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.

learn more… | top users | synonyms

0
votes
0answers
15 views

Showing an isomorphism using Mayer Vietoris

Let $M$ be a connected smooth manifold of dimension $n \geq 3$. For any $x \in M$ and $0 \leq p \leq n-2$, prove that the map $H_{dR}^p(M) \to H_{dR}^p(M\smallsetminus {x})$ induced by inclusion $M ...
0
votes
0answers
14 views

Prove that integration of a differential $k$-form is independent of choice of basis

This is Exercise 4 of Section 33 of Munkres' "Analysis on Manifolds" book: (Let $A$ be an open set in $\mathbb{R}^k$.) If $\eta$ is a $k$-form in $\mathbb{R}^k$ and if ...
0
votes
1answer
19 views

Reference about the space of closed curves in Riemann manifold

Some days ago, I listened a report about the width of Riemann manifold. I am interesting in the space of closed curves in Riemann manifold. It seemly has good topology and different construction.For ...
3
votes
1answer
155 views

Thom-Pontryagin construction of manifolds with boundary

Thom-Pontryagin construction gives the 1-1 correspondence between framed cobordism classes of $k$-dimensioanl sub-manifolds of $S^{n+k}$ and homotopy classes of maps from $S^{n+k}$ to $S^n$. Are ...
2
votes
0answers
28 views

Smoothing a continuous section in a fibre bundle.

Let $\pi: X \rightarrow M$ be a smooth fibre bundle and let $p^{1}_{0} : X^{(1)} \rightarrow X$ be its 1-jet bundle. Suppose there is a $\mathcal{C}^{1}$ section $h: M \rightarrow X$ such that it is ...
1
vote
0answers
20 views

On lifts of a trajectory of a quadratic differential

Let $X$ be a Riemann surface of genus $g\ge 2$ and $q$ a holomorphic quadratic differential on $X$. The differential $q$ defines a flat metric with conical singularities on $X$: if $q=f(z)dz^2$ ...
5
votes
1answer
55 views

Does the set of diffeomorphisms which are induced by flows form a group?

Let $M$ be a smooth manifold. Consider the set of diffeomorphisms which are induced by flows of vector fields. (which are not time-dependent) Is this set a subgroup of $\text{Diff}(M)$? (Note that ...
1
vote
0answers
14 views

Cobordism Groups an the Pontryagin-Thom Construction

I am confused by the statement that $\Omega^\text{framed}_1(S^3) \cong \mathbb{Z}$ which I came across as an application of the Pontryagin-Thom construction for showing that $\pi_3(S^2) \cong ...
3
votes
1answer
32 views

Does there exist a non-parallelisable manifold with exactly $k$ linearly independent vector fields?

For any positive integer $k$, is there a smooth, closed, non-parallelisable manifold $M$ such that the maximum number of linearly independent vector fields on $M$ is $k$? Note that any such $M$, ...
0
votes
0answers
14 views

Generate Lie-Algebra $su(N)$

Let $A \in \mathbb{C}^{n \times n}$ be a diagonal matrix with $Tr(A)=0$ and all eigenvalues (purely imaginary eigenvalues) are mutually different from each other. Hence, in particular $A \in ...
0
votes
1answer
17 views

Computing the degree of a one-variable map

Edit: This question originally contained a typo where the function $f$ specified below was equal to $x$, not $x^2$ as currently written, outside an interval $[-T,T]$, and the accepted answer was ...
9
votes
3answers
75 views

Can we recover a compact smooth manifold from its ring of smooth functions?

