg(f)= \sum_{f(x) = y}|Df(x)|,$$ where $| . |$ means the ...
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75 views

How can we assume the first homology group of the complement is zero when constructing a Casson handle?

I am currently working through Scorpan's Wild World of 4-Manifolds specifically the section on Casson Handles. On page 78, he says if $D$ is the core of the handle after $n$ stages we may assume $\...
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41 views

Bott's topological obstruction to integrability

Let $M$ be a smooth manifold, $E\subseteq TM$ be an integrable distribution and $\pi:TM\to Q=\frac{TM}{E}$ the quotient bundle with $\dim Q=q$. Bott showed that \begin{equation} Pont^k(Q)=0 \qquad \...
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1answer
54 views

Hodge theory on semi-Riemannian manifolds [reference request]

I need to learn a bit of Hodge theory on manifolds and I am looking for a reference which covers the case where the metric has arbitrary signature $(p,q)$. Most books I have found seem to focus on the ...
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148 views

Sard's Theorem Using Integration?

How exactly do we clean up this heuristic proof of Sard's theorem, from Schwarz' `Differential Topology for Physicists' using the Jacobian $J(f)(x)$: "A heuristic justification for Sard's theorem ...
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20 views

surgery of type $(\lambda,n-\lambda)$ on manifold, h-cobordism theorem by Milnor

I am reading Milnor's Lectures on h-cobordism theorem, and I am stuck on Milnor's definition on surgery of type $(\lambda,n-\lambda)$ on manifold, where the definition following can be found on page ...
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39 views

Is the space of rigid Riemannian metrics convex?

Let $M$ be a smooth manifold. Let $g_1,g_2$ be two rigid Riemannian metrics on it. (i.e with no isometries except the identity). Is it true that every convex combination of $g_1,g_2$ is also rigid? ...
3
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1answer
90 views

Show that $\Omega^*(M)\otimes \Omega^*(N)$ is isomorphic to $\Omega^*(M\times N)$

Let $M, N$ be compact manifolds and $\Omega^*$ its algebra exterior. How to prove that $\Omega^*(M)\otimes \Omega^*(N)$ is isomorphic to $\Omega^*(M\times N)$? I thought about the function $f(\omega,...
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1answer
67 views

Understanding Milnor's proof of the fact that the preimage of a regular value is a manifold

In the book "Topology from the Differential Viewpoint" (Milnor) he proves on page 11 the following lemma: If $f: M\to N$ is a smooth map between manifolds of dimension $m\geq n$ and if $y\in N$ is ...
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21 views

Is the Inverse of a Coordinate System on a Differentiable Manifold Differentiable?

In his book Calculus on Manifolds, Spivak shows that a differentiable k-dimensional manifold in $\mathbb{R}^n$ is a set $M$ such that $\forall x \in M$ there exists an open set $U$ containing $x$, an ...
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2answers
59 views

non-homotopic maps $\mathbb S^2\rightarrow \mathbb{P}_\mathbb{R}^2$

How can I find non-homotopic maps $\mathbb S^2 \rightarrow \mathbb{P}_\mathbb{R}^2$? I know that it is enought that the degree are different. But the canonical map given by the antipodal map has ...
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1answer
75 views

Why is $T$ in $\mathcal T_2^1(M)$?

Let $M$ be smooth manifold and $\nabla$ an affine connection on $M$. Then the torsion tensor of $\nabla$ is the map $T:\mathcal T(M)\times\mathcal T(M)\to\mathcal T(M)$ and $$T(X,Y)=\nabla_XY-\...
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1answer
47 views

Planar immersion of circle not approximate embedding

On a topology preliminary exam, students in past years were asked to find an immersion $f:S^1\to\Bbb R^2$ which cannot be approximated by an embedding (in the sense of the weak Whitney theorems). I ...
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1answer
36 views

Complete Vector field

I am reading "Geometry of Differential Forms". We want to show that on a smooth compact manifold, vector fields are complete. We claim that there is an interval $(-\epsilon ~ ~\epsilon)$ of time ...
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50 views

What is the index of a vector field with positive divergence?

Let $v:\Bbb R^n\to\Bbb R^n$ be a smooth vector field with an isolated zero at $z\in\Bbb R^n$. Suppose that $(\operatorname{div}v)(z)>0$. Can we say anything about the index of $v$ at $z$? I know ...
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46 views

What is the moment of inertia tensor of a hollow spheroid?

I am looking for exact or even approximate formulas for the moment of inertia of a hollow spheroid (oblate and prolate.) I have find formulae for a hollow spherical shell and for a filled ellipsoid ...
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38 views

Avoiding the spherical polar coordinate singularity on $S^2$ by using a double cover?

