Talk:Embeddings in Euclidean space: an introduction to their classification

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This page was created in 2010, improved since then, and submitted to Bulletin of the Manifold Atlas on 25 November 2016.

(April 2019) In my opinion, motivations are important for the reader. `Create a headache before offering aspirin' (Dan Meyer). In Remark 1.1 I humbly attempt to give some motivations for the Knotting Problem.

(April 2019; mostly superfluous after update of the page) The Browder-Novikov theorem of 1960s classifies higher-dimensional manifolds. (The statement in terms of Poincar\'e complexes is given e.g. in [Wall1999].) Analogously, the Browder-Wall theorem of 1960s, as well as the Goodwille-Weiss calculus of embeddings of 1990s, classifies embeddings of manifolds in codimension greater than 2. (The statement of the Browder-Wall theorem in terms of Poincar\'e complexes is given e.g. in M. Cencelj, D. Repov\v s and A. Skopenkov, On the Browder-Levine-Novikov embedding theorems, Proc. of the Steklov Math. Inst. 247 (2004) 280--290 and presumably in [Wall1999].) So a reader may wonder why mathematicians (including M. Kreck, his collaborators and students) are bothered by further research on classification of manifolds and their embeddings, and by its exposition in Manifold Atlas. This is explained in Remark 1.2 (Readily calculable classification).

(May 6, 2019) Without (c) and (ii) the notion of `readily calculable' is just empty. Because every classification would have to be readily calculable classification, because every classification implies General Position Theorem 2.1. Imagine I prove that embeddings of 4-manifolds into R^m are in 1-1 correspondence with pentuples with such and such properties up to such and such equivalence relation, explicit statement requiring several pages of formulation, the proof requiring several dozen pages, the only explicit corollary being General Position Theorem 2.1 (for 4-manifolds). Would you like to call my classification a readily calculable one?

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