Multiplicativity in fibre bundles

Abstract

The Milnor manifolds H _ij ⊂ ℙ_i × ℙ_j of the last section are total spaces of a beautiful fibre bundle. To see this, we consider the projection of ℙ_i × ℙ_j onto ℙ_i. This induces for i ≤ j a fibration of H _ij over ℙ_i with fibre ℙ_{j_1}, as one sees directly from the equation for H _ij . The manifold H _ij is therefore the total space of a projective bundle over ℙ_i. Since for even j every genus φ(ℙ_{j_1}) is zero, and since in the last section we saw that for elliptic genera also φ(H _ij ) = 0 for even j with j ≥ i, it follows that elliptic genera behave multiplicatively for these fibre bundles:

$$ \varphi ({H_{ij}}) = \varphi ({P_i})\cdot \varphi ({P_{j - 1}})\quad for\;j\;even,\;j\;\underline > \;i. $$

.