  The homotopy theory of the blow up construction in algebraic and symplectic
geometry is investigated via two approaches. The first approach introduces and
develops fibrewise surgery theory, for which the fibrewise framing is
characterized by the homotopy groups of a certain gauge group. This is used to
obtain a homotopy decomposition of the based loop space on a blow up that holds
$p$-locally for all but finitely many primes $p$ and holds rationally. The
second approach is purely homotopy theoretic and obtains a homotopy
decomposition of the based loop space on a blow up that holds integrally
provided a certain condition is satisfied by an associated homotopy action.
  As applications, we obtain $p$-local homotopy decompositions of the based
loop space of a focal genus $2$ manifold, improve an earlier result of the
authors on the homotopy of manifolds stabilized by projective spaces, and
obtain for blow-ups a refinement of the rational dichotomy.
