In Bott & Tu's marvelous book there is a derivation of the spectral sequence for a (smooth) fiber bundle for deRham cohomology done entirely in the realm of the smooth category. Unfortunately, as it's written there it is not clear how to generalize to other cohomology theories (they write down a double complex, and they use very much the fact that cohomology is given as the cohomology of this chain complex.)
Now, in Quillen's paper computing the complex cobordism ring, he introduces a geometric description of complex cobordism as a cohomology theory on smooth manifolds. I've been attempting, to no avail, to come up with a derivation of the Atiyah-Hirzebruch spectral sequence for a smooth fibration without using CW complex-esque techniques. My question is:
Does anyone know of a derivation of the Atiyah-Hirzebruch spectral sequence of a smooth fiber bundle for a generalized cohomology theory that does not leave the realm of manifolds?
So far, the most promising bet I have found is Segal's approach to this spectral sequence in the paper "Classifying spaces and spectral sequences." The trouble appears, however, in his use of a complex $BX_U$ for a covering $U$ of $X$. This is most definitely not a manifold, though it is homotopy equivalent to one for numerable covers. The question in this context, however, reduces to:
Does the natural filtration on $BX_U$ induce a filtration on $X$ that gives rise to the spectral sequence for a covering? If so, is there a nice description of this filtration using, say, just the data of $X$ and the numerable cover?
For this question I should be more specific about $BX_U$. This is defined as the geometric realization of the nerve of the topological category $X_U$ whose points are pairs $(x, U_{\sigma})$ where $U_\sigma$ is a finite intersection of elements of $U$ and $x \in U_\sigma$. Morphisms are inclusions $U_\sigma \subset U_\tau$. The filtration on $BX_U$ is given by looking at the images of $(NX_U)_n \times \Delta^n$ in $BX_U$, where $NX_U$ is the nerve.
