Introduction to Homology and Cohomology Theory

• Grant number: 25/11518-8
• Support Opportunities: Scholarships in Brazil - Scientific Initiation
• Start date: September 01, 2025
• End date: August 31, 2026
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Dahisy Valadão de Souza Lima
• Grantee: Alice Micheloni de Souza
• Host Institution: Centro de Matemática, Computação e Cognição (CMCC). Universidade Federal do ABC (UFABC). Ministério da Educação (Brasil). Santo André , SP, Brazil

Abstract

We propose the study of two fundamental topological invariants - homology groups and cohomology groups - with an emphasis on their formal properties, algebraic structures, and applications to smooth differentiable manifolds.We will begin with the construction of these invariants and their fundamental properties, such as homotopy invariance, along with the analysis of classical results, including the Long Exact Sequence of a Pair, the Excision Theorem, the Mayer-Vietoris Sequence, and the cohomological dual.Next, we will explore the theory of the de Rham cohomology, defined via differential forms on smooth manifolds, whose equivalence with singular cohomology with real coefficients is guaranteed by the de Rham's Theorem - establishing a bridge between differential analysis and algebraic topology. Finally, we will study Poincaré Duality, which provides an isomorphism between homology and cohomology groups of complementary degrees on compact, oriented manifolds, highlighting the intrinsic symmetry of these topological structures. (AU)