Is it possible to avoid the spherical polar coordinate singularity on $S^2$ by taking the coordinates as they originally are on $T^2$, i.e. ranging from $0$ to $2\pi$ mod $2\pi$? How would one ...
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33 views

A version of the regular value theorem [duplicate]

Assuming the regular value theorem, let $$f : \mathbb{R}^n\times \mathbb{R}^k \to \mathbb{R}^n.$$ Let $N = \{ x \in \mathbb{R}^n : f^1(x) = \ldots = f^{n-1}(x) = 0, ~~ f_n(x) \ge 0\}$. Supposing ...
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49 views

Is this map an immersion?

Let $g:\mathbb{R}^2\to \mathbb{R}^4,\ (x,y)\mapsto ((2+3\cos(2\pi x))\cos(4\pi y),\ (2+3\cos(2\pi x))\sin (4\pi x),\ 3\sin(2\pi x)\cos(2\pi y),\ 3\sin(2\pi x)\sin(2\pi y))$ I have to prove that for ...
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130 views

Is $n$-dim manifold which is immersed in $\mathbb{R}^n$ diffeomorphic to a ball?

Let $M$ be an $n$-dimensional smooth compact oriented manifold with boundary which is immersed in $\mathbb{R}^n$ (codimension zero). Must $M$ be diffeomorphic to a ball with boundary (the closed unit ...
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55 views

Non-homotopical manifolds with same de Rham cohomology

I am searching for manifolds $M$ and $N$ with different homotopy type such that their de Rham cohomology is isomorphic as rings. It would, of course, be enough to find $M$ and $N$ with different $\...
5
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67 views

For any smooth manifold, is it true that for any two points on the manifold, there exists a chart that covers the two points?

Some say that we can connect two points with a continuous curve and a small contractible neighborhood of the path together with the charts of the points can be regarded as the chart. But I don't know ...
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1answer
14 views

Vector fields- differential topology

Can anybody please explain me the reason for last 6th line. (The set of all vectors at all points...)
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1answer
21 views

Extending a section to a basis of sections in a trivial bundle.

Suppose we have a manifold $M$ of dimension $m$ and let $E$ be the rank $n$ trivial vector bundle on $M$. Let $\Gamma$ be a section of the trivial bundle $E$. My question is, are there some conditions ...
2
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1answer
52 views

Exact sequences of bundles and orientations

If we have an exact sequence of finite-dimensional vector spaces $$0\rightarrow E'\rightarrow E\rightarrow E''\rightarrow0$$ then an orientation of any two induces an orientation of the third. I have ...
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28 views

Applications of Banach's fixed point theorem on Differential Geometry

Does anyone know any simple application of Banach's fixed point theorem on Differential Geometry. I am looking for something involving manifolds.f
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64 views

Why does a tangent bundle have dimension 2n instead of n?

Let $n=dim(T_pM)$ for every $p\in M$, where $M$ is a smooth manifold. I understand that specifying $p$ is not enough to determine an element of $TM$, but what if do we specify only $v\in T_pM\subset ...
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54 views

Simple properties of wedge product [closed]

How to prove a) $\omega \wedge \eta =(-1)^{kl}\eta\wedge\omega, \omega$ is $k$-tensor and $\eta$ is $l$-tensor. b)$f^*(\omega \wedge \eta)=f^*(\omega)\wedge f^*(\eta)$ where $f:V\rightarrow W$ ...
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1answer
25 views

Definition of a cubic coordinate system

I'm looking at "Foundations of Differentiable Manifolds" by Frank Warner, and have a question about one of the basic definitions at the beginning of the book. He writes: A coordinate system $(U,\phi)$...
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29 views

Proving that every tangent bundle is direct summand of a trivial bundle.

I am trying to prove that there exists a formal immersion from a manifold $M$ of dim $m$ into $\mathbb{R}^{2m}$. Formal immersion is just an injective bundle map from $TM$ into $T\mathbb{R}^{2m}$ that ...
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1answer
31 views

What is the smoothness of a family of diffeomorphisms $t\mapsto \psi_t \in \text{Diff}(M)$ ? And how to interpret it intuitively?

First of all, we have to give the group $\text{Diff}(M)$ of all diffeomorphisms on $M$ a smooth-manifold structure. (To see this, it may be helpful to consider a easier problem: how to give the group $...
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1answer
38 views

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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1answer
20 views

Can I conclude $s$ is a submersion from these data?

