  We consider planning problems on a punctured Euclidean spaces, $\mathbb{R}^D
- \widetilde{\mathcal{O}}$, where $\widetilde{\mathcal{O}}$ is a collection of
obstacles. Such spaces are of frequent occurrence as configuration spaces of
robots, where $\widetilde{\mathcal{O}}$ represent either physical obstacles
that the robots need to avoid (e.g., walls, other robots, etc.) or illegal
states (e.g., all legs off-the-ground). As state-planning is translated to
path-planning on a configuration space, we collate equivalent plannings via
topologically-equivalent paths. This prompts finding or exploring the different
homology classes in such environments and finding representative optimal
trajectories in each such class.
  In this paper we start by considering the problem of finding a complete set
of easily computable homology class invariants for $(N-1)$-cycles in
$(\mathbb{R}^D - \widetilde{\mathcal{O}})$. We achieve this by finding explicit
generators of the $(N-1)^{st}$ de Rham cohomology group of this punctured
Euclidean space, and using their integrals to define cocycles. The action of
those dual cocycles on $(N-1)$-cycles gives the desired complete set of
invariants. We illustrate the computation through examples.
  We further show that, due to the integral approach, this complete set of
invariants is well-suited for efficient search-based planning of optimal robot
trajectories with topological constraints. Finally we extend this approach to
computation of invariants in spaces derived from $(\mathbb{R}^D -
\widetilde{\mathcal{O}})$ by collapsing subspace, thereby permitting
application to a wider class of non-Euclidean ambient spaces.
