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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 $\widetilde{\mathcal{S}}$)removed from it. We define and investigate the problem of finding ahomotopy-like class invariant ($\chi$-homotopy) for certain $(N-1)$-dimensionalcompact, closed and orientable sub-manifolds (which we call \emph{candidatemanifolds} and represent by $\omega$) of $\mathbb{R}^D \setminus\widetilde{\mathcal{S}}$, with special emphasis on computational aspects of theproblem. We determine a differential $(N-1)$-form,$\psi_{\widetilde{\mathcal{S}}}$, such that$\chi_{\widetilde{\mathcal{S}}}(\omega) = \int_\omega\psi_{\widetilde{\mathcal{S}}}$ is a class invariant for such candidatemanifolds. We show that the formula agrees with formulae from Cauchy integraltheorem and Residue theorem of complex analysis (when $D=2,N=2$), Biot-Savartlaw and Ampere's law of theory of electromagnetism (when $D=3,N=2$), and theGauss divergence theorem (when $D=3,N=3$), and discover that the underlyingequivalence relation suggested by each of these well-known theorems is the$\chi$-homotopy of sub-manifolds of these low dimensional punctured Euclideanspaces. We describe numerical techniques for computing$\psi_{\widetilde{\mathcal{S}}}$ and its integral on $\omega$, and givenumerical validations of the proposed theory for a problem in a 5-dimensionalEuclidean space. We also discuss a specific application from \emph{robot pathplanning problem}, when N=2, and describe a method for computing least costpaths with homotopy class constraints using \emph{graph search techniques}.

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