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I completely understand why a regular homotopy is required to pass through only immersions of the first manifold into the second, but I am not confident I understand why the second condition--it must extend continuously to a homotopy of the tangent bundles--is imposed. Are there any good examples to help me understand the presence of this second condition? What undesirable results would follow were this condition dropped?

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Your question is a little unclear. What definitions are you using? To me the terminology "regular homotopy equivalence" isn't a standard terminology, nor is the context of your question clear. – Ryan Budney Apr 11 '11 at 6:16
My apologies; I didn't realize this wasn't standard terminology. My use of "equivalence" may also have been incorrect. My question was prompted while I was reading Smale's paper that contains the proof that sphere eversion is Smale provides this definition in the first paragraph. It's also the same definition that appears on Wikipedia's page for "regular homotopy": – Zach Conn Apr 11 '11 at 7:43
"Regular homotopy" in that context means 1-parameter family of immersions. It isn't clear to me what this "second condition" is that you're talking about. I don't recall anything like it from Smale's work, and it appears to only be on the Wikipedia page you talk about. – Ryan Budney Apr 11 '11 at 15:07
I've updated the Wikipedia page to (I hope!) be a little less ambiguous. – Ryan Budney Apr 11 '11 at 15:15
Smale's paper says "A homotopy of an immersion is called regular if at each stage it is regular and if the induced homotopy of the tangent bundle is continuous." The second condition that I'm talking about is "if the induced homotopy of the tangent bundle is continuous." I think I am misunderstanding something but I am not sure what. Am I wrong in thinking that Smale's first condition that "at each state it is regular" ensures that it is a 1-parameter family of immersions and the second condition is an extra requirement, or am I misunderstanding some definitions here? – Zach Conn Apr 11 '11 at 19:19

1 Answer 1

Regular homotopy needs a condition that ensures that the derivatives of the curve are continuous. Some authors formulate this by saying the homotopy is smooth but there seem to be several other ways to do it, one of which is the formulation of Smale.

Without this condition the Whitney degree of a circle immersed in the plane is not conserved. See the discussion in: A contact geometric proof of the Whitney-Graustein theorem by H Geiges.

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