# Tagged Questions

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### Complexity of mappings (forms) in R. Thom's “Structural stability and morphogenesis”

In his "Structural stability and morphogenesis", R. Thom (especially in the chapter about dynamics of forms) among other things speculates about a notion of complexity of a "form" (mapping between ...

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166 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 ...

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161 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
...

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251 views

### Iterated Milnor fibrations and Thom's a_f condition

Ok so there's a lot of litterature about nearby cycles functor since it was introduced by Grothendieck and Deligne but I couldn't find any clear answer to the following natural question:
Problem: Let ...

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233 views

### Analogue of singularity theory in other categories

Whitney, Thom, Mather, Arnold and others develoved the singularity theory of smooth maps.
Does there exist any analogue of this theory in the category of TOP or PL (or Lipschitz) maps?
I mean notions ...

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105 views

### Is the space of degree $d$ curves with marked smooth points dense inside the space of curves with marked points?

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 ...

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**2**answers

359 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 ...

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443 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
...

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480 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 ...

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117 views

### Equivalence of Level Sets

Consider the zero level set of $f : \mathbb{R}^3 \to \mathbb{R}$, where $0$ is a regular value. Consider also the space of planes passing through the origin, i.e. $\mathbb{RP}^2$. For a fixed plane $P ...

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267 views

### Does a generic curve inside the space of curves with a node at a specific point have only finitely many nodes?

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 ...

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493 views

### Singular fibers of generic smooth maps of negative codimension

This is in some sense a follow-up to my question on submersions.
Let $f\colon\thinspace M\to N$ be a generic smooth map between closed manifolds of dimensions $m$ and $n$. Assume that the codimension ...

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393 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 ...

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253 views

### Thom's gradient conjecture and analyticity

Suppose we have an analytic function $f: U \to {\mathbb R}$, where $U\subset {\mathbb R}^n$ an open subset, with $0\in U$ a critical point of $f$. Thom conjectured that if a trajectory $x(t)$ of ...

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243 views

### What is the simplest way to show that a section of a vector bundle is transverse to the zero set

Let $\mathcal{D} \approx \mathbb{P}^{\delta_d}$, be the space of homogeneous
degree $d$ polynomials in three variables $[X,Y,Z] \in \mathbb{P}^2$ upto scaling, where
$\delta_d = \frac{d(d+3)}{2}$. ...

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795 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 ...

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**1**answer

246 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) ...