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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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Did you try the book by Eliashberg and Mishachev? You can also read the original papers. –  Misha Jan 6 '13 at 8:30
    
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. –  John Klein Jan 7 '13 at 13:48
    
You can also give a look on Vincent Borelli's page math.univ-lyon1.fr/~borrelli/Enseignement.html But every material is in French! –  Michele Triestino Mar 9 '13 at 12:19
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4 Answers

up vote 6 down vote accepted

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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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. –  John Klein Jan 7 '13 at 13:44
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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:

Rourke and Sanderson,

A master's thesis with another exposition

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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.

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