Abstract

In this presentation, we chronologically trace the foundational ideas that led to the emergence of topology, beginning with L. Euler’s celebrated formula:

Vertices – Edges + Faces = 2,

which applies to polyhedra homeomorphic to the sphere. We then examine how H. Poincaré uncovered the deeper meaning of this relation through the development of homology theory, grounded in basic linear algebra concepts.

Next, we explore M. Morse’s significant contribution, demonstrating how these topological insights extend to the dynamic analysis of gradient vector fields derived from smooth real-valued functions f defined on a surface M. We illustrate how the topology of regular level sets changes as f passes through its critical points.

Finally, we consider more recent developments in the field, highlighting applications to dynamical systems via a modern homotopy-theoretic framework—an intuitive “rubber sheet geometry”—as introduced by C. Conley.