  Symmetry protected topological (SPT) phases are gapped phases of matter that
cannot be deformed to a trivial phase without breaking the symmetry or closing
the bulk gap. Here, we introduce a new notion of a topological obstruction that
is not captured by bulk energy gap closings in periodic boundary conditions.
More specifically, we say two bulk Hamiltonians belong to distinct boundary
obstructed topological `phases' (BOTPs) if they can be deformed to each other
on a system with periodic boundaries, but cannot be deformed to each other for
symmetric open boundaries without closing the gap at at least one high symmetry
region on the surface. BOTPs are not topological phases of matter in the
standard sense since they are adiabatically deformable to each other on a torus
but, similar to SPTs, they are associated with boundary signatures in open
systems such as surface states or fractional corner or hinge charges. We show
that the double-mirror quadrupole model of [Science, 357(6346), 2018] is a
prototypical example of such phases, and present a detailed analysis of several
aspects of boundary obstructions in this model. In addition, we introduce
several three-dimensional models having boundary obstructions, which are
characterized either by surface states or fractional corner or hinge charges.
We also provide a general framework to study boundary obstructions in
free-fermion systems in terms of Wannier band representations (WBR), an
extension of the recently-developed band representation formalism to Wannier
bands. WBRs capture the notion of topological obstructions in the Wannier bands
which can then be used to study topological obstructions in the boundary
spectrum by means of the correspondence between the Wannier and boundary
spectra. This establishes a form of bulk-boundary correspondence for BOTPs by
relating the bulk band representation to the boundary topology.
