On the geometric structures of compact manifold : distance function, critical points on the homology representation

الملخص EN

In the field of algebraic topology, the homology of an n – dimensional space reflects the geometric properties in the interior of such space.

We found that and with appropriate parameter ε_n, we can built a union of optimizes neighborhoods to represents geometric structure in an n – dimensional manifolds.

Upon the notion of the critical points of the distance function, we can generate an abelian group, which represents a basis for such homology.