  One of the excitements generated by the cold atom systems is the possibility
to realize, and explore, varied topological phases stemming from
multi-component nature of the condensate. Popular examples are the
antiferromagnetic (AFM) and the ferromagnetic (FM) phases in the
three-component atomic condensate with effective spin-1, to which different
topological manifolds can be assigned. It follows, from consideration of
homotopy, that different sorts of topological defects will be stable in each
manifold. For instance, Skyrmionic texture is believed to be a stable
topological object in two-dimensional AFM spin-1 condensate. Countering such
common perceptions, here we show on the basis of a new wave function
decomposition scheme that there is no physical parameter regime wherein the
temporal dynamics of spin-1 condensate can be described solely within AFM or FM
manifold. Initial state of definite topological number prepared entirely within
one particular phase must immediately evolve into a mixed state. Accordingly,
the very notion of topology and topological stability within the sub-manifold
of AFM or FM become invalid. Numerical simulation reveals the linear Zeeman
effect to be an efficient catalyst to extract the alternate component from an
initial topological object prepared entirely within one particular
sub-manifold, serving as a potential new tool for "topology engineering" in
multi-component Bose-Einstein condensates.
