Splitting Madsen-Tillmann spectra II. The Steinberg idempotents and Whitehead conjecture

We show that, at the prime $p=2$, the spectrum ${\mathrm{\Sigma }}^{-n}D(n)$ splits off the Madsen-Tillmann spectrum $MTO(n)=BO(n{)}^{-{\gamma }_n}$ which is compatible with the classic splitting of $M(n)$ off $BO(n{)}_{+}$. For $n=2$, together with our previous splitting result on Madsen-Tillmann spectra, this shows that $MTO(2)$ is homotopy equivalent to $BSO(3{)}_{+}\vee {\mathrm{\Sigma }}^{-2}D(2)$.