Timeline for Vector bundle for prescribed Stiefel-Whitney classes
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 22, 2014 at 14:51 | comment | added | Oscar Randal-Williams | @IgorBelegradek: The simplest example is $B=S^3$ with $x_3$ the nontrivial element of $H^3(S^3;\mathbb{Z}/2)$. As $w_3 = \mathrm{Sq}^1(w_2)$ this cannot be realised. | |
| Apr 22, 2014 at 13:32 | comment | added | Igor Belegradek | @nsrt: nice example, and it does not require Wu formula: since $w_1=0$ the bundle is orientable, so $w_2\in H^2(B;\mathbb Z_2)\cong H^1(RP^2;\mathbb Z_2)\cong\mathbb Z_2$ must be a reduction of the Euler class which lives in $H^2(B)=H^1(PR^2)=0$. | |
| Apr 22, 2014 at 13:05 | comment | added | nsrt | @IgorBelegradek: There's no two-dimensional bundle over $\Sigma RP^2$ with $w_1=0,w_2\ne 0$. | |
| Apr 22, 2014 at 12:57 | comment | added | Igor Belegradek | This gives a potential obstruction. To make it into an answer one needs to produce an example of $B$ and $x_i$'s for which the Wu formula is not satisfied. | |
| Apr 22, 2014 at 8:42 | vote | accept | Oliver Straser | ||
| Apr 22, 2014 at 8:42 | comment | added | Oliver Straser | What is if they satisfy Wu's formula? | |
| Apr 22, 2014 at 8:38 | history | answered | Oscar Randal-Williams | CC BY-SA 3.0 |