# Closed manifolds are not absolute retracts?

A fundamental result in topology is that the $$n$$-sphere is not a retract of the $$n+1$$-ball. It implies that the $$n$$-sphere is not an absolute retract.

Is there a generalization from the sphere to closed manifolds (compact manifolds without boundary)? It would be the statement that no closed manifold is an absolute retract.

• Perhaps a bit more properties need to be assumed about the manifold? Do you know any sufficient conditions for your theorems to work? Mar 15, 2021 at 11:38

Closed manifolds are not contractible (if $$\dim M=n$$ look at $$H_n(M)$$) hence they are not ARs. Note however that contractibility is the only obstruction for manifolds, since every topological manifold is an ANR.
• Thank you! I see here that $H_n(M; \mathbb{Z}_2)=\mathbb{Z}_2$. Mar 15, 2021 at 12:07
• @mathieu: Yes, every closed topological manifold is $\mathbb{Z}/2\mathbb{Z}$-orientable. For example, you can find the result and its proof in Hatcher's book, p.235 in 2nd ed.