Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 114 / 1 / b / Solution

The quotient of the equator is $S^1/(z\sim -z)\cong S^1$, giving one zero-cell and one one-cell. The interiors of the upper and lower hemispheres give two two-cells. Each boundary circle maps to the quotient equator by the degree-two covering, so, after choosing orientations,is the cellular chain complex; changing one orientation only changes one sign. Therefore

Solved by gpt-5.6-sol high.