I am searching for a reference with information pertaining to the $\mathbb{Z}/{2^m}$ cohomology of ${\rm{BTOP}}(n)$, for $n \geq 8$ and $m=1,2$, where
$${\rm{TOP}}(n) = \{f \colon \mathbb{R}^n \to \mathbb{R}^n \mid f \text{ is a homeomorphism such that } f(0) = 0\}.$$
Note. The cohomology rings $H^*(B(S)O(n);\mathbb{Z}/{2^m})$$H^*(BSO(n);\mathbb{Z}/{2^m})$ for $n \in \mathbb{Z}_+$ or $n = \infty$, have been computed by Cadek and Vanzura in this paper.
Context. Using obstruction theory, I am searching for conditions under which the tangent microbundle of a closed topological manifold admits a rank $k$ trivial subbundle. I am aware that there are Stiefel-Whitney classes $w_i \in H^*(B{\rm{TOP}}(n);\mathbb{Z}_2)$ defined as the primary obstructions to obtaining a cross section of the fibration
$$V_{n,n-i+1}^{\rm{TOP}}\to B{\rm{TOP}}(i-1) \to B{\rm{TOP}}(n).$$
Hence, in the mod 4 cohomology of $B{\rm{TOP}}$ there should exist elements of the form
$$\theta_2(w_{i_1}\cdots w_{i_s}), \text{ for } 1 \leq i_1 < \dots < i_{r} \leq n/2, \text{ and }$$
$$\beta_4(w_{i_1}\cdots w_{i_r}), \text{ for } 1 \leq i_1 < \dots < i_{s} \leq n/2,$$ where $\theta_2$ is the map induced by the inclusion $\mathbb{Z}/2 \to \mathbb{Z}/4$, and $\beta_4$ is the mod 4 reduction of the integral Bockstein. I am uncertain, however, if there exist mod 4 reductions of Pontryagin classes and (when $n$ is even) the mod 4 reduction of an Euler class.
Known References. I am currently aware of the following relevant (but insufficient) references.
- "Dalian notes on rational Pontryagin classes," M. Weiss.
- "Rational Pontryagin classes of Euclidean fiber bundles," M. Weiss.
- The Classifying Spaces for Surgery and Cobordism of Manifolds, Ch. 10, Madsen and Milgram.
- "On topological and piecewise linear vector fields," Stern.
