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?

## 1 Answer

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 $d^3([M])$.