Homology and the fundamental group

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Related lectures (32)

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Covers the first properties of singular homology and the preservation of decomposition and path-connected components in topological spaces.

Bar Construction: Homology Groups and Classifying Space

Covers the bar construction method, homology groups, classifying space, and the Hopf formula.

Homology of Riemann Surfaces

Explores the homology of Riemann surfaces, including singular homology and the standard n-simplex.

Simplicial and Singular Homology Equivalence

Demonstrates the equivalence between simplicial and singular homology, proving isomorphisms for finite s-complexes and discussing long exact sequences.

Homology Groups: Basics

Introduces reduced homology groups and explains their properties and applications in topology.

Group Cohomology

Covers the concept of group cohomology, focusing on chain complexes, cochain complexes, cup products, and group rings.

Simplicial Homology: Structure and Complexes

Covers the structure of topological spaces with A-complexes and chain complexes.

Simplicial Homology Revisited

Covers the concept of simplicial homology, focusing on finite complexes and induced maps.

Homology: Introduction and Applications

Introduces homology as a tool to distinguish spaces in all dimensions and provides insights into its construction and applications.

Homotopy Invariance: Homology Groups

Explores homotopy invariance and its application to homology groups of quotients, showcasing isomorphism and chain homotopy.