= Solution
For a continuous self-map $f$ of a finite CW complex, its <Lefschetz number> over $\mathbb Q$ is
$$
L(f)=\sum_{q\geq0}(-1)^q
\operatorname{tr}\!\left(f_*:H_q(X;\mathbb Q)\to H_q(X;\mathbb Q)\right).
$$
The <Lefschetz fixed-point theorem> states that $L(f)\ne0$ 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, so
$$
\boxed{\chi(X)=n.}
$$
Equivalently, $X\simeq S^1\vee\bigvee_{j=1}^nS^2$, with rational homology dimensions $1,1,n$ in degrees zero, one, and two.
Back to article page
