# $n$-Fold Framed Functions

Suppose that $M$ is a manifold. One can consider a suitably constructed space of generalized framed Morse functions on $M$, let's call it $\mathrm{Fun}^\mathrm{fr}(M)$. This space is known to be contractible. This space also comes with an evaluation map $\mathrm{Fun}^\mathrm{fr}(M)\rightarrow C^\infty(M,\mathbb{R})$, which basically forgets the framing on generalized Morse functions.

I am wondering whether there is a generalization of this to case when we consider maps $M$ to $\mathbb{R}^n$, which we would call, say, $n$-fold framed functions. The way I would heuristically think about these functions is as follows. An $n$-fold framed function consists of a smooth map $f:M\rightarrow \mathbb{R}^n\simeq \mathbb{R}\times\mathbb{R}^{n-1}$ plus extra data. When we compose $f$ with the projection $p_1$ to $\mathbb{R}$, we get a generalized Morse function. Let us say that $x$ is a regular value for $p_1\circ f$, and let us denote by $M_x$ its preimage. Let $p_{n-1}:\mathbb{R}\times\mathbb{R}^{n-1}\rightarrow \mathbb{R}^{n-1}$ be the projection, and $i_x:M_x\rightarrow M$ be the inclusion. If for all such $x$, $$p_{n-1}\circ f\circ i_x:M_x\rightarrow \mathbb{R}^{n-1}$$ plus appropriate part of the extra data, is an $(n-1)$-fold framed function, then $f$ is an $n$-fold framed function.

The desire would be that one can construct a space of $n$-fold framed function, $\mathrm{Fun}^\mathrm{fr}_n(M)$, which is contractible and has an evaluation map $\mathrm{Fun}^\mathrm{fr}_n(M)\rightarrow C^\infty(M,\mathbb{R}^n)$. In the $n=1$ case we should recover framed functions.

Does anyone know whether such a notion exists?

Sorry, if the description is too vague.

• What is your reference for the contractibility of the framed function space? Oct 7 '14 at 12:39
• There is a paper by Eliashberg and Mishachev. They prove an extension theorem, which can be used to prove that $\mathrm{Fun}^\mathrm{fr}(M)$ is contractible. As a reminder, the space of framed functions is constructed as a geometric realization of a simplicial set. Oct 7 '14 at 13:56
• Thanks. I was aware that Eliashberg had proved a version of Igusa's theorem about the space of generalizaed Morse functions (without framing). I wasn't aware of the paper my Eliashberg-Mishachev. I read Lurie's paper when it first surfaced, but he had not stated that the framed function space was contractible at that time. Oct 7 '14 at 18:15