Concept

Reduced homology

Related lectures (32)

Bar Construction: Homology Groups and Classifying Space

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

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

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

Homotopy Invariance: Homology Groups

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

Cohomology: Cross Product

Explores cohomology and the cross product, demonstrating its application in group actions like conjugation.

Chain Maps: Homotopy Invariance

Covers chain maps, homotopy invariance, homology groups, and induced homomorphisms between cycles and boundaries.

Homology Groups: Basics

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

Relative Homology: Homotopy Invariance

Explains relative homology, n-cycles, n-boundaries, and exact sequences of chain complexes.

EML Spaces and Cohomology

Covers spaces, homology, chain groups, and abelianization in CW complexes and maps.

Induced Homomorphisms

Covers induced homomorphisms, homotopy invariance, and homology groups of quotients.

Homology and the fundamental group

Explores simplicial and singular homology, reduced homology groups, and their connection to the fundamental group.

Cohomology Groups: Hopf Formula

Explores the Hopf formula in cohomology groups, emphasizing the 4-term exact sequence and its implications.

Algebra: Chain Complexes

Explores chain complexes, abelian groups, homomorphisms, homology groups, and free abelian groups.

Zig Zag Lemma

Covers the Zig Zag Lemma and the long exact sequence of relative homology.

Homology of Projective Space

Covers the homology of projective space, focusing on cohomology and exact sequences.

A Lemma: Introduction to Homology Groups

Introduces the concept of homology groups and focuses on a lemma about free abelian groups.

Mayer-Vietoris Sequence

Explores the Mayer-Vietoris sequence, exact homomorphisms, embedded spheres, and path-connected spaces.

Homology Theorem

Covers the proof of Theorem A, discussing homology, quotients, and isomorphisms.

Topology Seminar: Tower Sequences and Homomorphisms

Explores tower sequences, homomorphisms, and their applications in topology, including the computation of homology and the construction of telescopes.

Homotopy Theory of Chain Complexes

Explores the homotopy theory of chain complexes, including path object construction and fibrations.