Timeline for Degree of sequence of mappings
Current License: CC BY-SA 3.0
10 events
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| May 21, 2012 at 18:41 | comment | added | Misha | @Marijan: You have to exclude the case $r= 1$ since degree makes sense only for maps between manifolds of the same dimension. (Of course, you can declare degree to be zero if dimensions are different, but then you already know that $deg(f)=0$.) | |
| May 21, 2012 at 17:54 | comment | added | Marijan | Thank you very much. I was not sure that the degree of the limiting function is one, because the sequence $r_n$ can tend to $1$ and therefore the limiting function is a singular mapping, for example the mapping $f(x)=x/|x|$? | |
| May 21, 2012 at 14:03 | comment | added | Misha | Degree of the limit is 1: Just stretch all target annuli to be the same. The new maps still converge and are eventually homotopic to limit rel. boundary. | |
| May 21, 2012 at 13:33 | history | edited | Marijan | CC BY-SA 3.0 |
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| May 21, 2012 at 13:33 | comment | added | Marijan | Of course all the mappings $f_n$ are sirjective and for einstance map the inner (outer) boundary onto inner (outer) boundary. | |
| May 21, 2012 at 13:14 | comment | added | Lee Mosher | Next question: without further restrictions, the degree is not well-defined. For instance, take $r=2$, $r_n=3$ for all $n$, and $f_n : A(1,2) \to A(1,3)$ to be the inclusion map, so $f_n$ converges to the inclusion map. Over all points of $A(1,3)$ of norm $<2$ the degree is $1$, and over all points of norm between $2$ and $3$ the degree is zero, so the degree of the limit is not well-defined. Do you have any further restrictions in mind to avoid examples like this? | |
| May 21, 2012 at 12:55 | history | edited | Marijan | CC BY-SA 3.0 |
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| May 21, 2012 at 12:54 | comment | added | Marijan | $A(r,1)$ is the set of points of norm $>1 $ and $<r$. | |
| May 21, 2012 at 12:51 | comment | added | Lee Mosher | What does your notation $A(1,r)$ mean? Do you mean the set of points in $R^d$ of norm $\ge 1$ and $\le r$? | |
| May 21, 2012 at 12:31 | history | asked | Marijan | CC BY-SA 3.0 |