2017/09/03 by Daniel Grady, Hisham Sati, Grady, Daniel +1
Mathematics · #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1709.00686
openalex publication_date 2017/09/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We develop a theory of parametrized geometric cobordism by introducing smooth Thom stacks. This requires identifying and constructing a smooth representative of the Thom functor acting on vector bundles equipped with extra geometric data, leading to a geometric refinement of the the Pontrjagin-Thom construction in stacks. We demonstrate that the resulting theory generalizes the parametrized cobordism of Galatius-Madsen-Tillman-Weiss. The theory has the feature of being both versatile and general, allowing for the inclusion of families of various geometric data, such as metrics on manifolds and connections on vector bundles, as in recent work of Cohen-Galatius-Kitchloo and Ayala.