Timeline for Chern classes of a mapping torus vector bundle in terms of the construction data
Current License: CC BY-SA 4.0
4 events
| when toggle format | what | by | license | comment | |
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| Oct 4, 2020 at 20:32 | vote | accept | Roberto Ladu | ||
| Oct 3, 2020 at 20:05 | comment | added | kiran | @RobertoLadu Good question! You go by induction. First you do the case $n=1$ by hand. Then you use the fiber bundle $U(n-1)\rightarrow U(n) \rightarrow S^{2n-1}$ find, by induction, that the $n$-th chern class is $ax_1y_{2n-1}$ for some integer $a$. To show that that integers is 1, you just need to exhibit a bundle constructed using your method whose $n$-th chern class is not divisible. For that, take the rank $n$ bundle over $(S^1)^{2n}$ which is the $n$-fold exterior direct sum of the line bundle over $(S^1)^2$ whose first chern class is a generator of $H^2$. | |
| Oct 3, 2020 at 17:59 | comment | added | Roberto Ladu | Thank you kiran. Please, may you explain , why the $k$-th Chern class of the universal bundle turns out to be $x_1 y_{2k-1}$? I guess this should be related to the homotopy class of the map $id:U(n)\to U(n)$ which is the obstruction to contract $U(n)$ to a point. | |
| Oct 3, 2020 at 8:20 | history | answered | kiran | CC BY-SA 4.0 |