 Solution

False. Take a bouquet of two circles and the two-dimensional torus. Both are connected polyhedra and have first homology group ${\mathbb{Z}}^2$. Their fundamental groups are respectively the nonabelian free group $F_2$ and the abelian group ${\mathbb{Z}}^2$, so they are not isomorphic. First homology only remembers the abelianization of the fundamental group.