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Whitney map

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The Whitney map establishes a relation between continuous differential forms over region $D \subset {\bf R}^n$ which is a realization of a cubical complex $K$ (or cell complex) and discrete differential forms, i.e., cochains, over $K$: $$W:C^k(K) \rightarrow \Omega ^k (D),k=0,1,...$$ defined via "extrapolation". Specifically, one extrapolates cochains by extrapolating the generators.

In dimension $1$, $0$-cochains are defined on ${...,-2,-1,0,1,2,3...}$ and the generators are: $$g_i(n)=\delta _{in}.$$ The corresponding $0$-form is a "tooth":

  • $W(g_i)(x)=0$ for $x<n-1$,
  • $W(g_i)(x)=x-n+1$ for $n-1 \le x \le n$,
  • $W(g_i)(x)=-x+n+1$ for $n \le x \le n+1$,
  • $W(g_i)(x)=0$ for $x \le n+1$.

Similar for $1$-forms. The result is an $L_2$-form.

In the simplicial case, the Whitney map is defined as follows. Let $\lambda _i$ denote the $i$th barycentric coordinate corresponding to $i$th vertex $v_i$ in simplicial complex $K$. Suppose we have a $k$-simplex $$\sigma =[v_0,...,v_k]$$ in $K$ and the corresponding $k$-cochain $\sigma '$. Then we define $$W(\sigma ')=k!\sum _{i=0}^k (-1)^i \lambda _i d\lambda _0 \wedge ... \wedge <d\lambda _i> \wedge ... \wedge d\lambda _k.$$ It is then extended by linearity.

The inverse link is given by the de Rham map: $$R:\Omega ^k (D) \rightarrow C^k(K),k=0,1,...$$ defined via integration.