Concept

Homotopy groups of spheres

Related lectures (32)

Left Homotopy as an Equivalence Relation: The Homotopy Relation in a Model Category

Explores the left homotopy relation as an equivalence relation in model categories.

Explores the construction of cylinder objects in chain complexes over a field, focusing on left homotopy and interval chain complexes.

Model Categories: Properties and Structures

Covers the properties and structures of model categories, focusing on factorizations, model structures, and homotopy of continuous maps.

Homotopy Theory of Chain Complexes

Explores the homotopy theory of chain complexes over a field, focusing on closure properties and decomposition.

Chain Homotopy and Projective Complexes

Explores chain homotopy, projective complexes, and homotopy equivalences in chain complexes.

Homotopy Theory: Cylinders and Path Objects

Covers cylinders, path objects, and homotopy in model categories.

Homotopy Equivalence in Chain Complexes

Explores homotopy equivalence in chain complexes, emphasizing path object construction and left/right homotopy characterization.

Attachment of a 2-cell

Covers the attachment of a 2-cell to a space and explores the concept of the attachment application f.

Algebraic Kunneth Theorem

Covers the Algebraic Kunneth Theorem, explaining chain complexes and cohomology computations.

Fundamental Groups

Explores fundamental groups, homotopy classes, and coverings in connected manifolds.

Homotopies & Equivalence Relations

Covers homotopies, fundamental group, and equivalence relations in applications.

Homotopy Classes and Group Structures

Explores wedges in pointed spaces, group structures, and the Echman-Hilton argument.

Universal Covering Construction

Introduces the concept of a universal covering construction with examples like Hawaiian rings.

Base B for the covering

Explores constructing a base B for a topology using homotopy classes and paths.

Quasi-Categories: Active Learning Session

Covers fibrant objects, lift of horns, and the adjunction between quasi-categories and Kan complexes, as well as the generalization of categories and Kan complexes.

Serre model structure on Top

Explores the Serre model structure on Top, focusing on right and left homotopy.

The Topological Künneth Theorem

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

Cohomology Representations: Lecture 14.1

Covers the concept of cohomology representations and the implications of reduced suspension operations on spaces.

Seifert van Kampen: strategy

Covers the Seifert van Kampen theorem, homotopy, and the repair of premie rectangles.

Cell Attachment and Homotopy

Explores cell attachment, homotopy, function existence, construction using universal property, and continuity.