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

Abstract

This work is motivated by the question of whether there are spaces X for which the Farber-Grant symmetric topological complexity TC S (X) differs from the Basabe-González-Rudyak-Tamaki symmetric topological complexity TC Σ (X). It is known that, for a projective space RP m , TC S (RP m ) captures, with a few potentially exceptional cases, the Euclidean embedding dimension of RP m . We now show that, for all m ≥ 1, TC Σ (RP m ) is characterized as the smallest positive integer n for which there is a symmetric Z 2 -biequivariant map S m × S m → S n with a "monoidal" behavior on the diagonal. This result thus lies at the core of the efforts in the 1970's to characterize the embedding dimension of real projective spaces in terms of the existence of symmetric axial maps. Together with Nakaoka's description of the cohomology ring of symmetric squares, this allows us to compute both TC numbers in the case of RP 2 e for e ≥ 1. In particular, this leaves the torus S 1 × S 1 as the only closed surface whose symmetric (symmetrized) TC S (TC Σ ) -invariant is currently unknown.2010 Mathematics Subject Classification: 55M30, 57R40.

Symmetrized topological complexity

“…In the end, the cohomology of symmetric squares proved to be the right setting for proving Theorem 6.1, and perhaps more besides. For instance, Jesús González has used calculations in the cohomology of symmetric squares of real projective spaces to calculate TC Σ (RP m ) when m is a 2-power [14]. The contents of the paper are as follows.…”

“…We wish to thank Don Davis and Jesús González for useful discussions and for sharing with us preliminary versions of their preprints [6] and [14].…”

Bidirectional Sequential Motion Planning

“…For the the torus T = S 1 ×S 1 the previous result yields T C Σ 2 (T ) ≥ T C β 4 (S 1 ) ≥ T C 4 (S 1 ) = 4. This lower bound was obtained in [6] using the cohomology of the symmetric product SP 2 (T ). Moreover, the subadditivity property in Proposition 10 and the value…”

“…The following calculations are based on Nakaoka's analysis [8], which is distilled in [6], and allow us to estimate higher symmetrized topological complexities. We record here in a brief way what we need from [6] and [8].…”

“…The second part of this latter result should be compared with [1]. Using cohomological lower bounds and building on the work in [6] we get the following results.…”