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

Abstract:

As a first goal, it is explained why the Goodwillie-Weiss calculus of embeddings offers new information about the Euclidean embedding dimension of $\mathrm {P}^m$ only for $m\leq 15$. 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 $m\geq 16$, the relation $\mathrm {TC}^S(\mathrm {P}^m)\geq n$ is translated into the triviality of a certain cohomotopy Euler class which, in turn, becomes the only Taylor obstruction to producing an embedding $\mathrm {P}^m\subset \mathbb {R}^n$. A speculative bordism-type form of this primary obstruction is proposed as an analogue of Davis’ $BP$-approach to the immersion problem of $\mathrm {P}^m$. A form of the Euler class viewpoint is applied to show that $\mathrm {TC}^S(\mathrm {P}^3)=5$, as well as to suggest a few higher-dimensional projective spaces for which the method could produce new information. As a second goal, the paper extends Farber’s work on the motion planning problem in order to develop the notion of a symmetric motion planner for a mechanical system ${\mathcal S}$. Following Farber’s lead, this concept is connected to $\mathrm {TC}^S(C({\mathcal S}))$, the symmetric topological complexity of the state space of ${\mathcal S}$. The paper ends by sketching the construction of a concrete $5$-local-rules symmetric motion planner for $\mathrm {P}^3$.