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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

   

 

Generalized blow-up of corners and fiber products


Authors: Chris Kottke and Richard B. Melrose
Journal: Trans. Amer. Math. Soc. 367 (2015), 651-705
MSC (2010): Primary 57R99; Secondary 14B05
Published electronically: June 18, 2014
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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
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

DOI: http://dx.doi.org/10.1090/S0002-9947-2014-06222-3
PII: S 0002-9947(2014)06222-3
Keywords: Manifold with corners, blow-up, generalized blow-up, fiber product, monoidal complex
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.
Article copyright: © Copyright 2014 Copyright retained by the authors