  Consider the configuration spaces of manifolds. An influential theorem of
McDuff, Segal and Church shows that the (co)homology of the unordered
configuration space is independent of number of points in a range of degree
called the stable range. We study the another important (and general) property
of unordered configuration spaces of manifolds (not necessarily orientable, and
not necessarily admitting non-vanishing vector field) that is homological
monotonicity in unstable part. We show that the homological dimension of
unordered configuration spaces of manifolds in each degree is monotonically
increasing. Our results show that the monotonicity property is not depend on
the differential structure and orientiability of manifold.
