Timeline for Is this subset of matrices contractible inside the space of non-conformal matrices?
Current License: CC BY-SA 4.0
9 events
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| Mar 11, 2020 at 14:44 | comment | added | Asaf Shachar | Thank you for this modification. I am not sure why the path you suggested in the first version should apply in this case as well. Can you say why do you think that this should be easier inside ${\cal NC}_+$ instead of ${\cal NC}$? | |
| Mar 4, 2020 at 11:21 | comment | added | Dmitri Panov | Asaf, I added some details - not 100% detailed, but this should work I guess. Let me know if you are happy with this. | |
| Mar 4, 2020 at 11:17 | history | edited | Dmitri Panov | CC BY-SA 4.0 |
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| Mar 4, 2020 at 10:19 | comment | added | Dmitri Panov | I see Asaf. I read the question too quickly, I thought that you want to contract this in the complement to conformal matrices in $SL(2,\mathbb R)$. Let me think how to fix the argument. I believe this can be done. At last this can be done very easily if you replace $\cal NC$ by ${\cal NC}_+$ of matrices of positive determinant | |
| Mar 4, 2020 at 9:32 | comment | added | Asaf Shachar | In other words, by restricting the domain of $V$ to $\text{SL}_2(\mathbb{R})$ your argument only applies to deformations of $\mathcal{F}$ in $\text{SL}_2(\mathbb{R}) \cap \mathcal{NC}$, while I am allowing arbitrary deformations in $\mathcal{NC}$. | |
| Mar 4, 2020 at 9:25 | comment | added | Asaf Shachar | Hi, after some further thinking I am not so sure that this argument settles the question. The question was whether or not $F$ is contractible inside $\mathcal{NC}:=\{ A \in M_2(\mathbb{R}) \, | \det A \ge 0 \, \,\text{ and } \, A \text{ is not conformal} \,\}$. But the point $(0,1)$ does lie in the image of your map $V:\mathcal{NC} \to M_2(\mathbb{R})$; take $Ae_1=2e_1, Ae_2=e_2$. (We allow matrices outside $\text{SL}_2(\mathbb{R})$ in $\mathcal{NC}$. I think that you have only proved that $\mathcal{F}$ is not contractible inside $\text{SL}_2(\mathbb{R}) \cap \mathcal{NC}$). | |
| Mar 3, 2020 at 14:38 | vote | accept | Asaf Shachar | ||
| Mar 2, 2020 at 16:20 | history | edited | Dmitri Panov | CC BY-SA 4.0 |
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| Mar 2, 2020 at 11:33 | history | answered | Dmitri Panov | CC BY-SA 4.0 |