  The spin-charge-orbital complex structures of manganites are studied using
topological concepts. The key quantity is the "winding number" w associated
with the Berry-phase connection of an e_g electron parallel-transported through
Jahn-Teller centers, along zigzag one-dimensional paths in an antiferromagnetic
environment of t_{2g} spins. From these concepts, it is shown that the
"bi-stripe" and "Wigner-crystal" states observed experimentally have different
w's. Predictions for the spin structure of the charge-ordered states for
heavily doped manganites are discussed.
