I'd like to learn the basics of Hirsch-Smale immersion theory. What sources are best for this? My background is mostly topological; however, many of the sources I've found on the internet focused on later work of Gromov on the h-principle which seems more analytic than I would like.
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1$\begingroup$ Did you try the book by Eliashberg and Mishachev? You can also read the original papers. $\endgroup$– MishaCommented Jan 6, 2013 at 8:30
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$\begingroup$ In the PL category one can consult Haefliger and Poenaru: "La classification des immersions combinatoires." In the TOP category, Kirby and Siebenmann is a standard reference. $\endgroup$– John KleinCommented Jan 7, 2013 at 13:48
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$\begingroup$ You can also give a look on Vincent Borelli's page math.univ-lyon1.fr/~borrelli/Enseignement.html But every material is in French! $\endgroup$– user47274Commented Mar 9, 2013 at 12:19
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4 Answers
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M. Weiss has a very good survey on his homepage: http://wwwmath.uni-muenster.de/u/mweis_02/papers.html called "Immersion theory for homotopy theorists". I also like very much M. Adachi's book "Embeddings and immersions". I think both are very good starters if you have a topological background.
J. Francis has also some notes on his homepage: http://www.math.northwestern.edu/~jnkf/classes/hprin/ on a course about the h-principle that can be helpful.
Also Eliashberg and Mischachev's book "introduction to the h-principle" is definitively a very good book.
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2$\begingroup$ Remark: I just checked John Francis notes. The crux of the matter, lecture 8, is not available, although most of the notes date from early 2011. Lecture 8 is supposed to have the key step in the proof of Smale-Hirsch theory. $\endgroup$ Commented Jan 7, 2013 at 13:44
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$\begingroup$ Weiss' survey has been expanded and published: gradmath.org/article/immersion-theory-for-homotopy-theorists $\endgroup$– skupersCommented Aug 30, 2020 at 2:15
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You may like the following lecture notes:
Weiss, M. Immersion theory for homotopy theorists
Francis, J. The h-principle in topology
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Immersion theory has been "explained" by the Compression Theorem, with new proofs arguably being much more elementary and intuitive:
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I think this is the most readable source: Haefliger, A. Lectures on the theorem of Gromov. Proceedings of Liverpool Singularities Symposium, II (1969/1970), pp. 128–141. Lecture Notes in Math., Vol. 209, Springer, Berlin, 1971.