All Questions
17 questions
8
votes
1
answer
304
views
Fibers of generic smooth maps between manifolds of equal dimension
I have heard that the following is a "well-known"
Claim. Let $M$ and $N$ be smooth manifolds with equal dimensions and $M$ compact. Then a generic smooth map $f\colon M\to N$ has finite ...
- 501
2
votes
0
answers
128
views
Making a continuous function into embedding by adding additional dimension
While doing my researches, I encountered the following problem.
Let $f:[0,1]^n\rightarrow \mathbb{R}^{n+k}$ be an arbitrary continuous function.
I want to make this function an embedding by perturbing ...
- 173
1
vote
0
answers
105
views
Codimension of cusp singularities in the space of 2-jets
In trying to prove Cerf's theorem about homotopies between Morse-functions I ended up thinking about the following problem.
For $n>2$, $a= (a_{i,j})\in GL(n-2)$, we define the polynomial map $C_a:\...
- 2,533
8
votes
1
answer
426
views
Orbifolds are Thom-Mather stratified spaces
Where can I find a proof of (or if it is even true) that an (effective) orbifold is a Thom-Mather stratified space?
edit: after some search, I found the proof should be contained in either
GIBSON, C....
- 803
2
votes
0
answers
191
views
Blowing up the zero section for "Chasse au Canard" (some new kind of geometric canards)
In this paper "Canard cycles and center manifolds" one encounters the blowing up of a non isolated set or manifold of singularities of a vector field or a singular foliation. This is a ...
- 356
1
vote
1
answer
738
views
Non-transverse intersection of submanifolds
What can we tell about non-transverse intersection points of (smooth) submanifolds?
Especially, in the case of complementary dimensions, how can we define and calculate the ''multiplicity'' of an ...
- 194
2
votes
0
answers
111
views
About the regularity of Thom's first isotopy theorem
Consider an abstract stratified set $(V, \Sigma)$ in the sense of Thom-Mather
(see Mather's note page 491-492 https://www.ams.org/journals/bull/2012-49-04/S0273-0979-2012-01383-6/S0273-0979-2012-01383-...
- 83
1
vote
1
answer
122
views
Tangent space to subspace of orbit in jet spaces
I consider a map germ $f: (\mathbb{R}^n,0) \to \mathbb{R} $ which is $k$-determined for some $k \in \mathbb N$, i.e. for all map germs $g: (\mathbb{R}^n,0) \to \mathbb{R} $ having the same $k$-jet as $...
- 113
25
votes
1
answer
4k
views
Can the constant rank theorem for smooth manifolds be generalized to nonconstant rank?
The constant rank theorem says that
if $f\colon M→N$ is a smooth map whose rank equals some fixed $k≥0$ at any point of $M$, then, locally with respect to $M$ and $N$, the map $f$ assumes the easiest ...
- 37.8k
7
votes
0
answers
229
views
Higher homotopy of diffeomorphism groups from singularities
In the case of a genus $g$ surface $\Sigma$, it is well known that $MCG(\Sigma) = \pi_0 \operatorname{Diff}^+(\Sigma)$ is generated by Dehn twists, which come from a Kahler degeneration with smooth ...
- 1,981
4
votes
0
answers
247
views
H-principle for smoothing
I'm trying to find algebraic embedding of singular curves into projective varieties that can be smoothed symplectically but not algebraically.
It's not hard (e.g. using the methods in Hartshorne-...
- 1,981
15
votes
1
answer
1k
views
Higher Cerf Theory
Morse functions on a manifold $M$ are defined as smooth maps $f:M \rightarrow \mathbb{R}$, such that at the critical points we can find local coordinates so that $$f(x_1,\dots,x_n)=-x_1^2-x_2^2-\dots-...
- 865
2
votes
1
answer
203
views
Understanding maps from M to R^n, for n>dim M
I am interested in "approximating" smooth maps from a compact smooth manifold $M$ of dim $m$ into $\mathbb R^n,$ for $n>m,$ by "nice" maps, with properties similar to those of Morse functions. Of ...
- 2,390
5
votes
0
answers
343
views
Stratification of a smooth map
So, this is an exercise. But from math.stackexchange I have been suggested to post this question here.
To find the Thom-Boardman stratification of the smooth map
$f(x,y,a,b,c,d)=x^2y+y^3+a(x^2+y^2)+...
- 231
13
votes
1
answer
637
views
Can a PDE constrain the degree of a $C^\infty$ map germ?
Let $\pi:E\to M$ be a smooth vector bundle over a smooth manifold, with $\text{rank}(E)=\text{dim}(M)$. For a section $\sigma$ of $E$ with a zero at $p\in M$, define the degree of the zero at $p$ to ...
- 3,212
7
votes
4
answers
973
views
I was wondering if the set of singular loops is a (somewhere) submanifold of loop space?
The set of all smooth maps $S^1\to M^n$ ($M$ is a smooth manifold) is a generalized manifold(see http://ncatlab.org/nlab/show/smooth+loop+space).
I was wondering if the set of singular loops (maps ...
- 5,063
2
votes
1
answer
430
views
Stable singularities of smooth map $\mathbb R^3\to \mathbb R^4$
Does anybody know any classification of stable singularities of smooth map $f:\mathbb R^3\to \mathbb R^4$?
It is clear that there are singularities which look like intersection of 2 (or 3 or 4) ...
- 5,063