  We analyze the global theory of boundary conditions for a constrained quantum
system with classical configuration space a compact Riemannian manifold $M$
with regular boundary $\Gamma=\partial M$. The space $\CM$ of self-adjoint
extensions of the covariant Laplacian on $M$ is shown to have interesting
geometrical and topological properties which are related to the different
topological closures of $M$. In this sense, the change of topology of $M$ is
connected with the non-trivial structure of $\CM$. The space $\CM$ itself can
be identified with the unitary group $\CU(L^2(\Gamma,\C^N))$ of the Hilbert
space of boundary data $L^2(\Gamma,\C^N)$. A particularly interesting family of
boundary conditions, identified as the set of unitary operators which are
singular under the Cayley transform, $\CC_-\cap \CC_+$ (the Cayley manifold),
turns out to play a relevant role in topology change phenomena. The singularity
of the Cayley transform implies that some energy levels, usually associated
with edge states, acquire an infinity energy when by an adiabatic change the
boundary condition reaches the Cayley submanifold $\CC_-$. In this sense
topological transitions require an infinite amount of quantum energy to occur,
although the description of the topological transition in the space $\CM$ is
smooth. This fact has relevant implications in string theory for possible
scenarios with joint descriptions of open and closed strings. In the particular
case of elliptic self--adjoint boundary conditions, the space $\CC_-$ can be
identified with a Lagrangian submanifold of the infinite dimensional
Grassmannian. The corresponding Cayley manifold $\CC_-$ is dual of the Maslov
class of $\CM$.
