Course

MATH-497: Topology IV.b - homotopy theory

Lectures in this course (27)

Introduction to Topology

Covers fundamental concepts of space, topology, groups, and homology theory.

CW Approximation Theorem

Explores the CW Approximation Theorem, constructing CW complexes from spaces to ensure isomorphism on homology groups.

Cellular Approximation: Homotopy and CW Complexes

Explores the cellular approximation theorem for CW complexes and its implications on homotopy groups.

Compact-open topology

Covers the compact-open topology, defining maps between spaces and discussing continuous maps and preimages in topology.

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.

Cofibrations: Extension Property and Recognition Principle

Explores the extension property of spaces and the recognition principle for cofibrations.

Exponential Law: Mapping Spaces and Compactness

Explores the exponential law for mapping spaces and the role of compactness in topological theory.

Pointed Mapping Spaces: Exponential Law and Adjunction

Explores pointed mapping spaces, the exponential law, adjunction properties, and homotopic classes.

EML Spaces and Cohomology

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

Homotopy Classes and Group Structures

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

Higher Homotopy Groups: Generalization and Structure

Explores the generalization and structure of higher homotopy groups, including their abelianness, historical context, and properties of H spaces.

Understanding Vibrations

Explores vibrations, their properties, and how any map can be transformed into a vibration through factorization.

Relative Homotopy Groups

Covers relative homotopy groups, establishing long exact sequences and defining boundary homomorphisms.

Fubini's Rule for Pushouts

Explores the Fubini Rule for pushouts, emphasizing the importance of transforming diagrams for maintaining pushout properties.

Long Exact Sequence in Homotopy

Explores the long exact sequence in homotopy, emphasizing the importance of sets and groups in the sequence.

Cellular Approximation Theorem

Focuses on proving the Cellular Approximation Theorem, introducing polyhedra and piecewise linear maps.

Homology and Homotopy

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

Homotopy Invariance: Push Out Product

Explores the push out product concept and its application in vibrations, setting the stage for understanding homotopy invariants.

Homotopy Pushouts: Standard Models and Equivalence Properties

Explores homotopy pushouts, standard models, equivalence properties, and the importance of strict commutativity in pushout constructions.

Hurewicz Theorem

Explores the proof of the Hurewicz Theorem and its applications to spheres and homotopy groups.