# Tag Info

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### 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
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### 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) ...

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 ...

### 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 $T^2$. ...
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 ...
### 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 ...