There is a theorem by Sullivan of the following form:



> 
> **Theorem:** There is an equivalence of $H$-spaces
>  $$ G/PL[\tfrac{1}{2}] \simeq BO\_{\otimes}[ \tfrac{1}{2} ]\ . $$
> 
> 
> 


It can be found for example in the book by Madsen and Milgram (Theorem 4.34 on page 97). I have two questions about this:


* Is the $H$-space structure on the left hand side that is meant in the theorem the geometric one coming from direct sum of block bundles?
* Both sides are also infinite loop spaces, is it known whether the map in the theorem is $A\_{\infty}$?



Sullivan's $H$-space equivalence between $G/PL[1/2]$ and $BO[1/2]$
Here is Dennis Sullivan's answer:
"Yes to the first.


For the second question I interpret it to mean does the H space map really mean a map of loop spaces? [this is synonymous with the A/infinity operad]


I read a paper by Milgram [from 81] where he is assuming these sorts of things for calculations 


This question goes beyond my expertise circa late 60's


My opinion..... now is that there are stronger natural infinite loop space or spectrum statements [these are synonymous with the E/infinity operad] that you might be able to address"


