Timeline for Homotopy type of spaces of functions with few critical points

5 events

when toggle format	what		by	license	comment
Jul 24, 2017 at 18:33	comment	added	Ryan Budney		Didn't Vassiliev study a question similar to this? Perhaps it was Morse functions with restricted numbers and types of critical points.
Jul 24, 2017 at 18:31	comment	added	Ryan Budney		Well, I should have been a little more detailed. In the k=0 case (which I did not think you were interested in) the Alexander argument gives you a deformation-retract to the linear subspace. That's a once-punctured dual space , homotopy-equivalent to a sphere as you say. When $k>0$ this Alexander argument gives a deformation-retract to the subspace with at most one critical point at the origin. Perhaps that has the homotopy type of $\Omega^n S^{n-1}$, or something like that.
Jul 24, 2017 at 11:21	comment	added	John Pardon		@RyanBudney: False. For example, $G_0(\mathbb R^n)$ is homotopy equivalent to $S^{n-1}$ (consider the "slope at the origin").
Jul 11, 2017 at 15:40	comment	added	Ryan Budney		One other small observation: $G_k(\mathbb R^n)$ is contractible, using an Alexander trick / straight line homotopy argument.
Jul 11, 2017 at 14:41	history	asked	John Pardon	CC BY-SA 3.0