# Classifying map of a simple circle bundle

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!

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