Lectures related to Simplicial Homology: Examples

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

Bar Construction: Homology Groups and Classifying Space

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

Topology: Classification of Surfaces and Fundamental Groups

Discusses the classification of surfaces and their fundamental groups using the Seifert-van Kampen theorem and polygonal presentations.

Group Cohomology

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

EML Spaces and Cohomology

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

Homology and Homotopy

Explores the comparison of long exact sequences for vibrations and the relationship between homotopy and homology groups.

Geometric realization: Simplicial sets

Covers the construction of a left adjoint to the singular set functor, comparing the homotopy theory of topological spaces with that of simplicial sets.

The Topological Künneth Theorem

Explores the topological Künneth Theorem, emphasizing commutativity and homotopy equivalence in chain complexes.

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.

Singular Homology: First Properties Log in to Mediaspace to watch this video

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

Simplicial and Singular Homology Equivalence Log in to Mediaspace to watch this video

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

Simplicial Homology: Structure and Complexes Log in to Mediaspace to watch this video

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

Homology Groups: Basics Log in to Mediaspace to watch this video

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

Simplicial Homology Revisited Log in to Mediaspace to watch this video

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

Induced Homomorphisms on Relative Homology Groups Log in to Mediaspace to watch this video

Covers induced homomorphisms on relative homology groups and their properties.

Homology and the fundamental group Log in to Mediaspace to watch this video

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

Homology Theorem Log in to Mediaspace to watch this video

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

Knot Theory: The Quadratic Linking Degree Log in to Mediaspace to watch this video

Covers the quadratic linking degree in knot theory, exploring its definitions, properties, and significance in algebraic geometry.

Homotopy Invariance: Homology Groups Log in to Mediaspace to watch this video

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

Transformations and Inversions: Laplace and Fourier Log in to Mediaspace to watch this video

Discusses Laplace and Fourier transformations, focusing on their inversion formulas and applications in solving differential equations.