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 $G=\mathrm{SU}(2)\simeq S^3$ ...
- 105k
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) ...
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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}(...
- 34.6k
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 ...
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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 ...
- 106k
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 \...
- 25.9k
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 ...
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7
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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 $T^2$. ...
- 65.7k
3
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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 ...
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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 ...
- 25.9k
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