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Last updated:

November 7, 2022

Duration:

Unlimited Duration

FREE

This course includes:

Unlimited Duration

Badge on Completion

Certificate of completion

Unlimited Duration

Description

This is a course on the singular homology of topological spaces.

Topics include: Singular homology, CW complexes, Homological algebra, Cohomology, and Poincare duality.

Course Curriculum

    • Introduction: Singular Simplices and Chains Unlimited
    • Homology Unlimited
    • Categories, Functors, Natural Transformations Unlimited
    • Categorical Language Unlimited
    • Homotopy, Star-shaped Regions Unlimited
    • Homotopy Invariance of Homology Unlimited
    • Homology Cross Product Unlimited
    • Relative Homology Unlimited
    • The Homology Long Exact Sequence Unlimited
    • Excision and Applications Unlimited
    • The Eilenberg Steenrod Axioms and the Locality Principle Unlimited
    • Subdivision Unlimited
    • Proof of the Locality Principle Unlimited
    • CW-Complexes Unlimited
    • CW-Complexes II Unlimited
    • Homology of CW-Complexes Unlimited
    • Real Projective Space Unlimited
    • Euler Characteristic and Homology Approximation Unlimited
    • Coefficients Unlimited
    • Tensor Product Unlimited
    • Tensor and Tor Unlimited
    • The Fundamental Theorem of Homological Algebra Unlimited
    • Hom and Lim Unlimited
    • Universal Coefficient Theorem Unlimited
    • Künneth and Eilenberg-Zilber Unlimited
    • Coproducts, Cohomology Unlimited
    • Ext and UCT Unlimited
    • Products in Cohomology Unlimited
    • Cup Product (cont.) Unlimited
    • Surfaces and Nondegenerate Symmetric Bilinear Forms Unlimited
    • Local Coefficients and Orientations Unlimited
    • Proof of the Orientation Theorem Unlimited
    • A Plethora of Products Unlimited
    • Cap Product and “Cech” Cohomology Unlimited
    • Cech Cohomology as a Cohomology Theory Unlimited
    • The Fully Relative Cap Product Unlimited
    • Poincaré Duality Unlimited
    • Applications Unlimited

About the instructor

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Massachusetts Institute of Technology