It is well-known that if $X$ is a reasonably nice topological space (compact Hausdorff, say) then we can recover $X$ from the ring $C(X)$ of continuous functions $X\to\mathbb R$; see this MSE question ...
2
votes
1answer
38 views

Example of a diffeomorphism from all of $\mathbb{R}$ to itself

I can think of diffeomorphisms from an interval to $(a,b)\rightarrow \mathbb{R}$, scaling the tangent function, and from the punctured plane, polar coordinates, or some odd polynomial, but does anyone ...
0
votes
0answers
15 views

O.D.E. in Homogeneity Lemma

Let $\psi: \mathbb{R}^{n} \to \mathbb{R}$ smooth such that $\psi(x) > 0$ for $x \in B(0,1)$ and $\psi(x) = 0$ for $x \notin B(0,1)$. Let $c \in S^{n-1}$ fix and arbitrary and consider the O.D.E. ...
5
votes
3answers
294 views

Application of Hodge decomposition

Hodge decomposition states any $p$ form can be decomposed into three orthogonal $L^2$ components: exact form, co-exact form and hamonic form. But actually we don't know how to decompose a general one. ...
0
votes
0answers
42 views

Computing relative de Rham cohomology?

Let $f: N \to M$ be a smooth map. Then in Bott and Tu they define a relative de Rham complex $\Omega^{\bullet}(f)$ where $\Omega^k(f) = \Omega^k(M) \oplus \Omega^{k-1}(N)$ with coboundary map given in ...
0
votes
1answer
224 views

Local Submersion Theorem - Differential Topology of Guillemin and Pollack

Local Submersion Theorem : Suppose that $f:X \to Y$ is a submersion at $x$, and $y=f(x)$. Then there exist local coordinates around $x$ and $y$ such that $f(x_1,...,x_k)=(x_1,...,x_l)$. That is, $f$ ...
3
votes
1answer
67 views

Cut out characteristic Submanifold N ($w_1(M)=w_1$(Normalbundle of N in M)). Remainder M-N is orientable? Orientation Character or CW-Structure?

So I try to understand the following (which is taken from Dold, "Structure of the cobordism ring", Page 3/274, in the paragraph "1. La suite exacte de Wall."): https://eudml.org/doc/109581 ): Giving a ...
5
votes
1answer
152 views

Quotients of $S^{2n+1}$

Any sphere $S^{2n+1} = SO(2n+2)/SO(2n+1)$ can be thought to be given as the zero-set in $\mathbb{C}^{n+1}$ of the equation, $\sum_{i=1}^{n+1} \vert z_i \vert ^2 = 1$ Now say one wants to quotient it ...
2
votes
1answer
292 views

“Completing” a vector field on a non-compact manifold $M$

Suppose I have a non-compact smooth manifold $M$ and an arbitrary nowhere-vanishing smooth vector field $X$ on $M$ which is not complete. Is there a way to create a smooth vector field $V$ that is ...
3
votes
2answers
57 views

Non-compact 3-manifold with incompressible boundary

Is there an orientable, irreducible, non-compact, 3-manifold $M$ with $\partial M\cong \Sigma_2$ , genus 2 orientable surface, with $\pi_1(M)\cong \pi_1(\Sigma_2)$ and $M$ not $\Sigma_2\times ...
6
votes
1answer
272 views

Question on “up to isotopy” when attaching two spaces

Let $M$, $N$, $A$, $B$ be topological spaces (or manifold) such that $A$ and $B$ are subspaces in $M$, $N$ respectively. Let $f: A \to B$ and $g:A \to B$ be homeomorphism and assume that $f$ and $g$ ...
0
votes
0answers
15 views

Which constructions on vector bundles satisfy a universal property?

I'm wondering whether constructions on vector bundles, such as the Whitney sum or tensor product of two bundles, satisfy some kind of universal property. What I mean by "some kind of universal ...
3
votes
0answers
21 views

How to keep symplecticity of a diffeomorphism after a coordinate rescaling and Taylor series expansion?

Let's take some, implicitly written, symplectic diffeomorphism $F$ of $\mathbb{R}^{2n}$, let it preserve the (possibly non-standard) symplectic form $\omega$. Suppose I introduce a small parameter ...
2
votes
2answers
64 views

What does $C^k$ at a single point mean?

