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Let $S^1$ denote the circle with the non-trivial spin structure, i.e. $0\neq[S^1]\in\Omega^{Spin}_1$. Considering characteristic numbers it is easy so see that $S^1\times\mathbb{H}P^3$ is spin null bordant, say via $W$. (If you wish you can take $W$ simply connected.) Now let $g$ be the standard metric of positive scalar curvature on $\mathbb{H}P^3$, then $dt^2\times g$ is a metric of positive scalar curvature on $\partial W=S^1\times\mathbb{H}P^3$.

Is it possible to extend $dt^2\times g$ to a metric of positive scalar curvature on $W$?

A possible answer may use the 'obstruction groups' $R_n$ introduced by Stolz (in: Concordance classes of positive scalar curvature metrics). However, there are hardly computable. In addition, eventual index obstructions vanish for dimensional reasons, as $\dim W\equiv6\mod{8}$.

Although I believe that this is a quite hard question I am grateful for any comments in advance.

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It seems like you expect the answer to be negative, since you're looking for obstructions? –  Dylan Thurston Jun 25 '12 at 15:55

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