# Thorough reference on regular homotopy

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?).

• 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. – Paul-Benjamin May 18 '16 at 0:18
• @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. – Mark Grant May 18 '16 at 6:14