# Questions tagged [differential-topology]

The study of differentiable manifolds and differentiable maps. One fundamental problem is that of classifying manifolds up to diffeomorphism. Differential topology is what Poincaré understood as topology or “analysis situs”.

**5**

votes

**0**answers

64 views

### Extension of Vector Field in the $\mathcal{C}^r$ topology

This question was previously posted on MSE.
Let $M\subset \mathbb{R}^n$ be a compact smooth manifold embedded in $\mathbb{R}^n$, we define $$\mathfrak{X}(M) := \{X: M \to \mathbb{R}^n;\ X\mbox{ is ...

**3**

votes

**0**answers

56 views

### Globalising fibrations by schedules

In a paper in Fund Math 130.2 (1988): 125-136. http://eudml.org/doc/211719 Dyer and Eilenberg give an account of the local-global theorem for fibrations by proving a "Schedule Theorem" that, given a ...

**12**

votes

**2**answers

674 views

### Examples of odd-dimensional manifolds that do not admit contact structure

I'm having an hard time trying to figuring out a concrete example of an odd-dimensional closed manifold that do not admit any contact structure.
Can someone provide me with some examples?

**0**

votes

**0**answers

9 views

### Boundary value problems for differential inclusions with fractional order

I'm having technical problems or just lack of knowledge problems, so I would appreciate your help.
the problem::
let $v_{*}\in F$, and for every $w \in F$, we have
$$|v_{n}-v_{*}| \leq |v_{n}-w|+|w-...

**1**

vote

**1**answer

93 views

### Classification of all equivariant structure on the Möbius line bundles

Is there a classification of all equivariant structures of the M?bius line bundle $\ell\to S^1$?.
For example the antipodal action of $\mathbb{Z}/2\mathbb{Z}$ on $S^1$ cannot be lifted to the total ...

**3**

votes

**0**answers

180 views

### Do smooth manifolds admit unique cubical structures?

It seems to me that a smooth manifold should admit the structure of a cubical complex by Morse theory, since handle attachments seem to be perfectly cubical maps.
Is this cubical structure "...

**3**

votes

**2**answers

305 views

### Is this a submanifold?

Let $(M,g)$ be a compact Riemannian manifold with an isometric action $\rho : G \to \mathrm{Iso}(M)$ by a compact Lie group $G$. There is a natural extension of $\rho$ to $TM$ given by:
$$\psi : G \...

**9**

votes

**1**answer

286 views

### Smooth vector fields on a surface modulo diffeomorphisms

Let $\Sigma$ be a two-dimensional connected smooth manifold without boundary. (Feel free to assume it is compact and orientable.)
Let $\mathcal{X}(\Sigma)$ denote the smooth vector fields on $\Sigma$...

**2**

votes

**1**answer

124 views

### $L^{2}$ Betti number

Let $\tilde{X}$ be a non-compact oriented, Riemannian manifold adimits a smooth metric $\tilde{g}$ on which a discrete group $\Gamma$ of orientation-preserving isometrics acts freely so that the ...

**1**

vote

**0**answers

207 views

### On the homotopy type of $\mathrm{Diff}(\mathbb{S}^3)$

I am confused with the following argument. I know I am doing something wrong but I can't find my mistake.
On one hand, one knows that if $M$ is a Lie group, then
$$\mathrm{Diff}(M)\simeq M\times\...

**0**

votes

**1**answer

127 views

### Homology of universal abelian cover of a manifold

If one define the universal abelian covering $M_0$ of a manifold $M$ as the abelian covering (i.e. normal covering with abelian group of deck transformations) that covers any other abelian covering, ...

**3**

votes

**0**answers

54 views

### Symplectic vector fields everywhere transverse to a co-dimension one hypersurface

Usually when speaking about vector fields transverse to a hypersurface in a symplectic manifold, we talk about Liouville vector fields, i.e. vector fields $X$ with the property that $\mathcal{L}_X\...

**3**

votes

**0**answers

191 views

### What mathematical background to i need in order to understand proofs of the h-cobordism theorem?

I am about to finish my undergraduate studies and I really enjoyed the topology and differential-geometry classes. I'd love to continue studying differentialtopology and i considered doing some ...

**2**

votes

**0**answers

62 views

### Transitivity of Diff on the space of embeddings of balls

Given 2 symplectic embeddings $g_0$ and $g_1$ of a 4-ball of radius $r \leq 1$ into the 4-ball of radius 1 (all equipped with the standard symplectic form coming from $\mathbb{R}^4$), does there exist ...

**8**

votes

**1**answer

235 views

### Different definitions of the linking number

Assume that
$$
\iota_1:\mathbb{S}^k\to\mathbb{R}^n,
\quad
\iota_2:\mathbb{S}^\ell\to\mathbb{R}^n,
\quad
k+\ell=n-1,
$$
are two embeddings of spheres with disjoint images $\iota_1(\mathbb{S}^k)\cap\...