All Questions
33
questions
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(\...
- 51
3
votes
1
answer
140
views
Homogeneous regular (= polynomial component) maps with odd degree and their being global homeomorphisms in dimensions higher than one?
Let $F:\mathbb{R}^m \to\mathbb{R}^m, F:=(F_1\dots F_m)$ be a regular map, i.e. with components $F_i$ that are polynomials.
Assume further that each $F_i$ is an odd degree (say $d$) homogenous ...
- 1,417
1
vote
0
answers
57
views
Given a proper submersion $f: X \setminus F_0 \to D \setminus 0$ which extends to $X \to D$, are cycles on $X$ which run around $0$ boundaries in $X$?
I have a proper map of complex manifolds
$$f: X \to D,$$
where $D \subset \mathbb C$ is the unit disc. By assumption, $f$ has connected fibers, is smooth over $D \setminus 0$, and a smooth fiber $F$ ...
- 1,141
-2
votes
1
answer
188
views
Topologies in the vicinity of Euclidean space
Given a smooth function $f:\mathbf R^n\to \mathbf R^m$ with $0$ as a regular value, I define the $(n-m)$ dimensional smooth manifold $M_f:=f^{-1}(0)$.
Let $f_0(x_1,...,x_n):=(x_1,...,x_m)$; $M_{f_0}$ ...
- 423
1
vote
0
answers
200
views
Question regarding affine fibre bundles
Let $f:X\to Y$ be a morphism of affine varieties such that it is a fibre bundle with fibre $F$. Let $\pi_1(Y)=\Gamma$ be a free group (non abelian) of finite rank and $\pi_1(F)$ is a finite group $G$ ...
- 585
5
votes
1
answer
406
views
A question related to fiber bundle
Let $f:\mathbb{C}^3 \to \mathbb{C}$ be a morphism of varieties such that it is a smooth fiber bundle. Can I say that the fiber is $\mathbb{C}^2$?
- 1,069
5
votes
0
answers
353
views
CW complex vs analytic manifold vs variety
I am looking to gain some intuition into the passage (or obstruction thereof) between different categories of objects one encounters in geometry and topology. To oversimplify things a bit, the ...
- 1,377
19
votes
1
answer
959
views
Can the product of a 3-dimensional lens space with a circle be diffeomorphic to another such product when the lens spaces aren't diffeomorphic?
This is a question that I need to answer in order to resolve an issue for my dissertation and I am looking for a reference. Here is the precise statement of the question.
Suppose we have two three-...
- 293
3
votes
1
answer
462
views
How to compute the index of a vectorfield defined by analytic formula?
An analytic local map (or map germ) $f: (\mathbb{R}^n, 0) \to (\mathbb{R}^n, 0) $ can be considered as a vector field with zero at the origin. Assume that the origin is an isolated zero of $f$. How ...
- 194
9
votes
1
answer
349
views
Smooth complex projective surface as the total space of a Serre fibration
Let $M$ be the underlying topological manifold of a smooth complex projective surface. Assume $\pi_1(M)=\{0\}$ and $\pi_2(M)\neq \mathbb{Z}^2$.
Is there a Serre fibration $M\to B$ where $B$ is a CW ...
user164740
4
votes
1
answer
233
views
Fundamental group of the complement of the arrangement of plane nodal curves
I want to calculate the fundamental group of the complement some collection of plane curves (specifically, two nodal cubics in a general position).
I've read about Severi problem (solved by Harris), ...
- 1,842
0
votes
1
answer
195
views
factorization morphism between projectives spaces
Please help me with this doubt:
Let $f:\mathbb{P}^1 \rightarrow \mathbb{P}^2$ be a non-constant morphism. Is there any factorization of $f$ as $$\mathbb{P}^{1} \overset{h}{\rightarrow}\mathbb{P}^{2}\...
- 25
3
votes
1
answer
529
views
Does the projectivization of a vector bundle have sections?
Let $E \to X$ be a homomorphic vector bundle over a projective variety $X$. Does $\mathbb{P}(E)$ always have holomorphic sections? If not what is the obstruction?
- 778
9
votes
3
answers
1k
views
Link of a singularity
I would like to understand the topological type of a link of a singularity in a simple example. Consider for instance the cone ${xy-z^2=0}\subset\mathbb{C}^3$.
If we set $x = x_1+ix_2, y = y_1+iy_2, z ...
user56259
1
vote
0
answers
120
views
Topological invariants of a certain "stratified" manifold, with pieces of different "dimensions"
Disclaimer: I don't fully understand what I'm talking about in the question below. I'm still trying to figure out the right question to ask. Quotations and question marks in brackets mean that I'm not ...
- 6,824