Cyclic point-gluing of two-spheres
= Cyclic point-gluing of two-spheres
Take $n$ labelled copies of $S^2$ and identify one marked point of copy $i$ with the antipodal marked point of copy $i+1$ cyclically. The resulting connected finite CW complex is homotopy equivalent to
$$
S^1\vee\bigvee_{i=1}^nS^2,
$$
so its rational homology has dimensions $1,1,n$ in degrees zero, one, and two, and its Euler characteristic is $n$.