# pair of injective morphisms of simplicial groups

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 ?

## 1 Answer

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.