Poincaré-Hopf Theorem and Its Applications

Abstract

This paper builds up and proves the Poincaré-Hopf Theorem for compact smooth manifolds. Beginning with the introduction of key concepts in differential topology, including smooth manifolds, coordinate charts, tangent spaces, and mappings, we prove foundational theorems like the Inverse Function Theorem and Sard’s Theorem. Using these results, we establish the properties of smooth homotopies, smooth isotopies, and the Homotopy and Homogeneity Lemmas, which lead to the definition of degree modulo 2 and Brouwer’s Degree for oriented manifolds. We define smooth vector fields and prove the invariance of indices of isolated zeros under diffeomorphism. We finally use Morse theory and constriction of the quality of zeros to prove that the sum of the indices at the isolated zeros of a smooth vector field equals the Euler characteristic of the manifold. This equivalence is then extended for closed manifolds and nondegenerate zeros to general compact manifolds. Then we demonstrate its application to the Hairy Ball Theorem on $S^2$ and the vanishing of the Euler characteristic for all compact, boundaryless, odd-dimensional manifolds.

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