Let $X$ be a topological space and $V$ and $N$ are subspaces of $X$. It is known that if $V$ is homotopy equivalent to $N$ then $X-V$ need not be homotopy equivalent to $X-N$, the Alexander horned sphere is an example. In the context when $N$ deformation retracts onto $V$ it is still true that $X-V$ needs not deformation retracts onto $X-N$ as illustrated by the example of $X$ is the disk and $V$ is the equator and $N$ is the upper half disk. Now looking into this paper

http://arxiv.org/pdf/1203.6097.pdf page 4 proof of Lemma 3.2

the author seems to use this fact twice!!

Indeed, for $Qp(n)$ the quaternion projective space he says "Let $A^1$ be the complement of an open tubular neighborhood of $QP(n−1) in QP(n)$" then he says "$A^1$ is a deformation retract of $QP(n) − QP(n − 1)$"

Is there any additional property that let this happen in this case.. thank you for the clarification!!