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Framed functions arose in the work of K. Igusa defining cohomology invariants for smooth manifold bundles (Igusa-Klein torsion). In the late 80's, he proved a strong connectivity result about the "space of framed functions" using Morse theory and conjectured that this space was, in fact, contractible. Jacob Lurie recently verified this conjecture using the language of higher category theory in the course of his proof of the Cobordism Hypothesis.

(1) Is there any progress on a more geometric proof of this? The relevant work of Igusa is very geometric and Lurie himself points out that a more direct proof should be found. Perhaps more realistically: is there a geometric heuristic for Lurie's result?

(2) Does anyone know of applications of framed functions beyond defining Igusa-Klein torsion and proving the Cobordism Hypothesis? Pointers into the literature on this topic would be much appreciated.

EDIT: A framed function on $M$ is a pair $(f,\xi)$ of a smooth function $f$ on $M$ having at most $A2$ singularities in the interior of $M$ and no singular points on the boundary, and an orthonormal framing $\xi$ of the negative eigenspace of the Hessian of $f$ at each critical point.

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    $\begingroup$ Should we request a definition of "framed function"? $\endgroup$
    – Gerald Edgar
    Commented Sep 10, 2010 at 20:14
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    $\begingroup$ Definition of "framed function" added. Note that framed functions occur in TQFT to handle transitions between slicing functions and slicing. See Walker's "On Witten's 3-manifold invariants". $\endgroup$
    – Daniel Moskovich
    Commented Nov 10, 2010 at 21:17

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It is possible to prove the contractibility directly (and thereby bypass the obstruction theory arguments sketched at the end of section 3 of my paper). The statement itself is an example of an h-principle: namely, one can show fairly easily that the framed function space of ${\mathbb R}^{n}$ is contractible, so the general result hinges on knowing that the framed function spaces also satisfy some sort of "local to global" principle for arbitrary manifolds. (If I remember correctly, Igusa told me that he and Eliashberg sketched out a proof along these lines, but never had the motivation to write it up.)

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  • $\begingroup$ Can anyone comment on what this '"local to global" principle for arbitrary manifolds' says exactly? $\endgroup$
    – Romeo
    Commented Sep 15, 2010 at 20:13
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    $\begingroup$ Romeo - one aspect of the h-principle concerns the situation where some sheaf of spaces on a manifold is also a homotopy sheaf in the sense that global sections can be recovered (up to homotopy) as a homotopy limit of sections over a cover -- the kind of statement needed to go from local contractibility to global contractibility. You can look at the sections on (micro)flexible sheaves in Gromov's book for information. $\endgroup$
    – Dustin Clausen
    Commented Dec 24, 2010 at 18:55
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    $\begingroup$ I suppose that proof of Eliashberg's was written up here? arxiv.org/abs/1108.1000 $\endgroup$
    – j.c.
    Commented Dec 9, 2014 at 15:52
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Eliashberg once told me that the framed function theorem should be a consequence of his work on wrinkled maps.

Igusa and I gave a fairly direct proof that the space of framed functions on the circle is contractible. The idea is that the assignment function $f \mapsto \Sigma(f)$ (= its singular set, suitably interpreted as a point of a configuration space) is is quasifibration with contractible fibers when $f$ varies through framed functions on $S^1$. The proof in this case appears in:

Igusa, Kiyoshi; Klein, John The Borel regulator map on pictures. II. An example from Morse theory. K-Theory 7 (1993), no. 3, 225–267.

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