We analyze form the topological perspective the space of all SLOCC
(Stochastic Local Operations with Classical Communication) classes of pure
states for composite quantum systems. We do it for both distinguishable and
indistinguishable particles. In general, the topology of this space is rather
complicated as it is a non-Hausdorff space. Using geometric invariant theory
(GIT) and momentum map geometry we propose a way to divide the space of all
SLOCC classes into mathematically and physically meaningful families. Each
family consists of possibly many `asymptotically' equivalent SLOCC classes.
Moreover, each contains exactly one distinguished SLOCC class on which the
total variance (a well defined measure of entanglement) of the state Var[v]
attains maximum. We provide an algorithm for finding critical sets of Var[v],
which makes use of the convexity of the momentum map and allows classification
of such defined families of SLOCC classes. The number of families is in general
infinite. We introduce an additional refinement into finitely many groups of
families using the recent developments in the momentum map geometry known as
Ness stratification. We also discuss how to define it equivalently using the
convexity of the momentum map applied to SLOCC classes. Moreover, we note that
the Morse index at the critical set of the total variance of state has an
interpretation of number of non-SLOCC directions in which entanglement
increases and calculate it for several exemplary systems. Finally, we introduce
the SLOCC-invariant measure of entanglement as a square root of the total
variance of state at the critical point and explain its geometric meaning.Comment: 37 pages, 2 figures, changes in the manuscript structur