TITLE: Differential of homological atiyah-Hirzebruch Spectral sequence for K-homology
QUESTION [6 upvotes]: The  first  non vanishing differential  $d_3$ of  the  cohomological Atiyah-Hirzebruch  spectral  sequence for  computing  (Complex) Topological  $K$-theory  out  of   ordinary  cohomology  has a  description  in terms  of the  cohomology operation $Sq^3_{\mathbb{Z}}$. Is  there  an explicit   formula  for  the  differential  of  the  homological  spectral  sequence   converging  to  complex  K-homology?

REPLY [4 votes]: One can use the fact that the AH spectral sequence for $KU$-homology is a module over the AH spectral sequence for $KU$-cohomology. For example, let $M$ be a closed orientable manifold with fundamental class $[M]\in H_n(M,\mathbb{Z})\cong H_n(M,KU_0)$. Consider a class $x\in H_p(M,KU_q)$, which we can write as $y\cap [M]$ for some $y\in H^{n-p}(M,\mathbb{Z})\cong H^{n-p}(M,KU^{-q})$. Then, we have $$d^3(x)=d^3(y\cap [M])=d_3(y)\cap [M]\pm y\cap d^3([M])=Sq_3(y)\cap[M]\pm y\cap d^3([M]).$$ This reduces the problem to computing $d^3([M])$.