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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”.
3
votes
0
answers
283
views
Do J-holomorphic curves "very nearly" fail to be an immersion near the bubbling points?
Let $u_{t}: \mathbb{P}^1 \longrightarrow \mathbb{P}^2$ be a family
of degree $2$ maps defined (for $t$ small and non zero) by
$$u_t([X,Y]) := [X^2, t Y^2, XY].$$
Note that as $t$ goes to zero, $u_t …
- 3,245
3
votes
1
answer
214
views
Is there a formula for the intersection of projectivized lines inside a projectivized vector...
Let $E\rightarrow D$ be a complex rank two vector bundle over a compact complex
one dimensional manifold $D$. Let $L_1, L_2 \subset E$ be rank one subbundles of E
(i.e. line bundles). Let
$$ n_1:= …
- 3,245
5
votes
1
answer
365
views
Is the space of real conics with a singular point an orientable manifold?
Consider the space of non zero real homogeneous degree $2$ polynomials in three variables upto scaling. This space is $\mathbb{R} \mathbb{P}^5$. The zero set
of such a polynomial gives a real curve i …
- 3,245
16
votes
5
answers
1k
views
Take contraction wrt a vector field twice and define kernel mod image. Does that give anythi...
First, we make the following observation: let $X: M \rightarrow TM $ be a vector
field on a smooth manifold. Taking the contraction with respect to $X$ twice gives zero, i.e.
$$ i_X \circ i_{X} =0.$ …
- 3,245
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
1
vote
1
answer
174
views
Is the space of degree $d$ curves with marked smooth points dense inside the space of curves...
Let $\mathcal{D} \approx \mathbb{P}^{\delta_d} $ be the space of nonzero
homogeneous degree $d$ polynomials in three variables upto scaling, where
$\delta_d = \frac{d(d+3)}{2} $
(basically degree $ …
- 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
4
votes
3
answers
562
views
What is the easiest way to show that three lines in two dimensional space do not intersect?
I have two similar questions:
1) Let $X$ and $Y$ be two measure spaces. Suppose for every point
$x \in X $ there exists a set $ \mathcal{U}_x \subset Y $ of full
measure in $Y$. Suppose $V \subset …
- 3,245
4
votes
2
answers
413
views
Is it impossible for the dimension of a topological space to increase under a smooth map?
First let me make a definition. Let $M$ be a smooth manifold and
$S \subset M $ a topological subspace of $M$. We say that $S$ has
"dimenion" at most $k$ if $S$ is a subset of
$$ X_1 \cup X_2 \ldots …
- 3,245
1
vote
1
answer
378
views
A version of implicit function theorem when sections are not everywhere smooth?
Let $V_1, V_2 \rightarrow M $ be smooth vector bundles over a manifold $M$ and $s_1: M \rightarrow V_1$
a smooth section transverse to the zero set and $s_2: M \rightarrow V_2$ a continuous section
…
- 3,245
2
votes
0
answers
957
views
Can one always extend a smooth section defined on a non compact submanifold to the whole man...
Let $V \rightarrow M$ be a smooth vector bundle over a smooth compact manifold $M$
(without boundary) and $X \subset M$ a smooth submanifold of $M$, that is not necessarily closed.
Suppose $s: X \ri …
- 3,245
4
votes
2
answers
553
views
Is there an analogous concept for the degree of a map, when the spaces are singular?
Let $M$ and $N$ be two smooth compact, oriented manifolds and
$X\subset M$ an oriented submanifold of $M$ of dimension $k$
(not necessarily closed). Suppose in addition that $\bar{X}-X$ is contained …
- 3,245
1
vote
0
answers
319
views
Does the closure of a ``nice'' smooth submanifold define a homology class?
Let $M$ be a smooth compact, oriented manifold. Let
$X$ be a submanifold which is of the following type
$$X := \{ p \in M: \psi(p) =0, ~~\varphi(p) \neq 0 \} $$
where
$$ \psi: M \rightarrow V, \q …
- 3,245
0
votes
1
answer
488
views
Does the closure of a smooth algebraic always define a homology class?
Let $X\subset \mathbb{C} \mathbb{P}^{N}$ be a smooth,
algebraic (locally closed) complex
submanifold of $\mathbb{C} \mathbb{P}^N$
of complex dimension $k$. More concretely, $X$ is of the
followin …
- 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