I would like to learn this topic of algebraic topology but I cannot find a relevant reference to answer my basic questions on the subject (for example, is there a Hurewicz theorem for regular homotopy groups? under which conditions can one homotop a non-zero class in a non-trivial higher homotopy group to an immersion?).


The Smale-Hirsch theorem says that the existence of an immersion in a given homotopy class is equivalent to the existence of certain bundle map, in turn equivalent to the existence of a section of a certain bundle. See Hirsch, Morris W. Immersions of manifolds; the Mathscinet review (MR0119214)by Kervaire is a good summary. Thus these questions are `reduced' to algebraic topology, in particular to obstruction theory. This is of course a great thing, but I doubt there's an answer to your question in any great generality, because obstructions are generally hard to compute.

The best hope for a general answer would be in the so-called stable range where the dimension of the source is less than 2/3 the dimension of the target; see Haefliger, André; Hirsch, Morris W. Immersions in the stable range. Ann. of Math. (2) 75 1962 231–241.

  • $\begingroup$ Thank you very much Prof. Ruberman, I will wait for some time before I accept your answer as I still hope at least a partial answer for my two basic questions. $\endgroup$ May 17 '16 at 17:02
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    $\begingroup$ @Paul-Benjamin Is there a specific example, or class of examples, that you are interested in? $\endgroup$ May 17 '16 at 18:21
  • $\begingroup$ Yes, for example, I would like to know if for a compact manifold $M$ (of dimension higher that two), there were some immersion of a sphere not homotopic to a constant. Or more optimistically, if $\pi_k(M)\neq 0$, and $f:S^k\rightarrow M$ represents a non-trivial class, can we homotop $f$ to an immersion? I posted a general question because I could not find a good reference book on the subject and thought this would be an occasion to learn it. $\endgroup$ May 18 '16 at 0:18
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    $\begingroup$ @Paul-Benjamin: If $\dim(M)\ge 2k$ then any map $f:S^k\to M$ can be homotoped to an immersion. This is because in these dimensions immersions are generic (the space of immersions is open and dense in the space of all maps). You can find these results in Hirsch's "Differential Topology" textbook. $\endgroup$
    – Mark Grant
    May 18 '16 at 6:14

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