Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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) hyperplanes in $\mathbb R^4$. But there are other ones.

Another type of stable singularity can be produced the following way: let $f$ maps $\mathbb R^3$ such way that for $x\in D^2$ $f(x)=f(-x)$ for $x\in \mathbb R^2, x=(x_1,x_2,0,0)$.

share|improve this question
add comment

1 Answer 1

up vote 2 down vote accepted

The codimension of $\Sigma^2$-points in the source for a codimension $k$ map is $2(k+2)$, in this case that's $6>3$, so you only get combinations of Morin and regular points. From the Morins, the cusp has codimension $2(k+1)=4$, so you have no cusps either and need to keep track only of the regular points (codimension $k=1$ in the target) and fold points (canonical form $(x,y,z) \mapsto (x^2,xy,y,z)$, codimension $2k+1=3$ in the target). So, you have the regular points with multiplicity 1 to 4, the fold curves and fold curves intersecting transversally a regular "branch", and that's all. What you wrote does not look like a stable singularity to me.

share|improve this answer
    
Could you give me a reference where I can find description of canonical form for Morin singularities? –  Nikita Kalinin Jul 18 '11 at 18:08
    
B. Morin: Formes canoniques des singularités d'une application différentiable, Comp. Rend. Acad. Sci. Paris 260 (1965), pp. 5662-5665 should have it –  Thorny Jul 21 '11 at 13:29
    
Thanks! But I have found this reference myself, problem in where I can find it in internet. –  Nikita Kalinin Jul 26 '11 at 14:03
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.