TITLE: Atiyah Bott-Shapiro orientation Vs Anderson-Brown-Peterson Splitting
QUESTION [9 upvotes]: Are  the  Atiyah-Bott-Shapiro Orientation and  the Anderson-Brown-Peterson Splitting compatible  in  any  sense? 
The  first guess is  that  the ABS-Orientation  is  related  to  the   projections  on the  $BO\langle 4(n(J))\rangle$, respectively $BO\langle 4(n(J))-2\rangle$ factors.

REPLY [8 votes]: According to  Michael Hopkins, Mark Hovey, Spin cobordism determines real K-theory, Mathematische Zeitschrift 210.1 (1992): 181-196, 4th page of the pdf file, the   Atiyah-Bott-Shapiro Orientation is just one of the Anderson-Brown-Peterson Splitting map $\pi ^0$.  Now, according to the paper by  Anderson-Brown-Peterson, the discussion between Theorem 1.3 and Corollary 1.4, this map is just the projection to the "bottom summand" $BO\langle 0 \rangle $.