Symmetric topological complexity of projective and lens spaces

Complex-projective and lens product spaces

Let $t$ be a positive integer. Following work of D. M. Davis, we study the topology of complex-projective product spaces, i.e. quotients of cartesian products of odd dimensional spheres by the diagonal $S^1$-action, and of the $t$-torsion lens product spaces, i.e. the corresponding quotients when the action is restricted to the $t^{\mathrm{th}}$ roots of unity. For a commutative complex-oriented cohomology theory $h^*$, we determine the $h^*$-cohomology ring of these spaces (in terms of the $t$-series for $h^*$, in the case of $t$-torsion lens product spaces). When $h^*$ is singular cohomology…

A Relation Between Existence of Real Symmetric Nonsingular Bilinear Maps and the Antisymmetric Index of Projective Spaces

We obtain a new proof, using integral cohomology and group actions, of an old embedding theorem for real projective spaces.

Symmetric bi-skew maps and symmetrized motion planning in projective spaces

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 $m\geq1$, $TC^Σ(RP^m)$ is characterized as the smallest positive integer $n$ for which there is a symmetric $\mathbb{Z}_2$-biequivariant map $S^m\times S^m\to S^n$ with a "monoidal" b…

Symmetric topological complexity as the first obstruction in Goodwillie's Euclidean embedding tower for real projective spaces

As a first goal, it is explained why Goodwillie-Weiss calculus of embeddings offers new information about the Euclidean embedding dimension of P^m only for m < 16. Concrete scenarios are described in these low-dimensional cases, pinpointing where to look for potential, but critical, high-order obstructions in the corresponding Taylor towers. For m > 15, the relation TC^S(P^m) > n-1 is translated into the triviality of a certain cohomotopy Euler class which, in turn, becomes the only Taylor obstruction to producing an n-dimensional Euclidean embedding of P^m. A speculative bordism-type form of …

On the symmetric squares of complex and quaternionic projective space

The problem of computing the integral cohomology ring of the symmetric square of a topological space has been of interest since the 1930s, but limited progress has been made on the general case until recently. In this work we offer a solution for the complex and quaternionic projective spaces $KP^n$, by taking advantage of their rich geometrical structure. Our description is in terms of generators and relations, and our methods entail ideas that have appeared in the literature of quantum chemistry, theoretical physics, and combinatorics. We deal first with the case $KP^\infty$, and proceed by …

关于32维Rosenfeld射影平面

On the 32-dimensional Rosenfeld projective plane

本文在已有代数拓扑相关研究基础上，对属于嘉当紧对称空间列表中EIII类的32维Rosenfeld射影平面进行了详细研究。