O teorema da fibração de Milnor para singularidades reais

Abstract

The main objective of this work is to present and prove the classic Milnor Fibration Theorem for Real Singularities. To provide a foundation for understanding this result, we first establish the study of differentiable manifolds, including the case of manifolds with boundary. Next, we review essential concepts of algebraic topology, such as contractible spaces, homology, cohomology, CW complexes, and homotopy groups. We present a review of algebraic sets and, as a fundamental tool for the main proof, we demonstrate the Curve Selection Lemma. We introduce the notions of fiber bundles and fibrations, also discussing Ehresmann’s Fibration Theorem for both manifolds without boundary and manifolds with boundary. Finally, we prove the Milnor Fibration Theorem for the real case and analyze the topology of the resulting fibers.