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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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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25 views

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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+100

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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50 views

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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21 views

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?
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27 views

Convex combination of projection operators

If $P_1, P_2: V \to V$ are linear projection operators on the vector space $V$ with $R := P_1(V) = P_2(V)$, is it true that any convex combination of $P_1$ and $P_2$ is again a projection operator $...
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Invariance of Linking numbers and critical values

So, I am trying to show that for a map $f: S^{2p-1} \rightarrow S^p$ , the linking number $l(f^{-1}(y),f^{-1}(z))$ of two framed submanifolds associated with regular values $y,z$ of $f$, defined as ...
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38 views

Area of a domain with Stokes' Theorem

This question came up on a preliminary exam: Define $$g(s,t)=(x(s,t),y(s,t))=(\cos(s)+\cos(t),\sin(s)+\sin(t)),$$ on the region $-\pi<s<\pi$, $s<t<s+\pi$. (The function $g$ is one-...
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66 views

Why every map $f : S^n \to T^n (n>1)$ has topological degree zero?

Why every map $f : S^n \to T^n (n>1)$ has topological degree zero? I don't know anything about covering spaces, and has been told to me that this assertion comes from this theory! I do appreciate ...
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41 views

Let $\gamma_1, \gamma_2 : I \to M$ two homotopic curves, $M$ a $C^{\infty}$ manifold. If $\omega$ is a closed $1-$form then:

Let $\gamma_1, \gamma_2 : I \to M$ two homotopic curves, $M$ a $C^{\infty}$ manifold. If $\omega$ is a closed $1-$form then: $$\int_{\gamma_1(I)} \omega = \int_{\gamma_2(I)}\omega.$$ Then, conclude ...
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44 views

2-connected 6 manifolds with boundary $S^5$

What are the 2-connected 6-manifolds that have boundary $S^5$? Are they all of the form $(\sharp_{i=1}^k S^3 \times S^3) \backslash D^6$ for some $k \ge 1$? Also, I think if $M^5$ is simply-...
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29 views

Explicit Dehn twist for $S^n\times S^n$

Fix $n\neq 2$ 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,\...
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33 views

Integration on manifolds with boundary

How can I define integral on manifolds with boundary? To use unity partition don't have I to deal with open sets of the same type, I mean, how can I be sure that there is a unity partition on my ...
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1answer
35 views

The group $\mathrm{Diff}(F)$ and transition functions of a fibre bundle.

Let $M$ and $F$ be differentiable manifolds, and let $F\to E\to M$ be a differentiable fibre bundle over $M$. A trivialising cover $\{(U_i,\phi_i)\,|\,i\in I\}$ of $M$ determines a set $\{t_{ij}:U_{ij}...
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78 views

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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1answer
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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421 views

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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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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70 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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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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1answer
88 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
51 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 $...
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247 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-...
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18 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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38 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? ...
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1answer
56 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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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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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
46 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
74 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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A problem possibly about the Whitney Embedding Theorem.

Problem. (1) Suppose $F:S^1 \to \mathbb{R}^3$ is a smooth immersion (i.e. $dF$ is nowhere zero). Prove that there is a unit vector $\mathbf v$ in $\mathbb{R}^3$ such that $\pi_{\mathbf v} \circ F : ...
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Is $T(M)$ necessarily equivalent to the normal bundle of $M$ in $\textbf{R}^{2n}$, if $M$ is totally real?

Consider real submanifolds, $M^n \subset \textbf{C}^n$. ($\textbf{C}^n \equiv (\textbf{R}^{2n}, J)$, where $J(x, y) = (-y, x)$). $M^n$ is totally real if $J(T_pM) \cap T_pM = 0$, for all $p \in M$. ...
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42 views

Discretizations of Differential, Geometric and Topological Notions

I have noticed a recurring theme in Graph Theory / Theoretical Computer Science (abbreviated GT and TCS throughout this post) in that notions typically belonging to differential calculus / geometry / ...
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1answer
31 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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42 views

Isomorphic modules of sections imply isomorphic bundles

For reference this is about a part of question 3-F in Characteristic Classes by Milnor and Stasheff, which discusses the $C(B)$-module structure of the space of sections of a topological vector bundle ...
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31 views

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 ...
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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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1answer
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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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48 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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123 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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36 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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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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1answer
49 views

$G$-structure defined by a tensor

Let $M$ be an $n$ dimensional manifold with its bundle of linear frames $\pi:L(M)\to M$. Suppose $T_0$ is a tensor on $\mathbb R^n$ and $u\in L(M)$. We may view $u$ as a linear map $u:\mathbb R^n\to ...
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1answer
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 $\...
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1answer
66 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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63 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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1answer
13 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
19 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 ...