  We study a two-dimensional (2D) tight-binding model of a topological
crystalline insulator (TCI) protected by rotation symmetry. The model is built
by stacking two Chern insulators with opposite Chern numbers which transform
under conjugate representations of the rotation group, e.g. $p_\pm$ orbitals.
Despite its apparent similarity to the Kane-Mele model, it does not host stable
gapless surface states. Nevertheless the model exhibits topological responses
including the appearance of quantized fractional charge bound to rotational
defects (disclinations) and the pumping of angular momentum in response to
threading an elementary magnetic flux, which are described by a mutual
Chern-Simons coupling between the electromagnetic gauge field and an effective
gauge field corresponding to the rotation symmetry. In addition, we show that
although the filled bands of the model do not admit a symmetric Wannier
representation, this obstruction is removed upon the addition of appropriate
atomic orbitals, which implies `fragile' topology. As a result, the response of
the model can be derived by representing it as a superposition of atomic
orbitals with positive and negative integer coefficients. Following the
analysis of the model, which serves as a prototypical example of 2D TCIs
protected by rotation, we show that all TCIs protected by point group
symmetries which do not have protected surface states are either atomic
insulators or fragile phases. Remarkably, this implies that gapless surface
states exist in free electron systems if and only if there is a stable Wannier
obstruction. We then use dimensional reduction to map the problem of
classifying 2D TCIs protected by rotation to a zero-dimensional (0D) problem
which is then used to obtain the complete non-interacting classification of
such TCIs as well as the reduction of this classification in the presence of
interactions.
