Are the [Atiyah-Bott-Shapiro Orientation](https://ncatlab.org/nlab/show/K-orientation#UniversalAtiyahBottShapiroOrientation) and the [Anderson-Brown-Peterson Splitting](https://projecteuclid.org/euclid.bams/1183527786) 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. 



Atiyah Bott-Shapiro orientation Vs Anderson-Brown-Peterson Splitting
According to Michael Hopkins, Mark Hovey, *[Spin cobordism determines real K-theory](https://ncatlab.org/nlab/files/HopkinsHoveyCobordismK.pdf),* 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 $.


