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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeJul 23rd 2010
   • (edited Jul 23rd 2010)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI am tinking that probably Andrew Stacey or somebody can give me an immediate answer to the following question, without me having to think about it: take $X$ a _convenient vector space_ , let $\Delta^n$ be the standard $n$-simplex regarded as a smooth manifold and consider the _mapping vector space_ $X^{\Delta^n}$. I want to consider the collection $C^\infty(X^{\Delta^n}, \mathbb{R})$ of all smooth functions on $X^{\Delta^n}$, and the collection $C^\infty(L_n X^{\Delta \bullet})$ of all "functions on the subspace of degenerate simplices" and ask if the restriction map $$ C^\infty(X^{\Delta^n}) \to C^\infty(L_n X^{\Delta^\bullet}) $$ is surjective. Here that collection of functions on degenerate simplices is the evident limit over function sets $C^\infty(X^{\Delta^k})$ for $k \lt n$. In other words: given an function on the set of all degenerate smooth $n$-simplices $\Delta^n \to X$ that is smooth when restricted to a function on $k$-simplices for all $k \lt n$, does it always have an extension to a smooth function on the space of all $n$-simplices?

   I am tinking that probably Andrew Stacey or somebody can give me an immediate answer to the following question, without me having to think about it:

   take $X$ a convenient vector space , let $\Delta^n$ be the standard $n$-simplex regarded as a smooth manifold and consider the mapping vector space $X^{\Delta^n}$.

   I want to consider the collection $C^\infty(X^{\Delta^n}, \mathbb{R})$ of all smooth functions on $X^{\Delta^n}$, and the collection $C^\infty(L_n X^{\Delta \bullet})$ of all “functions on the subspace of degenerate simplices” and ask if the restriction map

   $$C^\infty(X^{\Delta^n}) \to C^\infty(L_n X^{\Delta^\bullet})$$

   is surjective.

   Here that collection of functions on degenerate simplices is the evident limit over function sets $C^\infty(X^{\Delta^k})$ for $k \lt n$.

   In other words: given an function on the set of all degenerate smooth $n$-simplices $\Delta^n \to X$ that is smooth when restricted to a function on $k$-simplices for all $k \lt n$, does it always have an extension to a smooth function on the space of all $n$-simplices?

2. • CommentRowNumber2.
   • CommentAuthorAndrew Stacey
   • CommentTimeJul 23rd 2010
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   Author: Andrew Stacey
   Format: MarkdownItexMy guess would be that it does extend, and that an extension is given by something like $$ f(x_0, \dots, x_n) = (-1)^n \sum (-1)^{|I|} f_I(x_I) $$ The signs may be wrong, of course. The simplest case is considering the positive quadrant in the $xy$-plane and two functions $f_x, f_y \colon [0,\infty) \to \mathbb{R}$ with $f_x(0) = f_y(0)$. Then define $f \colon [0,\infty)\times [0,\infty) \to \mathbb{R}$ by $f(x,y) = f_x(x) + f_y(y) - f_{0}(0)$, where $f_{0} \colon \{0\} \to \mathbb{R}$ is $f_0(0) = f_x(0)$.

   My guess would be that it does extend, and that an extension is given by something like

   $$f(x_0, \dots, x_n) = (-1)^n \sum (-1)^{|I|} f_I(x_I)$$

   The signs may be wrong, of course.

   The simplest case is considering the positive quadrant in the $xy$-plane and two functions $f_x, f_y \colon [0,\infty) \to \mathbb{R}$ with $f_x(0) = f_y(0)$. Then define $f \colon [0,\infty)\times [0,\infty) \to \mathbb{R}$ by $f(x,y) = f_x(x) + f_y(y) - f_{0}(0)$, where $f_{0} \colon \{0\} \to \mathbb{R}$ is $f_0(0) = f_x(0)$.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeJul 23rd 2010
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   Author: Urs
   Format: MarkdownItexThanks. Let's see, you are thinking of maps that embed the simplex with straight edges and faces, right? so that only the images of the vertices need to be specified? I was thinkig of all smooth maps $\Delta^n \to X$.

   Thanks. Let’s see, you are thinking of maps that embed the simplex with straight edges and faces, right? so that only the images of the vertices need to be specified? I was thinkig of all smooth maps $\Delta^n \to X$.

4. • CommentRowNumber4.
   • CommentAuthorAndrew Stacey
   • CommentTimeJul 23rd 2010
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   Author: Andrew Stacey
   Format: MarkdownItexShould have made it clearer: the $I$ in the summation is a multi-index. So you sum over all proper subsets of $\{0, \dots, n\}$. The idea being an extension of the example given with two variables.

   Should have made it clearer: the $I$ in the summation is a multi-index. So you sum over all proper subsets of $\{0, \dots, n\}$. The idea being an extension of the example given with two variables.