Timeline for Classifying map of a simple circle bundle
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
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| Mar 12, 2019 at 16:26 | history | edited | BrianT | CC BY-SA 4.0 |
added 1516 characters in body
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| Mar 12, 2019 at 16:02 | comment | added | BrianT | Thanks for your answer. I will modify a little bit my text in order to make clearer what I am trying to do. | |
| Mar 12, 2019 at 15:12 | comment | added | Jason Starr | I should have written "character lattice", not "cocharacter lattice". This is defined to be the group of group homomorphisms from the torus $\mathbb{K}$ to the $1$-dimensional torus $S^1$. The valuewise product of any two group homomorphisms is another group homomorphism. This valuewise product makes the set of such group homomorphisms into an Abelian group. In fact it is a free Abelian group of rank equal to the dimension of $\mathbb{K}$. The cohomology of $\mathbb{K}$ is canonically isomorphic to the polynomial algebra on $\Lambda(\mathbb{K})$ generated in degree $2$. | |
| Mar 12, 2019 at 12:30 | comment | added | BrianT | @Jason Starr, thank you. Could you explain to me what the cocharacter lattice is ? What is the notation $\text{Hom}_{\text{tori}}(\mathbb{K},S^1)$ ? | |
| Mar 12, 2019 at 12:27 | comment | added | Jason Starr | The cohomology of $B\mathbb{K}$ is a free (graded) commutative algebra on the cocharacter lattice $\Lambda(\mathbb{K}) = \text{Hom}_{\text{Tori}}(\mathbb{K}, S^1)$ concentrated in degree $2$. The cocharacter lattice is contravariant. The class $f(\text{Eu})$ is a generator for the kernel of the pullback homomorphism $\Lambda(\mathbb{K}) \to \Lambda(\mathbb{K}_0)$. Does that answer your question? | |
| Mar 12, 2019 at 11:49 | history | asked | BrianT | CC BY-SA 4.0 |