Past exam of the mathematics course of the University of Cambridge / 2011 / ii / Paper 1 / 21H (Algebraic Topology)

True. A path $u:x\to y$ gives the basepoint-change isomorphism of fundamental groups $[\gamma ]\mapsto [\overline{u}\ast \gamma \ast u]$. Concatenating paths is compatible with homotopy and loop multiplication. Using the reversed path gives its inverse, because $u\ast \overline{u}$ and $\overline{u}\ast u$ contract relative to endpoints.

False. Changing the connecting path changes the isomorphism by an inner automorphism. Take $x=y$, one path constant and the other representing $a$ in the Klein bottle group $⟨a,b:ab{a}^{-1}=b^{-1}⟩$. The first induced isomorphism is the identity, whereas the second sends $b$ to $b^{-1}\ne b$. The element $b$ has infinite order, as is also witnessed by mapping it to a translation and $a$ to a reflection in the infinite dihedral group. A fixed path to a different endpoint can be appended if distinct basepoints are desired.

True. Coordinate projections define ${\pi }_1(X\times Y,(x,y))\to {\pi }_1(X,x)\times {\pi }_1(Y,y)$. A pair of based loops gives the inverse by $t\mapsto ({\gamma }_X(t),{\gamma }_Y(t))$. Homotopies combine and project coordinatewise, and loop concatenation does too. These maps are well-defined inverse group homomorphisms, requiring no local connectedness assumptions.

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.