# Third differential in the homology AHSS

I need some guidance in identifying the third differential in the homology AHSS for $$\Omega_{\ast}^{\text{Spin}^c}(X)$$ in degrees $$\leq 4$$.

Remember that $$\pi_0(M\text{Spin}^c)=\Bbb Z$$, $$\pi_2(M\text{Spin}^c)=\Bbb Z$$, $$\pi_4(M\text{Spin}^c)=\Bbb Z\oplus \Bbb Z$$ and zero in degree $$1,3$$. So if we consider the (homologically graded) AHSS $$E_{p,q}^2 = H_p(X;\Omega_{q}^{\text{Spin}^c} ) \Rightarrow \Omega_{p+q}^{\text{Spin}^c}(X)$$ we immediately see that there are no second differentials and the third page looks like this (in the portion I'm interests in)

Can I conclude something akin to the cohomology AHSS (See here) where the first non-trivial differential is a stable cohomology operation? what bothers me is that having to deal with integer coefficients homology I don't have (in general) a perfect pairing with cohomology where the linked question would apply.

Any comments/observations are really appreciated! Thanks a lot!

• If your $X$ is an orientable manifold, then maybe you can use that the homology SS is a module over the cohomology SS, as described in Ben Antieau's answer here: mathoverflow.net/a/186421/8103 – Mark Grant May 8 at 16:19
• Wow, sadly the space I had in mind is the classifying space of some finite group, far from being closed/oriented (and even a manifold :( ) – Riccardo May 8 at 17:12
• Can you work over $\mathbb{F}_p$ for each $p$ and then reassemble the results to recover what happens with integer coefficients? – John Palmieri May 8 at 20:53
• I don't know how does it work. It would be really interesting though! Do you have any example/references where I can check out how does this reasoning work? – Riccardo May 8 at 22:12
• A standard tool for relating mod $p$ homology to integral homology is the Bockstein spectral sequence. See pages.vassar.edu/mccleary/files/2011/04/MC10.fin_.pdf for example. – John Palmieri May 9 at 19:51