2 removed "projective"

Given any smooth complex surface, you can blow up any point on the surface:

http://en.wikipedia.org/wiki/Blowing_up

This gives you another smooth, projective complex surface with isomorphic fundamental group but with the second Betti number increased by 1. (There is, in some sense, essentially one more curve on the new surface than there was on the old one, called the "exceptional divisor".)

Starting with any one simply connected smooth complex surface (e.g. the projective plane) and repeatedly blowing up shows that your question has a negative answer.

1

Given any smooth complex surface, you can blow up any point on the surface:

http://en.wikipedia.org/wiki/Blowing_up

This gives you another smooth, projective complex surface with isomorphic fundamental group but with the second Betti number increased by 1. (There is, in some sense, essentially one more curve on the new surface than there was on the old one, called the "exceptional divisor".)

Starting with any one simply connected smooth complex surface (e.g. the projective plane) and repeatedly blowing up shows that your question has a negative answer.