A Homotopy-like Class Invariant for Sub-manifolds of Punctured Euclidean Spaces

Abstract

We consider the $D$-dimensional Euclidean space, ${\mathbb{R}}^D$, with certain $(D-N)$-dimensional compact, closed and orientable sub-manifolds (which we call \emph{singularity manifolds} and represent by $\tilde{\mathcal{S}}$) removed from it. We define and investigate the problem of finding a homotopy-like class invariant ($\chi$-homotopy) for certain $(N-1)$-dimensional compact, closed and orientable sub-manifolds (which we call \emph{candidate manifolds} and represent by $\omega$) of ${\mathbb{R}}^D\setminus \tilde{\mathcal{S}}$, with special emphasis on computational aspects of the problem. We determine a differential $(N-1)$-form, ${\psi }_{\tilde{\mathcal{S}}}$, such that ${\chi }_{\tilde{\mathcal{S}}}(\omega )={\int }_{\omega }{\psi }_{\tilde{\mathcal{S}}}$ is a class invariant for such candidate manifolds. We show that the formula agrees with formulae from Cauchy integral theorem and Residue theorem of complex analysis (when $D=2,N=2$), Biot-Savart law and Ampere's law of theory of electromagnetism (when $D=3,N=2$), and the Gauss divergence theorem (when $D=3,N=3$), and discover that the underlying equivalence relation suggested by each of these well-known theorems is the $\chi$-homotopy of sub-manifolds of these low dimensional punctured Euclidean spaces. We describe numerical techniques for computing ${\psi }_{\tilde{\mathcal{S}}}$ and its integral on $\omega$, and give numerical validations of the proposed theory for a problem in a 5-dimensional Euclidean space. We also discuss a specific application from \emph{robot path planning problem}, when N=2, and describe a method for computing least cost paths with homotopy class constraints using \emph{graph search techniques}.

Submitted to Springer, Discrete & Computational Geometry

References in corpus (1)

• Perelman's proof of the Poincaré conjecture: a nonlinear PDE perspective