## New answers tagged differential-topology

4
votes

Accepted

### Are there always flat connections?

Just so there'll be an answer: Whether every vector bundle over $G/\Gamma$ admits a flat connection depends on the group $G$ and the subgroup $\Gamma$.
For example, if $G=\mathrm{SU}(2)\simeq S^3$ ...

1
vote

Accepted

### Reference request and prerequisites for understanding the Sphere Theorem and the Loop Theorem in 3-manifold theory

On Marc Lackenby’s webpage you can find notes on 3-manifold topology (Michaelmas 1999). The proof of the loop theorem in Chapter 9 uses special hierarchies (instead of Papakyriakopoulos’ towers) ...

7
votes

### Realizing integral homology classes on non-orientable manifolds by embedded orientable submanifolds

Here are some comments that don't really answer the question, but are too long for the comment box.
Firstly, the Poincaré dual of $\nu\in H_n(M;\mathbb{Z})$ is a twisted integer class $D\nu\in H^{m-n}(...

6
votes

Accepted

### Connectivity of the space of transverse vector fields

Let $F(M)$ be this space of vector fields. Let's work this out for an even-dimensional sphere $M=S^n=S^{2p}$, so $n=2p$ and $k=2p-1$. I claim that it is not rationally $3$-connected, if $n\ge 3$.
This ...

3
votes

Accepted

### Recovering the openness of a map from the openness of its scalar projections

$\newcommand\R{\mathbb R}\newcommand\C{\mathbb C}$The answer to Question 1 is no. Here is a counterexample: $X=\C$, $m=2$, $\R^2$ is identified with $\C$, and
$$F(z):=\exp\Big(-\frac1{|z|\,(2\pi-\arg ...

4
votes

Accepted

### Is a local diffeomorphism with nice boundary values a diffeomorphism?

This is true and follows from a more general fact. Note that in the dimension $n=2$ the unbounded component of $f(\partial\mathbb{D})$ is simply connected.
Theorem. Let $f: \bar{\mathbb{B}}^n \to \...

3
votes

Accepted

### Does a $C^1$ perturbation induces diffeomorphic level set?

I am not completely sure but (at least if $f,g$ are $C^2$) for me the answer is yes.
Indeed, there is a bounded neighborhood $U$ of $f^{-1}(c)$ and $\varepsilon >0$ such that $f,g$, defined as in ...

7
votes

### Embedded 2-tori in $S^1\times S^4$

I think this is true (homotopy implies isotopy). Consider a generic smooth torus embedding $f: T^2 \to S^1\times S^4$, then the projection to $S^1$ gives a circle-valued Morse function on $T^2$. ...

3
votes

### Mapping class groups are finitely generated

For closed connected smooth manifolds $N$ of dimension $d \geq 6$ with $\pi_1(N)$ finite, it is known that $\pi_0(\mathrm{Diff}(N))$ has a classifying space which is a finite CW-complex. This is due ...

2
votes

### Does a $C^1$ perturbation induces diffeomorphic level set?

In general, if we do not assume that $f$ is proper (I missed the word "proper" when I read the question), $c$ need not be a regular value of $g$ for any $\epsilon>0$. For example $0$ is a ...

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