TITLE: Is it possible for a closed manifold to deformation retract onto a proper subset of itself?
QUESTION [14 upvotes]: Let $M$ be a closed (compact, without boundary) topological manifold. Is it possible for there to exist a subset $A$ of $M$ such that $M$ deformation retracts onto $A$?

REPLY [2 votes]: By a corollary of the generalized Poincaré duality, referred as Poincaré Alexander Lefschetz duality in the book Geometry and Topology of Bredon, you can derive the following tool :

Corollary 8.5 (Bredon - G. & T.) If $L$ is a proper compact subset of an orientable connected $n$-manifold $M$, then $\check H^n(L; G) = 0$ for any coefficient group $G$.

The same is true with $\mathbb Z/2$ coefficients if $M$ is not orientable. If $M$ retracts on a proper subspace $L$, then $L$ is closed.
But for any proper closed subset $L$ of $M$, by this property from Bredon, we have $\check H^n(L;\mathbb Z/2)=0$. This implies that $L$ can't be a deform retract of $M$, otherwise $L$ would have a non vanishing top homology, as $M$ does.