Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 114 / 4 / c / Solution

Write $\alpha =pa+qb$. In the Gysin sequence of a sphere bundle for the oriented circle bundle $S(E_{\alpha })\to S^2\times S^2$, multiplication by $\alpha$ isin degrees $0$ to $2$, andin degrees $2$ to $4$. If $\alpha \ne 0$ and $d=\gcd (|p|,|q|)$, taking the relevant kernels and cokernels givesWhen $\alpha =0$, the bundle is trivial and the groups in degrees $0$ through $5$ have ranks $1,1,2,2,1,1$, respectively. These are precisely the groups recorded in integral cohomology of a circle bundle over a product of two spheres.

The additive cohomology for nonzero $\alpha$ depends only on $d$. On the other hand, part (a) shows that the homeomorphism group acts on $(p,q)$ only by signed permutations. For example, $(1,0)$ and $(1,1)$ both have $d=1$, so their sphere bundles have isomorphic additive cohomology, but no signed permutation carries one Euler class to the other. The cohomology therefore does not determine the homeomorphism-group orbit of $\alpha$.