Symmetrized topological complexity

Abstract

Abstract. We present upper and lower bounds for symmetrized topological complexity TC Σ (X) in the sense of Basabe-González-Rudyak-Tamaki. The upper bound comes from equivariant obstruction theory, and the lower bounds from the cohomology of the symmetric square SP 2 (X). We also show that symmetrized topological complexity coincides with its monoidal version, where the path from a point to itself is required to be constant. Using these results, we calculate the symmetrized topological complexity of all odd spheres.

Symmetric Bi-Skew Maps and Symmetrized Motion Planning in Projective Spaces

“…For instance, Mark Grant has noticed that TC Σ (X) is bounded from below by the cup-length of H * (SP 2 (X)) -compare to Proposition 4.6 below. The latter observation can be used (with X = S 1 , see Corollary 4.8 below) to reprove, in a slightly streamlined way, the fact (first noticed in [6,17]) that TC Σ (S 1 ) = 2.…”

“…The author would like to thank Mark Grant and Kee Lam for illuminating email discussions on the topics of this paper, and Don Davis and Mark Grant for sharing with the author of this paper early versions of their preprints [6,17].…”

“…As noted in [16,Example 2.6], e 0,1 is a τ -fibration, so TC Σ (X) can equivalently be defined as secat τ (e 0,1 ), the τ -equivariant sectional category of e 0,1 : P(X) → X × X.…”

“…hold for any reasonable space X (see [2,Proposition 4.2]). The equality TC Σ (X) = TC S (X) is known to hold for a number of spaces: spheres (see [2, Example 4.5] for even dimensional spheres and [16] for odd dimensional spheres), simply connected closed symplectic manifolds (as follows from [12,proof of Proposition 10] and [13,Theorem 1]; see [15,Theorem 6.1] for the case of complex projective spaces) and all closed surfaces with the potential exceptional case of the torus S 1 × S 1 (see Remark 4.2 below). Surprisingly, except for homotopically uninteresting situations [2,Example 4.4], no example of a space X with TC Σ (X) = TC S (X) is known.…”

“…Proof of the inequality sb(m) ≤ TC Σ (RP m ) in Theorem 1.4. In view of[17, Theorem 5.2], we can start with an open covering U 0 , . .…”