We have the following CW-complex inside a Complex Projective space:
One $0$-cell,
Three $2$-cells A,B,C. Their closure is made by adding the $0$-cell,
Three $4$-cells. The closure of the first one is made by adding A,B and the $0$-cell, the closure of the second one is made by adding A,C and the $0$-cell, the closure of the third one is made by adding B,C and the $0$-cell,
One $6$-cell, its closure is made by adding everything listed above.
We want to know:
$1)$ Can I deduce this is not a smooth variety since there are three $2$-cells attached to a single $0$-cell?
$2)$ Can I deduce the homology groups with coefficients in $\mathbb Q$? I don't understand if their dimensions listed from bottom to top only in even degrees are $(1,0,0,1)$ or $(1,3,3,1)$.
Thanks!