All Questions
Tagged with at.algebraic-topology differential-topology
503
questions
12
votes
0
answers
213
views
When can we extend a diffeomorphism from a surface to its neighborhood as identity?
Let $M$ be a closed and simply-connected 4-manifold and let $f: M^4 \to M^4$ be a diffeomorphism such that $f^*: H^*(M;\mathbb{Z})\to H^*(M;\mathbb{Z})$ is the identity map. Moreover, let $\Sigma \...
4
votes
1
answer
330
views
Bott & Tu differential forms Example 10.1
In Bott & Tu's "Differential forms", Example 10.1 states:
$\textbf{Example 10.1}$ Let $\pi: E \to M$ be a fiber bundle with fiber $F$. Define a presheaf on $M$ by $\mathcal F(U) = H^q(\...
2
votes
0
answers
138
views
Compute the Euler class of tautological $C$-bundle over $CP^1$
$\DeclareMathOperator\SO{SO}$This might be an old question. But since I have not found an explicit answer to this question, I put the question here.
The background is that we need to use a similar ...
1
vote
0
answers
55
views
extendability of fibre bundles on manifolds with same dimensions
Let $M$ be an $m$-manifold. Let $M'\subseteq M$, where
$M'$ is also an $m$-manifold.
Let $N$ be an $n$-manifold. Let $N'\subseteq N$, where
$N'$ is also an $n$-manifold.
Suppose there is fibre ...
6
votes
0
answers
128
views
Are there isospectrally equivalent exotic spheres?
Let $X$ and $Y$ be two different exotic spheres. Are there metrics $g$ and $h$ on $X$ and $Y$, respectively, such that the laplacians of $(X,g)$ and $(Y,h)$ have the same spectrum?
I would be happy ...
8
votes
1
answer
434
views
On the definition of stably almost complex manifold
According to Adams' paper "Summary on complex cobordism", a manifold is
stably almost-complex if it can be embedded in a sphere of sufficiently high dimension with a normal bundle which is a ...
2
votes
0
answers
94
views
lifting a family of curves to a family of sections of a vector bundle?
This is a question in obstruction theory. It should be basic but I can't find a reference.
Let's stick to the $C^\infty$ category, so all objects mentioned are smooth. Let $\pi: E \to M$ be a vector ...
1
vote
0
answers
153
views
If $X$ is a strong deformation retract of $\mathbb{R}^n$, then is $X$ simply connected at infinity?
Let $X \subseteq \mathbb{R}^n$, and assume there is a strong deformation retract from $\mathbb{R}^n$ to $X$. Is $X$ necessarily simply connected at infinity?
(Edit) Follow up question: if there is a ...
0
votes
1
answer
121
views
Local embedding and disk in domain perturbation
Consider say $M=(\mathbb{S}^1\times\dotsb\times \mathbb{S}^1)-q$ ($n$-times). Assume that $B$ is an $n$ disk in $M$ (for instance, thinking of $\mathbb{S}^1$ as gluing $-1$ and $1$, the cube $B=[-\...
4
votes
1
answer
259
views
Is the wildness of 4-manifolds related to the diversity of their fundamental groups?
$n = 4$ is the smallest dimension such that the fundamental group of a closed $n$-manifold can be any finitely-presentable group (leading e.g. to various undecidability results stemming from the ...
5
votes
1
answer
374
views
Linking number and intersection number
Consider a disjoint union of two circles $A$ and $B$ smoothly embedded in $\mathbb{R}^3$ with linking number more than $1$. Suppose we know that there exists a disc $D$ in $\mathbb{R}^3$ such that $\...
13
votes
0
answers
292
views
Is there an analogue of Steenrod's problem for $p>2$?
An element $\alpha \in H_k(X; \mathbb{Z})$ is said to be realisable if there is a $k$-dimensional connected, closed, orientable $k$-dimensional submanifold $Y$ such that $\alpha = i_*[Y]$. The ...
0
votes
1
answer
135
views
Vector bundles over a homotopy-equivalent fibration
I think this question is related to what is known as "obstruction theory", but I'm not very familiar with this field of mathematics, so I am asking here.
Let $\pi:N\rightarrow M$ be a smooth ...
4
votes
1
answer
382
views
Criteria for extending vector field on sphere to ball
Below is a theorem that is equivalent to Brouwer fixed-point theorem, which I found quite interesting. The proof is in this PDF file.
Let $v: \mathbb S^{n-1} \to \mathbb R^n$ be a continuous map, ...
3
votes
1
answer
192
views
Does the group of compactly supported diffeomorphisms have the homotopy type of a CW complex?
It is known that the group of diffeomorphisms of a compact manifold with the natural $C^{\infty}$ topology has the homotopy type of a countable CW complex. See for instance this thread: Is the space ...