I'm reading textbooks on manifolds. I usually see a saying that "function $f$ is $C^k$ at a point $p \in M$". I am confused with this. What does this mean? Furthermore if $f$ is $C^k$ at $p$, is it ...
1
vote
0answers
12 views

Why do we need the $S \otimes L^{1/2}$ bundle product to determine a $Spin_c$ structure?

I am reading Marino's book on topological field theory and 4-manifolds and I am very confused in the construction of the $Spin_c$ structures for manifolds that do not admit a $Spin$ structure. In the ...
1
vote
0answers
20 views

Construct a lifting from the the total space of a fiber bundle to its spoace of 1-jets

I am reading the book "Convex Integrtion Theory by D.Spring" and need some help. Let $\pi:X \rightarrow V$ be a smooth fiber bundle over a smooth base manifold $V$. Let $h\in\Gamma^{0}(X)$ and ...
16
votes
0answers
357 views

On the Constant Rank Theorem and the Frobenius Theorem for differential equations.

Recently I was reading chapter $4$ (p. $60$) of The Implicit Function Theorem: History, Theorem, and Applications (By Steven George Krantz, Harold R. Parks) on proof's of the equivalence of the ...
0
votes
0answers
16 views

Derivative of vector field on a manifold sends the tangent plane to itself

A vector field $\vec{v}$ on a (smooth) manifold $X \subset \mathbb{R}^N$ is a map $\vec{v}:X \to \mathbb{R}^N$ such that $\vec{v}(x)\in T_x(X)$ for every $x \in X$. Suppose $\vec{v}(x)=0$, show that ...
0
votes
1answer
27 views

Let $f : M\to N$ be a submersion with $M$ compact and $N$ connected. The $f$ is surjective.

I have no idea how to do this. I tried think in, once $N$ is connected and locally path connected it has to be path connected, but, this does not help. Any hints, solutions, will be very appreciate... ...
1
vote
1answer
77 views

compactification of $\Bbb R^n$

I want to show that $n$ dimensional real projection space is compactification of $\Bbb R^n$, how can I do that? I can show PR^n is compact, and R^n is localy compact, but I have problem to show R^n ...
0
votes
1answer
41 views

Example of a disconnected manifold where the tangent space is not the dimension of the manifold?

Wikipedia says that the tangent spaces of a connected manifold all have the same dimension, equal to that of the manifold. Well, is there an example of a simple disconnected manifold that doesn't ...
1
vote
1answer
42 views

Two proof of Petersen's 'Manifold'

Picture below is from the 5 page of Petersen's Manifold. First, why diffeomorphism is forced to be a lieanr isomorphism? The define of diffeomorphism accords to Wiki. Second , what space the point ...
1
vote
0answers
24 views

De Rham cohomology ring of flag bundles/manifolds in Bott and Tu

I'm trying to understand the result for the de Rham cohomology ring of flag manifolds in Differential Forms in Algebraic Topology by Bott and Tu. I'm sort of starting from the result and working ...
1
vote
0answers
30 views

$f: M \to Y$ $C^{1}$ map between manifolds $M$ e $N$ with dimension $m$ and $n$ respectively then $f$ is locally proper

We say that $f:f: M \to N$ is locally proper if for all $x \in M$ there exist $V \ni x $ open in $M$ such that $f|_{\overline{V}}:\overline{V} \to Y$ is proper. I know two ways to prove it. First, ...
2
votes
1answer
49 views

Degree theory and Invariance of domain

We'll use the Proposition (F) to show that: (Invariance of domain) Let $f: M \to N$ be a proper smooth mapping of two oriented, boundaryless, smooth manifolds of dimension $m$; furthermore, $N$ is ...
0
votes
1answer
24 views

Bundle that is isomorphic to the bundle of Whitney sum

I am involved with one question that a friend of mine asked. If a vector bundle $(E,\pi,M)$ has two sub-bundles, $(F,\xi,M)$, $(L,\psi,M)$, and $\pi^{-1}(x) \cong \xi^{-1}(x) \oplus \psi^{-1}(x),$ ...
0
votes
1answer
25 views

