Generalized blow-up of corners and fiber products
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- by Chris Kottke and Richard B. Melrose PDF
- Trans. Amer. Math. Soc. 367 (2015), 651-705
Abstract:
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 generalized boundary blow-up 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 basic monoidal complex 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 binomial variety. 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.
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Additional Information
- Chris Kottke
- Affiliation: Department of Mathematics, Brown University, Providence, Rhode Island 02912
- Address at time of publication: Department of Mathematics, Northeastern University, Boston, Massachusetts 02115
- MR Author ID: 771111
- Email: ckottke@math.brown.edu, c.kottke@neu.edu
- Richard B. Melrose
- Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
- Email: rbm@math.mit.edu
- Received by editor(s): April 10, 2012
- Received by editor(s) in revised form: May 15, 2013
- Published electronically: June 18, 2014
- Additional Notes: The second author was supported in part by NSF grant DMS-1005944.
- © Copyright 2014 Copyright retained by the authors
- Journal: Trans. Amer. Math. Soc. 367 (2015), 651-705
- MSC (2010): Primary 57R99; Secondary 14B05
- DOI: https://doi.org/10.1090/S0002-9947-2014-06222-3
- MathSciNet review: 3271273