[Paper Review] Motion planning in connected sums of real projective spaces

Daniel C. Cohen, Lucile Vandembroucq|arXiv (Cornell University)|Jul 26, 2018

Robotic Path Planning Algorithms10 references4 citations

TL;DR

This paper establishes that the topological complexity of the connected sum of $ g \geq 2 $ copies of $ \mathbb{R}P^n $ for $ n \geq 2 $ is exactly $ 2n $, demonstrating that motion planning in these higher-dimensional nonorientable manifolds achieves the maximal possible complexity. Using algebraic topology tools—specifically, the integral cohomology and the ideal-valued topological complexity invariant—the authors prove that the standard motion planner construction based on CW skeleta is optimal, extending prior results from surfaces to higher dimensions.

ABSTRACT

The topological complexity ${\sf TC}(X)$ is a homotopy invariant of a topological space $X$, motivated by robotics, and providing a measure of the navigational complexity of $X$. The topological complexity of a connected sum of real projective planes, that is, a high genus nonorientable surface, is known to be maximal. We use algebraic tools to show that the analogous result holds for connected sums of higher dimensional real projective spaces.

Motivation & Objective

• To determine the topological complexity of connected sums of $ g \geq 2 $ copies of $ \mathbb{R}P^n $ for $ n \geq 2 $, extending known results from surfaces to higher dimensions.
• To establish that the topological complexity of these spaces is maximal, equal to $ 2n $, using algebraic invariants.
• To generalize the motion planning construction from cell complexes to higher-dimensional nonorientable manifolds, proving optimality via cohomological methods.
• To extend techniques from prior work on nonorientable surfaces to higher-dimensional real projective spaces using the ideal-valued topological complexity and Steenrod squares.

Proposed method

• The authors use the ideal-valued topological complexity invariant $ \operatorname{{\sf TC}}(X) $, defined as one less than the minimal number of local domains in a motion planner for a space $ X $.
• They analyze the cohomology of the product space $ X \times X $, particularly in $ \mathbb{Z}_2 $-coefficients, to detect non-vanishing cup products that obstruct lower complexity.
• The key tool is the evaluation of the $ \mathfrak{v}^{2n} $-invariant on the diagonal class $ \Delta_{*} $, which detects the nontriviality of the top-dimensional cohomology class.
• The proof relies on decomposing the diagonal class $ \mathbf{c}_n \times \mathbf{c}_n $ in terms of generators $ \mathbf{a}_n, \mathbf{b}_n $, and analyzing the image under the $ \mathfrak{v}^{2n} $-map in the tensor product of group rings.
• By projecting to subalgebras $ I(Y;\mathbb{Z}_2) \otimes I(Z;\mathbb{Z}_2) $, they isolate the non-vanishing components corresponding to $ (y-1)^{\otimes n-2} \otimes (z-1)^{\otimes n-2} $, which are shown to survive under the $ \mathfrak{v}^{2n} $-map.
• They verify the non-vanishing of $ \mathfrak{v}^4(\mathbf{a}_2 \times \mathbf{b}_2 + \mathbf{b}_2 \times \mathbf{a}_2) $ via reduction to a 3-fold wedge product in $ \bigwedge^3 I(D;\mathbb{Z}_2) $, confirming the complexity bound.

Experimental results

Research questions

• RQ1 What is the topological complexity of the connected sum of $ g \geq 2 $ copies of $ \mathbb{R}P^n $ for $ n \geq 2 $?
• RQ2 Does the standard motion planner construction based on CW skeleta yield an optimal motion planner for these higher-dimensional nonorientable manifolds?
• RQ3 Can the cohomological techniques used for surfaces be extended to prove maximality of topological complexity in higher dimensions?
• RQ4 Is the topological complexity of $ \mathcal{P}_g^n = \mathbb{R}P^n \# \cdots \# \mathbb{R}P^n $ equal to $ 2n $?
• RQ5 What is the role of the $ \mathfrak{v}^{2n} $-invariant in detecting the non-vanishing of the diagonal class in the cohomology of $ X \times X $?

Key findings

• The topological complexity of the connected sum of $ g \geq 2 $ copies of $ \mathbb{R}P^n $ is exactly $ 2n $ for all $ n \geq 2 $, confirming that it is maximal.
• The standard motion planner construction based on the CW skeleta of $ \mathcal{P}_g^n $ is optimal, as it achieves the upper bound $ 2n $.
• The non-vanishing of the $ \mathfrak{v}^{2n} $-invariant on the diagonal class $ \Delta_* $ in $ H_{2n}(X \times X; \mathbb{Z}_2) $ confirms that $ \operatorname{{\sf TC}}(\mathcal{P}_g^n) = 2n $.
• The key cohomological component arises from the terms $ \mathbf{a}_{n-2} \times \mathbf{b}_{n-2} $ and $ \mathbf{b}_{n-2} \times \mathbf{a}_{n-2} $, which project nontrivially to $ I(Y;\mathbb{Z}_2)^{\otimes n-2} \otimes I(Z;\mathbb{Z}_2)^{\otimes n-2} $.
• The non-vanishing of $ \mathfrak{v}^4(\mathbf{a}_2 \times \mathbf{b}_2 + \mathbf{b}_2 \times \mathbf{a}_2) $ is confirmed via reduction to a nonzero element in $ \bigwedge^3 I(D;\mathbb{Z}_2) $, completing the proof.
• The result generalizes prior work on nonorientable surfaces ($ n=2 $) to higher dimensions, showing that the complexity remains maximal at $ 2n $.