On computing the Differential of a Smooth Map

In class, we proved Jacobi's formula for the differential of the determinant using the following formula for the differential of a smooth map $F$ between manifolds $M$ and $N$ $$D_A F(B) = ...
0
votes
1answer
22 views

Coset spaces of a lie group and fiber bundles

I am reading Steenrod. He writes: Another example of a bundle is a Lie group $B$ operating as a transitive group of transformations on manifold $X$. The projection is defined by selecting a point ...
0
votes
0answers
24 views

Topologies of partially exponentiated lie algebras, especially in regard to $SU(2)$

Consider the fundamental respresentation of $\mathfrak{su}(2)$ given in terms of the Pauli matrices as $\mathfrak{su}(2) = \langle ...
0
votes
0answers
18 views

Normal bundle of sub-manifold is a manifold

This is exercise 2.3.12 of Guillemin and Pollack: Let $Z$ be a sub-manifold of $Y$, where $Y \subset \mathbb{R}^M$. Define $N(Z;Y)=\{(z,v):z\in Z, v\in T_z(Y), v \perp T_z(Z)\}$. Prove that $N(Z;Y)$ ...
3
votes
1answer
28 views

How to show that is a submanifold or How to derivate the determinant function?

I am trying to show that the space of $2\times 2$ matrix with rank equals $1$ is a submanifold of $\mathbb{R}^4 - \{0\}$ whoose the dimension equals $3$. To do this, I have defined $\det : ...
-1
votes
2answers
65 views

Covering spaces as fiber bundles

I am reading Steenrod. I think there is something wrong or sloppy in his definition of a covering space as a fiber bundle. He writes: A fibre bundle consists of: (i) A topological space $B$ (ii) a ...
2
votes
2answers
75 views

Doubt in definition of symmetric continuous function and norm in Kupka's paper

In this article "Counterexample to the Morse-Sard theorem in the case of infinite-dimensional manifolds" of I. Kupka has the following passage: For $H=l^{2}$ "Let $H^{*}$ be the dual of $H$. A base ...
2
votes
1answer
22 views

The tangent space of the moduli space of connection?

I'm reading one of Floer's paper. (An Instanton-Invariant for 3-Manifold). Let $M$ be a $3$-manifold. A principal $SU_2$-bundle P over $M$ must be trivial. Fixed a trivialization $P \cong M \times ...
2
votes
2answers
51 views

Representable homology classes on smooth manifolds

Let $X$ be a closed (compact without boundary) smooth manifold. We can consider its singular homology $H_*(X,\mathbb{Z})$. Let $H_{k}(X,\mathbb{Z})$ be the $k$-th singular homology group of $X$ and ...
0
votes
0answers
15 views

covariant derivative of a helicoid

Given a helicoid $S$ parametrized by $x(u,v)=(v\cos(u),v\sin(u),u)$, a point $p=(1,0,0)$ on the helicoid, a tangent vector $v=(2,1,1)$ on $T_pS$ and a tangent vector field ...
0
votes
0answers
22 views

When the boundary of a manifold is orientable?

I am not sure whether the boundary of some manifold is definitely a manifold, but let's assume it is anyway. Then in what case the boundary is an orientable manifold. Maybe when the manifold can be ...
1
vote
0answers
23 views

Construct manifold from vector fields and point in $\mathbb{R}^n$

Suppose one is given a set of vector fields $X_{\alpha} \in \mathbb{R}^n$. Is it generally possible to construct a manifold embedded in $\mathbb{R}^n$ going through a particular point and whose ...
2
votes
0answers
40 views

What is an integral differential form and how do we recognize it as such?

I am reading about embedding theorems of various types of manifolds (Kodaira's embedding theorem being one of them), and one condition that repeats in all of them is that the manifolds should be ...