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.



Closed manifolds are not absolute retracts?
A metrizable space is an absolute retract (AR) if and only if it is an absolute neighbourhood retract (ANR) and it is contractible.


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.


