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We develop the properties of the
sequential topological complexity
,
a homotopy invariant introduced by the third author as an extension
of Farber’s topological model for studying the complexity of motion
planning algorithms in robotics. We exhibit close connections of
to the Lusternik–Schnirelmann category of cartesian powers
of , to the cup length of
the diagonal embedding
,
and to the ratio between homotopy dimension and connectivity of
. We fully compute the
numerical value of
for products
of spheres, closed
–connected
symplectic manifolds and quaternionic projective spaces. Our study includes two symmetrized
versions of
. The
first one, unlike Farber and Grant’s symmetric topological complexity, turns out to be a homotopy
invariant of
;
the second one is closely tied to the homotopical properties of the configuration space of
cardinality-
subsets of .
Special attention is given to the case of spheres.