It should be noted that potentially one needn't be so abstract about this. One can also proceed recursively on successive graph approximations to the fractal--supposing such graphs exist and are suitably well-behaved--by considering more or less discrete de Rham cohomology on the approximating graphs. This is not unlike the simplicial cohomology of a simplicial complex, and perhaps there is a limiting procedure here not unlike that which prevails in Cech cohomology and which could produce the final de Rham cohomology of the fractal itself. Certainly it is possible to define a notion of $k$-form on the graph level and to, for instance, derive algorithms for the recursive construction of, say, harmonic $1$-forms on graphs, where "harmonic" is defined in reference to the Laplacian $d\delta + \delta d$, where $\delta$ is the codifferential, thereby mimicking classical Hodge theory. Moreover, the full classical Hodge decomposition would hold here, whereas only a partial Hodge decomposition is obtained in the paper above. Of course, one must find a way to transition to the limit.
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Here's an attempt at an answer to your more general question about differential forms on a fractal. Suppose $K$ is a suitable fractal on which there is given a Dirichlet form, so we can single out an algebra $F$ of real-valued functions on $K$ of finite energy. There is a space of "universal" 1-forms $\Omega^1 F$ characterized by the following universal mapping property: there exists a derivation $d : F \rightarrow \Omega^1 F$ such that given any derivation $D : F \rightarrow E$ into an $F$-bimodule $E$ there exists a unique bimodule morphism $\phi_D : \Omega^1 F \rightarrow E$ so that $D = \phi_D\circ d$. It suffices to take the map $d : F \rightarrow F\otimes F$ defined by $df := 1\otimes f - f\otimes 1$. This is indeed a derivation since $d(fg) = 1\otimes (fg) - (fg)\otimes 1 = f\otimes g - (fg)\otimes 1 + 1\otimes(fg) - f\otimes g = f(dg) + (df)g.$ With this choice, we can take $\Omega^1 F$ to be the sub-$F$-bimodule of $F\otimes F$ generated by all elements of the form $f\text{ }dg$. More explicitly, $\Omega^1 F$ is the kernel of the multiplication map $F\otimes F \rightarrow F$ defined by $f\otimes g \mapsto fg$. It can be verified that this is the solution to the universal mapping problem posed above. Define $\overline{F} := F/\mathbb{R}$ and write $\overline{f}$ for the image of $f \in F$ in $\overline{F}$. Then $\Omega^1 F = F\otimes \overline{F}$ by the identification $f\otimes \overline{g} \mapsto f\text{ }dg$. To build the space of universal 2-forms, we define $\Omega^2 F := \Omega^1 F \otimes_F \Omega^1 F = (F\otimes \overline{F}) \otimes_F (F\otimes \overline{F}) = F \otimes \overline{F} \otimes \overline{F}$. More generally, $\Omega^n F := F\otimes \overline{F}^{\otimes n}$. The differential $d : F\otimes \overline{F}^{\otimes n} \rightarrow F\otimes \overline{F}^{\otimes (n+1)}$ is the shift $d(f_0 \otimes \overline{f}_1\otimes\dots\otimes\overline{f}_n) := 1\otimes \overline{f}_0 \otimes \overline{f}_1 \otimes \dots \otimes \overline{f}_n.$ And $d^2 = 0$ is immediate since $\overline{1} = 0$. Now this does not solve the posed question. It only provides a starting framework. In particular, one must wonder whether, say, these universal one-forms admit finite integrals over paths inside $K$ with respect to some self-similar measure on $K$. As far as I know, a general approach here does not exist and one must look at the specific fractal in great detail. Here, for instance, is a paper of four authors which works out the details in the case that $K$ is the two-dimensional Sierpinski gasket, the prototypical post-critically finite fractal. You can see there that the authors ultimately consider only certain quotients of the full space of universal $1$-forms. They also appeal, somewhat subtly, to some heavy-duty results which permit one to construct a derivation which acts, basically, as a differential square root of the Dirichlet form. |
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