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In this paper N.Teleman constructs the signature operator on an arbitrary (closed, oriented) Lipschitz manifold with coefficients in a vector bundle $\xi$. In particular the notion of a differential $k$-form is introduced (see page 5 in the above paper): these forms are required to have coefficients in $L^2$ and one restricts to the space of all forms $\omega$ such that $d\omega$ has also coefficients in $L^2$.

In this manner one gets the notion of a differential $k$ form for any $k$ and also one does not need the notion of the tangent bundle in order to define these forms.

However in the another paper N.Teleman seems to use the notion of the tangent bundle when he deals with Hirzebruch polynomial $L(M)$ for $M$-he assumes that $M$ is only a topological manifold and uses the result of Sullivan that any topological manifold of dimension $\neq 4$ possess (essentially unique) Lipschitz structure. I'm aware of the theory of Milnor microbundles which is a substitute for the notion of the tangent bundle in the context of the topological manifolds, however I suspect that N.Teleman does not use them since he works in the Lispchitz category which is regular enough to provide the existence of derivatives almost everywhere. So my first Question is:

Question 1: Is the tangent bundle well defined for an arbitrary Lipschitz manifold?

Let us now move to a richer category of quasiconformal manifolds. In yet another paper N.Teleman together with A.Connes and D.Sullivan consider the space of differential forms on an arbitarary (closed, oriented) quasiconformal manifold. We find the following sentence ,,The tangent bundle of a quasiconformal manifold is a measurable real vector bundle'' which suggest the affirmative answer to the question above and gives rise to the

Question 2: How is the tangent bundle of a quasiconformal manifold defined?

As I understood in the context of quasiconformal manifold it is not possible to define $k$-forms which are $L^2$ but it is only possible to define $k$ forms which are $L^{\frac{n}{k}}$.

Problem 1: I would like to have a clear picture of how these forms are defined.

To be more precise I would like to understand:
a) do we need the notion of the tangent bundle in order to define them? Why we cannot proceed as in the Lipschitz case
b) what are the obstructions to define $L^2$ forms of any order $k$ or in other words, why for $k$-forms we have to change our $L^p$-norms?
c) as the exterior derivative increases order of a differntial form by $1$ it seems to me that we are not able to define $d$ as an operator acting between Hilbert spaces: we need $L^2$ forms but this forces the dimensional of the underlying manifold to be $2l$ and the order of the forms to be $l$. Am I right?

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