5
$\begingroup$

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 ?

$\endgroup$
2
+50
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.