TITLE: fundamental group and complete invariant of irreducible 3-manifolds 
QUESTION [8 upvotes]: I heard that,by Perelman's work,we can get that the fundamental group is a
complete invariant of irreducible 3-manifolds (except for lens spaces).
can someone help explain this.Thank you!

REPLY [2 votes]: An addendum to BM's answer: you might want to start with the Kneser-Milnor theorem which gives a unique decomposition of a 3-manifold as a connected sum and hence its fundamental group as a free product to reduce the question to prime (and then irreducible) 3-manifolds. The Poincare conjecture says you don't have any summands "hidden" from $\pi_1$. 
Also, you have to be careful with what kind of 3-manifolds you consider, for example  if $F_1$ is a 3 punctured sphere and $F_2$ a punctured torus, then    $F_1\times S^1$ and $F_2\times S^1$ (or the complement of the square and granny knots for a more subtle example) have the same fundamental group but aren't homeomorphic.  So perhaps you want to first consider closed 3-manifolds. The issues for general compact 3-manifolds can get technical, but they are   understood. A good starting point is Hempel's book.