Symmetry protected topological phases of one-dimensional spin systems have
been classified using group cohomology. In this paper, we revisit this problem
for general spin chains which are invariant under a continuous on-site symmetry
group G. We evaluate the relevant cohomology groups and find that the
topological phases are in one-to-one correspondence with the elements of the
fundamental group of G if G is compact, simple and connected and if no
additional symmetries are imposed. For spin chains with symmetry
PSU(N)=SU(N)/Z_N our analysis implies the existence of N distinct topological
phases. For symmetry groups of orthogonal, symplectic or exceptional type we
find up to four different phases. Our work suggests a natural generalization of
Haldane's conjecture beyond SU(2).Comment: 18 pages, 7 figures, 2 tables. Version v2 corresponds to the
published version. It includes minor revisions, additional references and an
application to cold atom system
