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Results tagged with differential-topology
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user 4463
The study of differentiable manifolds and differentiable maps. One fundamental problem is that of classifying manifolds up to diffeomorphism. Differential topology is what Poincaré understood as topology or “analysis situs”.
2
votes
2
answers
603
views
Does the Bertini Theorem imply that there exists $k$ points such that passing through them i...
Consider the space of all homogeneous degree $d$ polynomials in three variables
$[X,Y,Z]$, i.e.
$$ f[X,Y,Z] = f_{d00} X^d + f_{d10} X^{d-1}Y + \ldots .$$
This can be thought of as a section of the l …
- 3,245
0
votes
3
answers
360
views
If you take the closure of two smooth varieties and then take their intersections, is the s...
Let
$$ X, Y \subset \mathbb{P}^N$$
be two non singular algebraic varieties of dimensions $k$ and $l$ that
intersect transversally. Is it true that the ``dimension'' of the variety
$\overline{X} \cap …
- 3,245
3
votes
1
answer
394
views
Does passing through a point in general position cut down the dimension by one?
Let $\mathcal{D} \approx P^{\delta_d}$ be the space of homogeneous degree $d$
polynomials in three variables (up to scaling), where $\delta_d = \frac{d(d+3)}{2}$. A point
$p\in \mathbb{P}^2$ gives u …
- 3,245
3
votes
2
answers
808
views
Can you ``perturb'' a submanifold to intersect transversally with any other smooth submanifo...
Let $\mathcal{D} \approx \mathbb{P}^{\delta_d}$ be the space of homogeneous
degree $d$ polynomials in three vriables, where
$\delta_d = \frac{d(d+3)}{2}$. Let
$$ X \subset \mathcal{D} \times \mathb …
- 3,245
2
votes
0
answers
957
views
Can one always extend a smooth section defined on a non compact submanifold to the whole man...
Let $V \rightarrow M$ be a smooth vector bundle over a smooth compact manifold $M$
(without boundary) and $X \subset M$ a smooth submanifold of $M$, that is not necessarily closed.
Suppose $s: X \ri …
- 3,245
9
votes
1
answer
935
views
Do partitions of unity exist if we impose additional conditions on the derivatives?
Let $ ~~\cup_{k=-1}^{\infty} U_k = \mathbb{R} $ be an open covering of
$\mathbb{R}$. It is a well known fact that partitions of unity subbordinate to
the cover exists, i.e. there exists smooth
func …
- 3,245
3
votes
1
answer
360
views
General position argument for reasonable spaces
Let $\mathcal{D} \approx \mathbb{P}^{\delta_d}$ be the space of non zero homogeneous degree $d$ polynomials in three variables up to scaling, where
$\delta_d:= \frac{d(d+3)}{2} $. Given a point $p \i …
- 3,245
2
votes
1
answer
709
views
General position argument
Let $\mathcal{D} \approx \mathbb{P}^{\delta_d}$ be the space of homogeneous degree $d$
polynomials in three variables (up to scaling), where $\delta_d = \frac{d(d+3)}{2}$.
Define $\mathcal{A}$ to be …
- 3,245
1
vote
0
answers
319
views
Does the closure of a ``nice'' smooth submanifold define a homology class?
Let $M$ be a smooth compact, oriented manifold. Let
$X$ be a submanifold which is of the following type
$$X := \{ p \in M: \psi(p) =0, ~~\varphi(p) \neq 0 \} $$
where
$$ \psi: M \rightarrow V, \q …
- 3,245
2
votes
2
answers
367
views
Does a generic curve inside the space of curves with a node at a specific point have only fi...
Let $\mathcal{D} \approx \mathbb{P}^{\delta_d}$ be the space of homogeneous degree $d$
polynomials in three variables (up to scaling), where $\delta_d = \frac{d(d+3)}{2}$.
Define $\mathcal{A}$ to be …
- 3,245
1
vote
1
answer
188
views
If you perturb a polynomial by a smooth function, then is the signed number of small zeros o...
Let $f: \mathbb{C} \rightarrow \mathbb{C} $ be a function of the form
$$ f(z) = z^n + z^{n+ 1} g(z) $$
where $g$ is a $\textbf{smooth}$ function (not necessarily holomorphic).
Is it true that the …
- 3,245
8
votes
2
answers
414
views
Are there analogous statements for the number of zeros of a section in terms of the Euler cl...
Let $V \rightarrow M$ be an oriented rank $k$ vector bundle over a compact orientd manifold $M$. Let $X \subset M$ be a
compact topological subspace of $M$ that is a smooth oriented submanifold of
d …
- 3,245
16
votes
5
answers
1k
views
Take contraction wrt a vector field twice and define kernel mod image. Does that give anythi...
First, we make the following observation: let $X: M \rightarrow TM $ be a vector
field on a smooth manifold. Taking the contraction with respect to $X$ twice gives zero, i.e.
$$ i_X \circ i_{X} =0.$ …
- 3,245
41
votes
3
answers
3k
views
Can one recover the smooth Gauss Bonnet theorem from the combinatorial Gauss Bonnet theorem ...
First let me state two known theorems.
Theorem 1 (for smooth manifolds): Let $(M,g)$ be a smooth compact two dimensional Riemannian manifold. Then
$$ \int \frac{K}{2 \pi} dA = \chi (M) $$
where $K$ i …
- 3,245
9
votes
3
answers
2k
views
Is there a natural form representing the Thom class of a vector bundle, which when pulled ba...
Let $V \rightarrow M$ be an oriented vector bundle over a compact
oriented manifold $M$ equipped with a metric $h$ (the metric $h$
is a metric on the Vector bundle $V$, not on the manifold $M$).
Is …
- 3,245