Geodesic
The Map SJ → SF Does Not Deloop Mod 2
Canadian journal of mathematics, Tome 27 (1975) no. 4, pp. 737-745

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It has been widely conjectured that there exists a homotopy commutative Diagram where J is the stable Whitehead f-homomorphism and BSJ is the space constructed in [3]. In [4], Stasheff and the author proved that this conjecture is false. However, Quillen's proof of the Adams conjecture in [7] has as a corollary the existence of the homotopy commutative diagram 
Clough, Robert R. The Map SJ → SF Does Not Deloop Mod 2. Canadian journal of mathematics, Tome 27 (1975) no. 4, pp. 737-745. doi: 10.4153/CJM-1975-082-2
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