2017/02/17 by Jesús González, González, Jesús
Mathematics · #55M30 #55S15 #57R40 #68T40 #Advanced Combinatorial Mathematics #Algebraic Geometry and Number Theory #Algebraic Topology (math.AT) #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #math.AT #msc:55M30 #msc:55S15 #msc:57R40 #msc:68T40
paper · pdf · doi:10.48550/arxiv.1702.05457
15 pages. New features: (1) A full unrestricted characterization of TC^Σ of RP^m in terms of symmetric Z_2-biequivariant maps with a "monoidal" behavior in the diagonal. (2) A computation of TC^Σ of RP^m for m a 2-power. (3) A computation of TC^Σ of S^1 reproving the recent announcements by D. M. Davis and M. Grant
openalex publication_date 2017/02/17 · arxiv created 2017/03/22 · arxiv updated 2017/03/24 · openalex created_date 2022/08/07 · openalex updated_date 2026/07/28
This work is motivated by the question of whether there are spaces X for which the Farber-Grant symmetric topological complexity TCS(X) differs from the Basabe-González-Rudyak-Tamaki symmetric topological complexity TCΣ(X). It is known that, for a projective space RPm, TCS(RPm) captures, with a few potentially exceptional cases, the Euclidean embedding dimension of RPm. We now show that, for all m≥1, TCΣ(RPm) is characterized as the smallest positive integer n for which there is a symmetric ℤ2-biequivariant map Sm× Sm→ Sn 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 RP2e for e≥1. In particular, this leaves the torus S1× S1 as the only closed surface whose symmetric (symmetrized) TCS (TCΣ) -invariant is currently unknown.