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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)

enter image description here

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!

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    $\begingroup$ 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 $\endgroup$ – Mark Grant May 8 at 16:19
  • $\begingroup$ 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 :( ) $\endgroup$ – Riccardo May 8 at 17:12
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    $\begingroup$ Can you work over $\mathbb{F}_p$ for each $p$ and then reassemble the results to recover what happens with integer coefficients? $\endgroup$ – John Palmieri May 8 at 20:53
  • $\begingroup$ 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? $\endgroup$ – Riccardo May 8 at 22:12
  • $\begingroup$ 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. $\endgroup$ – John Palmieri May 9 at 19:51

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