Pointed Mapping Spaces: Exponential Law and Adjunction

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.