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 $H^*(BSO(n);\mathbb{Z}/{2^m})$ for $n \in \mathbb{Z}_+$ or $n = \infty$, have been computed by Cadek and Vanzura in [this][1] 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,"][2] M. Weiss.
 - ["Rational Pontryagin classes of Euclidean fiber bundles,"][3] M. Weiss.
 - *[The Classifying Spaces for Surgery and Cobordism of Manifolds,][4]* Ch. 10, Madsen and Milgram. 
 - ["On topological and piecewise linear vector fields,"][5] Stern.


  [1]: https://projecteuclid.org/journals/journal-of-mathematics-of-kyoto-university/volume-35/issue-1/The-cohomology-rings-of-BO-n-and-BSO-n-with/10.1215/kjm/1250518838.full
  [2]: https://arxiv.org/abs/1507.00153
  [3]: https://msp.org/gt/2021/25-7/p02.xhtml
  [4]: https://www.maths.ed.ac.uk/~v1ranick/papers/madmil.pdf
  [5]: https://bpb-us-e2.wpmucdn.com/faculty.sites.uci.edu/dist/3/246/files/2011/03/4_TopandPLVectorFields.pdf