Let $M$ be a smooth manifold. Morse functions on $M$ are smooth functions $M \to \mathbb{R}$ with only very nice singularities.
Fact: The space of Morse functions on $M$ is not, in general, connected.
The space of "generalized Morse functions" includes Morse functions and those with slightly worse singularities.
Theorem: The space of generalized Morse functions on $M$ is connected. (This is a fundamental result of Cerf theory.)
But it is not in general simply-connected. So there are loops in the space of Morse functions which are not contractible.
Finally, there is the space of "framed functions" on $M$. Let $f$ be a smooth function which has only reasonable singularities. A framing on $f$ is a smooth choice of frame for the negative eigenspace of the Hessian of $f$ at each of its singular points.
Theorem: (Igusa) The space of framed functions on an $n$-dimensional manifold is $(n-1)$-connected.
So for a surface, the space of framed functions is simply-connected. One ought to be able to find a loop of Morse functions on a closed surface which is not contractible in the space of generalized Morse functions but which is contractible in the space of framed functions (after adding framing data).
My question is: are concrete examples of this kind of thing available anywhere in the literature? Or are there standard examples known to experts?
(Thanks to Lurie, as well as Eliashberg and Mishachev, we know that the space of framed functions on a manifold is in fact contractible, but understanding $\pi_1$ seems like a good first goal.)
Word on the street is that the following is a good candidate, so I think I should include it. The manifold at hand is $S^2$. The diagrams below each show (schematically) a Morse function on $S^2$ with three $2$-handles, two $1$-handles, and a single $0$-handle. Each diagram has three circles (representing the $2$-handles) and two arcs (the cores of the $1$-handles). So the index two critical points lie above your screen and the index one and zero critical points lie below it.
The diagram shows a sequence of three handleslides between the 1-handle. The last function is of course isotopic through Morse functions to the first.
Rumor: This loop of Morse functions is not contractible in the space of generalized Morse functions. After endowing the one-handles with framings (i.e. orienting the arcs) this path is no longer a loop, but the path given by traversing it TWICE is a contractible loop.
