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Explore Algebraic Topology at Andrews University, covering cell complexes, homotopy, homology, and cohomology in a comprehensive online course.
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.
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
24 hours and 3 minutes of online video course
Exercises to train your knowledge
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