2023/11/15 by Steven Hurder, Hurder, Steven
Mathematics · #57R20 #57R32 #57R50 #58D05 #Algebraic Geometry and Number Theory #Algebraic Topology (math.AT) #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2311.09160
openalex publication_date 2023/11/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this work we relate the known results about the homotopy type of classifying spaces for smooth foliations, with the homology and cohomology of the discrete group of diffeomorphisms of a smooth compact connected oriented manifold. The Mather-Thurston Theorem forms a bridge between the results on the homotopy types of classifying spaces and the various homology and cohomology groups that are studied. We introduce the algebraic K-theory of a manifold M that is derived from the discrete group of diffeomorphisms of M, and observe that the calculations of homotopy groups in this work are about these K-theory groups. We include a variety of remarks and open problems related to the study of the diffeomorphism groups and their homological invariants using the Mather-Thurston Theorem.