Math 655: Differential Topology

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Catalog Information

Title

Algebraic Topology 1.

Credit Hours

3

Prerequisite

Math 313, 342, and Math 553 and 554 or Instructor's consent.

Description

An introduction to manifolds and smooth manifolds.

Desired Learning Outcomes

Prerequisites

Knowledge of basic point topology from Math 553, 554 will be assumed. This includes topological spaces, basis and countability, metric spaces, quotient spaces, fundamental group, and covering maps. We also assume a course in linear algebra (Math 313) and one in introductory analysis (Math 341 and 342).

Minimal learning outcomes

  1. Manifolds
    • Topological and smooth manifolds
    • Manifolds with boundary
    • Tangent vectors
    • Tangent bundles
    • Vector bundles and bundle maps
    • Cotangent bundles
  2. Submanifolds
    • Submersions, immersions, embeddings
    • Inverse and implicit function theorems
    • Transversality
    • Embedding and approximation theorems
  3. Differential forms and tensors
    • Wedge product
    • Exterior derivative
    • Orientations
    • Stoke's Theorem
    • DeRham cohomology

Additional topics

Courses for which this course is prerequisite