Free online courseAlgebraic Topology

Duration of the online course: 24 hours and 3 minutes

New course

Explore Algebraic Topology at Andrews University, covering cell complexes, homotopy, homology, and cohomology in a comprehensive online course.

Course Description

Explore the fascinating world of Algebraic Topology with this comprehensive course offered by Andrews University. Delve into key concepts and discover the foundation of Algebraic Topology, covering a range of intriguing subjects. Begin with the study of Cell Complexes and understand how they form the building blocks of topological spaces.

Progress to investigating Homotopy Equivalence and grasp how different shapes and structures can be transformed into each other without tearing or gluing. This leads into an understanding of the Fundamental Group, a crucial concept that examines loops within a space and their role in classifying topological spaces.

The course introduces the Brouwer Fixed Point Theorem, a compelling result that asserts the existence of fixed points under certain conditions. You will also explore the concept that homeomorphic spaces have isomorphic fundamental groups, enhancing your understanding of space classification.

Gain insights into the Seifert-Van Kampen Theorem and its application in calculating fundamental groups, followed by an examination of Covering Spaces and their distinct properties. Learn about Deck Transformations and how they relate to fundamental groups.

Expand your knowledge through Simplicial and Singular Homology, identifying how these sophisticated tools measure and classify spaces. The course delves into Exact Sequences in homological algebra, illustrating their significance in topological applications.

A special focus is placed on the relationship between Singular Homology and Simplicial Homology, as well as the application of Complex Homology in understanding geometric and algebraic structures. Explore the Mayer-Vietoris sequence to see how it assists in computing homology groups for complex shapes.

Conclude with an introduction to Category Theory, providing a broader context for understanding mathematical structures and relationships. The course wraps up with an in-depth look into Cohomology and the intriguing concept of the Cup Product, concluding with real-world applications on constructs like the Torus.

This Algebraic Topology course offers a profound exploration of topology, providing invaluable insights for students pursuing studies in algebra or related fields.

Conteúdo do Curso

  • Video class: Algebraic Topology 0: Cell Complexes

    1h08m

  • Video class: Algebraic Topology 1: Homotopy Equivalence

    1h08m

  • Video class: Algebraic Topology 2: Introduction to Fundamental Group

    1h05m

  • Video class: Algebraic Topology 3: Fundamental Group is a Group!

    1h01m

  • Video class: Algebraic Topology 4: Brouwer Fixed Point Theorem

    1h06m

  • Video class: Algebraic Topology 5: Homeomorphic Spaces have Isomorphic Fundamental Groups

    1h07m

  • Video class: Algebraic Topology 6: Seifert-Van Kampen Theorem

    1h16m

  • Video class: Algebraic Topology 7: Covering Spaces

    1h00m

  • Video class: Algebraic Topology 8: Properties of Covering Spaces

    1h08m

  • Video class: Algebraic Topology 9 : Deck Transformations of Covering Spaces

    1h12m

  • Video class: Algebraic Topology 10: Simplicial Homology

    1h26m

  • Video class: Algebraic Topology 11: What is homology measuring?

    1h00m

  • Video class: Algebraic Topology 12: Intro to Singular Homology

    0h55m

  • Video class: Algebraic Topology 13: Homotopy Equivalence Preserves Homology

    1h06m

  • Video class: Algebraic Topology 14: Exact Sequences

    0h54m

  • Video class: Algebraic Topology 15: Exact Sequence of Homology and Excision

    0h57m

  • Video class: Algebraic Topology 16: Singular Homology = Simplicial Homology

    0h42m

  • Video class: Algebraic Topology 17: Degree and Cellular Homology

    1h06m

  • Video class: Algebraic Topology 18: Mayer-Vietoris

    0h58m

  • Video class: Algebraic Topology 19: Category Theory

    1h00m

This free course includes:

24 hours and 3 minutes of online video course

Exercises to train your knowledge

Certificate of course completion

100% free, from content to certificate

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