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STRING TOPOLOGICAL ROBOTICS 2
Keywords:
string topology, topological robotics, Gerstenhaber algebra, Batalin-Vilkovisky algebraDOI:
https://doi.org/10.17654/0972555522031Abstract
We aim to link two well-known theories; namely the string topology (founded by Chas and Sullivan in [1]) and the topological robotics (founded by Farber some few years later, in [4]). For our purpose, we consider a compact Lie group $G$ acting on a path connected $n$-manifold $M$. On the set $\mathcal{M}^{L P}(M)$ of the loop motion planning algorithms, we define a string loop motion planning product and extend it to a kind of a string loop motion planning product, which endows the shifted homology $\mathbb{H}_*\left(\mathcal{M}^{L P}(M)\right):=H_{*+2 n}\left(\mathcal{M}^{L P}(M)\right)$, with a structure of a graded commutative and associative algebra on $\mathbb{H}_*\left(\mathcal{M}^{L P}(M)\right)$. We show that it yields a structure of Gerstenhaber and Batalin-Vilkovisky algebras.
Received: January 15, 2022
Accepted: March 3, 2022
References
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