Concept

Homology sphere

Related lectures (32)

Simplicial and Singular Homology Equivalence

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

Homotopy Invariance: Homology Groups

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

Singular Homology: First Properties

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

Jordan Curve Theorem

Covers the proof of the Jordan Curve Theorem and the properties of embedded spheres.

Chain Maps: Homotopy Invariance

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

Geodesics on Surfaces

Explores geodesics on surfaces, focusing on minimizing distances and properties of paths, with examples like great circles on spheres.

The Jordan Curve Theorem

Explores the Jordan Curve Theorem, illustrating the division of a plane by a simple closed curve into two regions.

Closed Surfaces and Integrals

Explains closed surfaces like spheres, cubes, and cones without covers, and their traversal and removal of edges.

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.

Conformal Maps and Hyperbolic Geometry

Explores conformal maps, hyperbolic geometry, and isometric properties of disks.

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.

Cohomology Groups: Hopf Formula

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

Relative Homology: Homotopy Invariance

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

Riemann Sphere: Correlations and Conditions

Covers the Riemann sphere, focusing on its conditions and correlation functions in mathematical and physical contexts.

Homology groups: Quotients

Covers homology groups of quotients, homotopy invariance, and exact sequences.

Gaussian Curvature and Geodesics

Explores the derivative of curve lengths, fixed-end deformations, geodesics, surface point typologies, and sphere parametrization.

Sphere, Cone, and Paraboloid

Explores the intersection of a sphere, cone, and paraboloid in 3D space.

Zig Zag Lemma

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