Generalized blow-up of corners and fiber products
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 mani...
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| Format: | Article |
| Language: | en_US |
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2016
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| Online Access: | http://hdl.handle.net/1721.1/104634 https://orcid.org/0000-0002-1494-8228 |
| _version_ | 1826213006378270720 |
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| author | Kottke, Chris Melrose, Richard B |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Kottke, Chris Melrose, Richard B |
| author_sort | Kottke, Chris |
| collection | MIT |
| description | 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. |
| first_indexed | 2024-09-23T15:41:37Z |
| format | Article |
| id | mit-1721.1/104634 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T15:41:37Z |
| publishDate | 2016 |
| record_format | dspace |
| spelling | mit-1721.1/1046342022-10-04T05:32:06Z Generalized blow-up of corners and fiber products Kottke, Chris Melrose, Richard B Massachusetts Institute of Technology. Department of Mathematics Melrose, Richard B 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. 2016-09-30T22:02:43Z 2016-09-30T22:02:43Z 2014-06 Article http://purl.org/eprint/type/JournalArticle 0002-9947 1088-6850 http://hdl.handle.net/1721.1/104634 Kottke, Chris, and Richard B. Melrose. “Generalized Blow-up of Corners and Fiber Products.” Transactions of the American Mathematical Society 367.1 (2014): 651–705. https://orcid.org/0000-0002-1494-8228 en_US http://dx.doi.org/10.1090/S0002-9947-2014-06222-3 Transactions of the American Mathematical Society Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. application/pdf American Mathematical Society |
| spellingShingle | Kottke, Chris Melrose, Richard B Generalized blow-up of corners and fiber products |
| title | Generalized blow-up of corners and fiber products |
| title_full | Generalized blow-up of corners and fiber products |
| title_fullStr | Generalized blow-up of corners and fiber products |
| title_full_unstemmed | Generalized blow-up of corners and fiber products |
| title_short | Generalized blow-up of corners and fiber products |
| title_sort | generalized blow up of corners and fiber products |
| url | http://hdl.handle.net/1721.1/104634 https://orcid.org/0000-0002-1494-8228 |
| work_keys_str_mv | AT kottkechris generalizedblowupofcornersandfiberproducts AT melroserichardb generalizedblowupofcornersandfiberproducts |