Differential Topology and the Poincaré-Hopf Theorem

Authors

• Tara Saini, Bellevue High SchoolFollow

Abstract

In this paper, we will develop the ideas needed to understand and prove the Poincaré–Hopf Theorem, which connects the local behavior of smooth vector fields to global topological properties. We will begin by introducing smooth manifolds and smooth maps, which are the basis of differential topology. We will then define derivatives of smooth maps through tangent spaces and use these to classify points. To build toward the theorem, we will introduce orientation, degree, and smooth vector fields. These concepts will culminate in a proof of the Poincaré–Hopf Theorem, aided by Brouwer’s Fixed Point Theorem. Finally, we will apply the result to the famous Hairy Ball Theorem and other results related to the Euler characteristic.

Author Bio

Tara is a high school student from Bellevue, Washington. She is interested in pursuing pure math research. She has taken classes in linear algebra, differential equations, and vector calculus, and has also studied algebraic topology and abstract algebra at advanced math programs. She is currently researching combinatorial graph theory and applications of the Poincaré-Hopf Theorem.

Recommended Citation

Saini, Tara (2025) "Differential Topology and the Poincaré-Hopf Theorem," Rose-Hulman Undergraduate Mathematics Journal: Vol. 26: Iss. 2, Article 5.
Available at: https://scholar.rose-hulman.edu/rhumj/vol26/iss2/5