Lecture

Pointed Mapping Spaces: Exponential Law and Adjunction

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In course

MATH-497: Topology IV.b - homotopy theory

We propose an introduction to homotopy theory for topological spaces. We define higher homotopy groups and relate them to homology groups. We introduce (co)fibration sequences, loop spaces, and suspensions. We study long exact sequences. We construct Eilen ...

Description

This lecture focuses on pointed mapping spaces, where all spaces are pointed and locally compact. The exponential law for mapping spaces is applied to study pointed mapping spaces, which preserve the base point. The left adjoint to the pointed mapping space is the smashed product, providing a unique map from X times D to Y. The lecture explores the relationship between pointed mapping spaces and smashed products, identifying homomorphisms and adjunction properties. Specific cases, such as quotients and loop spaces, are discussed, highlighting the adjunction between the loop space and the reduced suspension. The lecture concludes by examining the exponential law for mapping spaces to loop spaces, emphasizing the importance of understanding homotopic classes in well-pointed spaces.

Instructor

Jérôme Scherer

Teaching & PhD PhD Students Virgile Simon Constantin Past EPFL PhD Students Samuel Lavenir Past EPFL PhD Students as codirector Kay Remo Werndli, Lyne Moser Courses Topology IV.a - Cohomology rings MATH-506 Singular cohomology is defined by dualizing the singular chain complex for spaces. We will study its basic properties, see how it acquires a multiplicative structure and becomes a graded commutative algebra. We study an algebraic version, namely group cohomology, and compare both approa

Official source

Ontological neighbourhood

Mathematics

Algebra: Abstract algebra

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