Timeline for A question on classification of almost complex structures on $4$-manifolds
Current License: CC BY-SA 2.5
11 events
| when toggle format | what | by | license | comment | |
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| Aug 21, 2011 at 9:12 | vote | accept | Dmitri Panov | ||
| Aug 20, 2011 at 23:55 | answer | added | Peter Teichner | timeline score: 22 | |
| Mar 29, 2011 at 12:36 | history | edited | Dmitri Panov | CC BY-SA 2.5 |
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| Mar 29, 2011 at 12:28 | comment | added | Dmitri Panov | Paul, thanks! I it seems that your are right with $e+\tau=0$ and $4CP^2$ does not have an almost complex structure. I'll change this | |
| Mar 29, 2011 at 11:37 | comment | added | Paul | Dmitri: You're right. make that $4 CP^2$. | |
| Mar 29, 2011 at 9:34 | comment | added | Dmitri Panov | Paul $5+3=0$ mod $4$, so your first sentence is wrong – | |
| Mar 29, 2011 at 7:09 | history | edited | Dmitri Panov | CC BY-SA 2.5 |
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| Mar 29, 2011 at 2:52 | comment | added | Paul | (make that $\pi_3S^2$ in the last sentence) | |
| Mar 29, 2011 at 2:50 | comment | added | Paul | If $M$ admits a almost complex structure, $e+\tau=0$ mod 4 ($e$=Euler characteristic, $\tau$=signature).So eg $CP^2\# CP^2\# CP^2$ doesnt admit an almost complex structure, even though $c=(3,3,1)$ satisfies $c^2=19= 2e+3\tau$. Also,if I′m not mistaken,$[S^1\times S^3, S^2]=\pi_3(S^2)\oplus \pi_4(S^2)$: the cofibration sequence $$S^1 v S^3\to S^1\times S^3\to S^4 $$ gives the exact sequence $[S^4,S^2]\to [S^1\times S^3,S^2]\to [S^1 v S^3,S^2]$, and $[S^1 v S^3, S^2]=\pi_2S^2 $ (I think), so the Hopf invariant misses the $Z/2=\pi_4(S^2)$. | |
| Mar 28, 2011 at 18:33 | history | edited | Dmitri Panov | CC BY-SA 2.5 |
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| Mar 28, 2011 at 11:50 | history | asked | Dmitri Panov | CC BY-SA 2.5 |