Let $\mathbb{K}_0 \subset \mathbb{K}$ be two tori (subtori of $(S^1)^n$). We suppose that $\mathbb{K}_0$ is obtained from $\mathbb{K}$ by the following procedure: consider, on the lie algebra $\text{Lie}(\mathbb{K})$ of $\mathbb{K}$, a linear function $p : \text{Lie}(\mathbb{K}) \to \mathbb{R}$. If $p$ is "rational", in the sense that $\ker p \subset \text{Lie}(\mathbb{K})$ closes into a torus via the exponential map, then we can define $\mathbb{K}_0$ as the codimension $1$ subtorus of $\mathbb{K}$ with Lie algebra $\ker p$.

We denote $EG \to BG$ the universal principal bundle associated with a Lie group $G$, and will consider cohomology with coefficients in $\mathbb{C}$. I am trying to understand the classifying map of the circle bundle $$ \pi: B\mathbb{K}_0 \overset{\mathbb{K} / \mathbb{K}_0}{\longrightarrow} B\mathbb{K}, $$ or more precisely its induced map in cohomology. Below are a few arguments in this direction.

First of all, the cohomology $H^*(B(\mathbb{K} / \mathbb{K}_0), \mathbb{C})$ can be identified with $$ \mathbb{C}[(\text{Lie}(\mathbb{K}) / \text{Lie}(\mathbb{K}_0))^*] \simeq \mathbb{C}[p]. $$ Thus, the classifying map in cohomology is of the form $$ f : \mathbb{C}[p] \to H^*(B \mathbb{K}, \mathbb{C}). $$

Let now $I$ denote the vanishing ideal of $\text{Lie}(\mathbb{K}) \otimes \mathbb{C}$, that is, the ideal generated by the polynomials on $\mathbb{R}^n \otimes \mathbb{C}$ which vanish on $\text{Lie}(\mathbb{K})$, and denote $I_0$ that of $\text{Lie}(\mathbb{K}_0) \otimes \mathbb{C}$. If $(u_1,...,u_n)$ is a basis for $\mathbb{R}^{n*}$, then we have $$ H^*(B \mathbb{K}) \simeq \mathbb{C}[u_1,...,u_n] / I, \quad H^*(B \mathbb{K}_0) \simeq \mathbb{C}[u_1,...,u_n] / I_0. $$

Now, the two above cohomology groups are related by the Gysin sequence $$ ... \longrightarrow H^*(B \mathbb{K}) \overset{\cup f(p)}{\longrightarrow} H^{*+2}(B \mathbb{K}) \overset{\pi^*}{\longrightarrow} H^{*+2}(B \mathbb{K}_0) \overset{\pi_*}{\longrightarrow} H^{*+1}(B \mathbb{K}) \longrightarrow ... \ . $$ In particular, the image of the cup-product by $f(p)$ is the kernel of the projection $$ \mathbb{C}[u_1,...,u_n] / I \to \mathbb{C}[u_1,...,u_n] / I_0. $$

Claim: $\cup f(p)$ is the multiplication by $p$ !

Could someone help me ? Thanks to all for your help!

  • 1
    $\begingroup$ 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? $\endgroup$ – Jason Starr Mar 12 at 12:27
  • $\begingroup$ @Jason Starr, thank you. Could you explain to me what the cocharacter lattice is ? What is the notation $\text{Hom}_{\text{tori}}(\mathbb{K},S^1)$ ? $\endgroup$ – BrianT Mar 12 at 12:30
  • $\begingroup$ 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 $2$. $\endgroup$ – Jason Starr Mar 12 at 15:12
  • $\begingroup$ Thanks for your answer. I will modify a little bit my text in order to make clearer what I am trying to do. $\endgroup$ – BrianT Mar 12 at 16:02

Your Answer

By clicking "Post Your Answer", you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.