Timeline for Maps inducing zero on homotopy groups but are not null-homotopic
Current License: CC BY-SA 2.5
4 events
| when toggle format | what | by | license | comment | |
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| Apr 4, 2010 at 18:15 | comment | added | Ying Zhang | Sorry about my last sentence, from the above sequence I got a map from $RP^{\infty}\to K(Z/2,2)$. To get a map from $RP^{\infty}\to CP^{\infty}$, we should look at the coefficient sequence $0\to Z\to Z\to Z/2Z\to 0$ which does the job. Geometrically it is not too hard to find an interesting map from $RP^{\infty}$ to $CP^{\infty}$ just by quotient out more things. | |
| Apr 4, 2010 at 4:05 | comment | added | Ying Zhang | Thanks, this argument is very nice! I guess it is not very hard to find a space with non trivial Bockstein, e.g.$M=RP^2$, we have $H^i(M,Z/2)=H^i(M,Z/4)=Z/2$for i=1,2, and if I look at the reduced cohomology long exact sequence $H^0(M,pt,Z/2)=H^0(M,pt,z/4)=0$. Then the Bockstein from $H^2$to $H^1$will not be trivial, otherwise we will have $0\to Z/2\to Z/2\to Z/2\to 0$ contradiction. Although I used the reduced cohomology, since I only use maps to K(z/2,2) and K(Z/2,1) to represent $H^1$ and $H^2$ I am fine. From this example, we get a non null-homotopic map from $CP^{\infty}$ to $RP^\infty$! | |
| Apr 4, 2010 at 3:40 | vote | accept | Ying Zhang | ||
| Apr 4, 2010 at 3:07 | history | answered | Chris Schommer-Pries | CC BY-SA 2.5 |