Exercise 2, § 5

Let H and G be groups, let $\mathscr{H}$ and $\mathscr{G}$ be the groupoids associated with them; let $\varphi$ and $\psi$ be morphisms of groupoids from $\mathscr{H}$ into $\mathscr{G}$.

(a) Prove that the set of vertices of the cohomotopy coequalizer (resp. of the coequalizer) of the pair $(\varphi , \psi )$ is reduced to one element; denote by K (resp. L) its isotropy group at this point.

(b) Prove that the group K is the quotient of the free product $G*\mathbf{Z}$ by the smallest normal subgroup of $G*\mathbf{Z}$ which contains the elements $\varphi (g)t\psi (g)^{-1}t^{-1}$, for $g\in G$, where $t$ denotes the element 1 of $\mathbf{Z}$.

(c) Prove that the group L is the quotient of the group G by the smallest normal subgroup containing the elements $\varphi (g)\psi (g)^{-1}$, for $g\in G$.