Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 4 / 21F / Solution

For a continuous self-map $f$ of a finite CW complex, its Lefschetz number over $\mathbb{Q}$ isThe Lefschetz fixed-point theorem states that $L(f)\ne 0$ implies that $f$ has a fixed point. Consequently every fixed-point-free self-map has Lefschetz number zero.

The space in the question is a cyclic point-gluing of two-spheres. Before gluing, the $n$ spheres have total Euler characteristic $2n$. Each of the $n$ point identifications lowers the Euler characteristic by one, soEquivalently, $X\simeq S^1\vee {\bigvee }_{j=1}^n{S}^2$, with rational homology dimensions $1,1,n$ in degrees zero, one, and two.