Let $M$ and $N$ be smooth manifolds ($C^\infty$). Let $s\in C^\infty(M, N)$ and $u\in C^\infty(N, M)$ be maps satisfying: $u$ is an embedding; $s\circ u=\textrm{id}_N$; $(u\circ s)^2=u\circ s$. ...
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Proving that an $E$-oriented manifold has an $E$-oriented normal bundle

This is the setting we are working in: $M$ is a closed, smooth $n$-manifold embedded in $\mathbb{R}^{n+k}$ with a chosen embedding $e\colon M^n\to \mathbb{R}^{n+k}$. It is $E$-oriented, for $E$ a ...
0
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1answer
46 views

What is the topological degree of the constant map?

What is the topological degree of the constant map? To me it does not make any sense, once $f$ being the constant map has no regular values. So, how to proceed?
0
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1answer
43 views

Proper map and manifolds

Let $M$ and $N$ two manifolds which have the same dimension, $f:M\to N$ a map $\mathcal{C^\infty}$. We suppose that $M$ is compact and we have $b$ a regular value of $f$. First, I have to prove that $...
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60 views

Why $h$ has zero topological degree?

I am trying to prove that $f,g : M^n \to S^n$, both $C^1$ (indeed just $C^0$ is enough) with the same topological degree are homotopic. I saw on a book that the trick is as follows: Take $W = M\...
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1answer
29 views

There is no smooth map from $R^3$ to $R$ such that there i

I have to prove there is no smooth map $f:\mathbb{R}^3 \to \mathbb{R}$ and no regular value $y$ of $f$ such that $f^{-1}(y)$ is the projective space of dimension 2. From the pre-image theorem, $f^{-...
4
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1answer
72 views

Relation between tangent spaces of (un)stable manifolds in Morse theory

After asking this question about signs in the Morse complex, I realised that my confusion is really about how tangent spaces to different (un)stable manifolds are related. So suppose we have a Morse ...
2
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1answer
91 views

On the proof of that fixed point set of an involution is a submanifold

Let $M$ be a smooth manifold, and let $f:M\to M$ be a smooth involution (i.e. $f^2=\text{id}$). If we introduce a Riemannian metric on $M$ so that $f$ is isometry, we can prove easily that the fixed ...
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Looking for two non diffeotopic embeddings $\mathbb{R}\rightarrow\mathbb{R}$

im studyin the book "Introduction to differential topology" by Th. Bröcker and K. Jänich and I'm stuck with two of the exercices in Chapter 9 on isotopy. A short overview (for those who are familiar ...
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42 views

Radius of $\mathbb{CP}^n$

The question I am asking is basically Is it possible/usual to define a "radius" for certain metrics on $\mathbb{CP}^n$ in analogy to the case for $S^n$? To provide some more context about what I ...
5
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1answer
57 views

Explicit verification of signs in Morse complex

I'm trying to check by hand that the signs in the Morse complex, defined via choices of orientations on the unstable manifolds, lead to $\partial^2=0$. The books I've looked in seem to say either ...
4
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2answers
56 views

What is the relationship between diffeomorphisms of the sphere modulo isotopy and exotic spheres?

In his "Classification of (n-1)-connected 2n-dimensional manifolds and the discovery of exotic spheres", Milnor observes that since his exotic 7-spheres admit a Morse function with only two critical ...
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1answer
38 views

$1$-forms $\omega$ on $S^1$ are the differential of functions provided $\int_{S_1}\omega=0$

Prove that a (smooth) $1$-form $\omega$ on $S_1$ such that $\int_{S_1}\omega=0$ is the differential of some $f:S^1\to\mathbb{R}$. Hint: Let $h:\mathbb{R}\to S^1$ defined by $h(t)=(\cos(t),\sin(t))...
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1answer
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Restriction of an $n$-form to a submanifold

I am currently studying exterior calculus on manifolds and am not sure if I understand things correctly. In my textbook (R.W.R. Darling, Differential Forms and Connections) there is an example of a $2$...
5
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1answer
116 views

Poincare lemma for compact vertical supports in Bott & Tu

I'm trying to work out Proposition (6.16) from Bott and Tu which states that $H^*_{cv}(M\times\mathbb{R}^n)$ is isomorphic to $H^{*-n}(M)$. The first group being forms compactly supported along the ...
0
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1answer
52 views

Inversion of Sphere

I was reading about inversion of sphere. Wikipedia defines it as: Let $f: S^2\to R^3$ be the standard embedding; then there is a regular homotopy of immersions $f_t\colon S^2\to R^3$ such that $...
27
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2answers
265 views

Orientability of $m\times n$ matrices with rank $r$

I know that $$M_{m,n,r} = \{ A \in {\rm Mat}(m \times n,\Bbb R) \mid {\rm rank}(A)= r\}$$is a submanifold of $\Bbb R^{mn}$ of codimension $(m-r)(n-r)$. For example, we have that $M_{2, 3, 1}$ is non-...