  We show that the electronic structure of the low-energy bands in the small
angle-twisted bilayer graphene consists of a series of semi-metallic and
topological phases. In particular we are able to prove, using an approximate
low-energy particle-hole symmetry, that the gapped set of bands that exist
around all magic angles has what we conjecture to be a stable topological index
stabilized by a magnetic symmetry and reflected in the odd winding of the
Wilson loop in the Moir\'e BZ. The approximate, emergent particle-hole symmetry
is essential to the topology of graphene: when strongly broken, non-topological
phases can appear. Our paper underpins topology as the crucial ingredient to
the description of low-energy graphene. We provide a $4$-band short range
tight-binding model whose $2$ lower bands have the same topology, symmetry, and
flatness as those of the twisted graphene, and which can be used as an
effective low-energy model. We then perform large-scale ($11000$ atoms per unit
cell, 40 days per $\bf k$-point computing time) ab-initio calculations of a
series of small angles, from $3^\circ$ to $1^\circ$, which show a more complex
and somewhat qualitatively different evolution of the symmetry of the
low-energy bands than that of the theoretical Moir\'e model, but which confirms
the topological nature of the system. At certain angles, we find no insulating
filling in graphene at $-4$ electrons per Moir\'e unit cell. The ab-initio
evolution of gaps tends to differ from that of the continuum Moir\'e model.
