4
$\begingroup$

In Kobayashi and Nomizu's book Foundations of Differential geometry they introduce the concept of connection on a principal $G$ bundle. In this book, they use connection on a principal bundle to define (among other things) the characteristic classes of a (complex) vector bundle.

Let $G$ be a Lie group and $P(M,G)$ be a principal $G$ bundle. Let $\Gamma$ be a connection on $P(M,G)$. This defines what is called a Weil homomorphism $I(G)\rightarrow H^*(M,\mathbb{R})$.

Given a complex vector bundle $E\rightarrow M$ with fibre $\mathbb{C}^r$ they consider associated principal $Gl(r,\mathbb{C})$ bundle $P\rightarrow M$ and define $k$-th Chern class of $E$ to be image of some element of $I(G)$.

But, it seems this Weil homomorphism can do more than constructing Characteristic classes.

Is Weil homomorphism introduced (and used) only to construct Characteritic classes? If not, where else do we use this Weil homomorphism?

Edit (2 Nov 2019) : While searching for something, I have found what is called a refined Chern-Weil homomorphism, which has something to do with secondary characteristic classes.

Improve this question
$\endgroup$
6
  • $\begingroup$ Characteristic classes are not defined via Chern–Weil theory, rather the C–W theory gives another way to construct them. If you haven't seen it, the classic Characteristic classes by Milnor and Stasheff is worth checking out. $\endgroup$
    – David Roberts ♦
    Commented Nov 2, 2019 at 7:26
  • $\begingroup$ @DavidRoberts I mean characteristic classes are also defined using connection and for this Chern-Weil homomorphism is used... I am not 100% sure if this also is grammatically correct :) please feel free to edit to make it look correct... $\endgroup$
    – Praphulla Koushik
    Commented Nov 2, 2019 at 8:07
  • 1
    $\begingroup$ This is a comment in addition to David's comment. If you want to define Chern classes via C-W-theory you "only" get the images of the Chern classes defined in Milnor's book under the coefficient homomorphism $H^k(M;\mathbb Z) \to H^k(M;\mathbb R)$. In particular C-W cannot see torsion phenomena. $\endgroup$
    – Panagiotis Konstantis
    Commented Nov 2, 2019 at 11:26
  • $\begingroup$ @PanagiotisKonstantis Is that so, I did not observe it.. can you kindly explain why it does not see torsion? Any reference would be helpful.. Thanks for the comment :) $\endgroup$
    – Praphulla Koushik
    Commented Nov 2, 2019 at 12:44
  • $\begingroup$ @PraphullaKoushik E.g. if a Chern class is a torsion element in $H^k(M;\mathbb Z)$ then it has to be mapped to zero under the homomorphism $H^k(M;\mathbb Z) \to H^k(M;\mathbb R)$ since the latter group do not contain torsion. But with C-W theory you construct exactly this zero class in $H^k(M;\mathbb R)$ (where $H^k(M;\mathbb R)$ is understood to be the deRham cohomology) $\endgroup$
    – Panagiotis Konstantis
    Commented Nov 2, 2019 at 13:05

0

Reset to default

You must log in to answer this question.