2011/07/31 by Chris Kottke, Richard Melrose, Richard B. Melrose · 2 citations
Mathematics · #Advanced Combinatorial Mathematics #Boundary (topology) #Context (archaeology) #Geometric and Algebraic Topology #Geometry #Gravitational singularity #Homogeneity (statistics) #Homotopy and Cohomology in Algebraic Topology #Manifold (fluid mechanics) #Mathematical analysis #Mathematics #Product (mathematics) #Pure mathematics #Transversality #Variety (cybernetics) #math.DG #math.GT #msc:14B05 #msc:57R99
paper · pdf · doi:10.1090/s0002-9947-2014-06222-3
published in Transactions of the American Mathematical Society 367(1), 651-705 (American Mathematical Society) · 53 pages, to appear in Transactions of the AMS. Includes revisions suggested by the referee
arxiv created 2013/06/24 · openalex publication_date 2014/06/18 · arxiv updated 2014/11/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Real blow-up, including inhomogeneous versions, of boundary faces of a manifold (with corners) is an important tool for resolving singularities, degeneracies and competing notions of homogeneity. These constructions are shown to be particular cases of <italic>generalized boundary blow-up</italic> in which a new manifold and blow-down map are constructed from, and conversely determine, combinatorial data at the boundary faces in the form of a refinement of the <italic>basic monoidal complex</italic> of the manifold. This data specifies which notion of homogeneity is realized at each of the boundary hypersurfaces in the blown-up space. As an application of this theory, the existence of fiber products is examined for the natural smooth maps in this context, the b-maps. Transversality of the b-differentials is shown to ensure that the set-theoretic fiber product of two maps is a <italic>binomial variety</italic> . Properties of these (extrinsically defined) spaces, which generalize manifolds but have mild singularities at the boundary, are investigated, and a condition on the basic monoidal complex is found under which the variety has a smooth structure. Applied to b-maps this additional condition with transversality leads to a universal fiber product in the context of manifolds with corners. Under the transversality condition alone the fiber product is resolvable to a smooth manifold by generalized blow-up and then has a weaker form of the universal mapping property requiring blow-up of the domain.