pair of injective morphisms of simplicial groups

Question

Let $X$ and $Y$ be two simplicial groups such that there exists $f:X\rightarrow Y$ and $g:Y\rightarrow X $ two injective morphisms of simplicial groups. Suppose that $X$ is contractible, does it follow that $Y$ is contractible ?

Answer 1

Pick pointed topological spaces $A$ and $B$ which admit pointed injective continuous maps $A \to B$ and $B \to A$ for which $A$ is contractible but $B$ has nonvanishing reduced homology. For example, we could take $A = [0,1]$ and $B = [0,1] \cup \{2\}$ pointed at $0$.

Now apply the functor $\mathbb{Z}[\mathrm{Sing}(-)]/\mathbb{Z}[\mathrm{Sing}(*)]$ to obtain simplicial (abelian) groups $X$ and $Y$ and maps $X \to Y$ and $Y \to X$, which are still injective. We have $\pi_*(X) = \tilde H_*(A)$ and $\pi_*(Y) = \tilde H_*(B)$, so $X$ is contractible but $Y$ is not.