Singularities of maps, characteristic classes, and intersection homology
Introduction to algebraic topology for the study of singular varieties
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Introduction to Intersection Homology
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| Author(s): |
Hana Marinho Lucena
Total Authors: 1
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| Document type: | Master's Dissertation |
| Press: | São Carlos. |
| Institution: | Universidade de São Paulo (USP). Instituto de Ciências Matemáticas e de Computação (ICMC/SB) |
| Defense date: | 2024-08-08 |
| Examining board members: |
Nivaldo de Góes Grulha Junior;
Thaís Maria Dalbelo;
Denise de Mattos;
Thiago de Melo
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| Advisor: | Nivaldo de Góes Grulha Junior |
| Abstract | |
The Homology and Cohomology Theories constitute fundamental concepts in Algebraic Topology, with the purpose of distinguishing and establishing relationships between topological spaces. Its applications, both inside and outside Mathematics, are vast. One of the most notable results that connects these theories is the famous Poincaré Duality. This duality allows isomorphisms to be established between homology and cohomology groups, providing a deep understanding of the topology of spaces. However, it is important to highlight that the validity of Poincaré Duality, in general, is conditioned by the absence of singularities in the topological structure of the studied space. When the space in question presents singularities, Poincaré Duality does not apply universally. In 1974, Mark Goresky and Robert MacPherson developed a Homology and Cohomology Theory specific to dealing with singular cases, known as Intersection Homology. This approach allows capturing relevant information about the contributions of singularities, expanding the scope of application of these theories. In this work, we explore the theory of homology and cohomology, present Poincaré Duality, and finally discuss the definitions, results, and basic examples of intersection homology. For instance, we introduce an adaptation of Poincaré Duality in the singular context. Thus, we provide a brief introduction to this theory, including examples and its relationship with classical homology. (AU) | |
| FAPESP's process: | 22/10020-8 - Introduction to Intersection Homology |
| Grantee: | Hana Marinho Lucena |
| Support Opportunities: | Scholarships in Brazil - Master |