  The same bulk two-dimensional topological phase can have multiple distinct,
fully-chiral edge phases. We show that this can occur in the integer quantum
Hall states at $\nu=8$ and 12, with experimentally-testable consequences. We
show that this can occur in Abelian fractional quantum Hall states as well,
with the simplest examples being at $\nu=8/7, 12/11, 8/15, 16/5$. We give a
general criterion for the existence of multiple distinct chiral edge phases for
the same bulk phase and discuss experimental consequences. Edge phases
correspond to lattices while bulk phases correspond to genera of lattices.
Since there are typically multiple lattices in a genus, the bulk-edge
correspondence is typically one-to-many; there are usually many stable fully
chiral edge phases corresponding to the same bulk. We explain these
correspondences using the theory of integral quadratic forms. We show that
fermionic systems can have edge phases with only bosonic low-energy excitations
and discuss a fermionic generalization of the relation between bulk topological
spins and the central charge. The latter follows from our demonstration that
every fermionic topological phase can be represented as a bosonic topological
phase, together with some number of filled Landau levels. Our analysis shows
that every Abelian topological phase can be decomposed into a tensor product of
theories associated with prime numbers $p$ in which every quasiparticle has a
topological spin that is a $p^n$-th root of unity for some $n$. It also leads
to a simple demonstration that all Abelian topological phases can be
represented by $U(1)^N$ Chern-Simons theory parameterized by a K-matrix.
