\b[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.

Solution

Past exam of the mathematics course of the University of Cambridge / 2011 / ii / Paper 1 / 21H / iv / Solution

 Solution (source code)