References for properties of Atiyah-Hirzebruch Spectral Sequence for a spectrum $X$ and generalised homology theory $MSpin_*$

Question

Currently I'm working on the following version of the AHSS $$ E^2_{pq}\cong H_p(M\eta; MSpin_q(\ast))\Rightarrow MSpin_{p+q}(M\eta)$$ where $\eta \colon B \to BSO$ is a stable vector bundle, and $M\eta$ denotes its Thom Spectrum.

My experience with the AHSS is "graduate level", therefore most properties and, let's say, tricks I'm aware of it is for the simple version with $E^2_{pq}(X)\Rightarrow h_*(X)$ where $X$ is a CW space, NOT a spectrum.

So these are the questions I'm looking for an answer (and a reference for it)

1. Third differentials for $ E^2_{pq}\cong H_p(M\eta; MSpin_q(\ast))\Rightarrow MSpin_{p+q}(M\eta)$.

I was able to find a reference for the second differentials (https://math.berkeley.edu/~teichner/Papers/Signature.pdf) Prop 1 page $750$, but nothing for the third differentials. In my specific case, there is a possibly non-trivial $d_3\colon E^3_{5,0}\to E^3_{2,2}$. Asking around it seems that it's general knowledge that the $d_3$ starting from the $0$th row is a secondary cohomology operation associated to $Sq^2Sq^2=0$. I'm not able to find any reference for this, and my little knowledge in secondary cohomology operations prevents me from proving it on my own.

2. Edge homomorphisms: Are there some characterisations for the edge homomorphisms in this case?

I'm aware that the characterisation can't be something easy, but if there is something it might prove useful sometimes.

3. Is there some multiplicative structure on the cohomological version?

I'm referring to something on the lines of Kochman page 34 def. $2.2.1$, in particular differentials are derivations. I think there should be, but I'm looking for some opinion from someone more experienced than me.

4. Is there a pairing between the AHSS's for $MSpin_*(M\eta)$ and $MSpin^*(M\eta)$ like in the case for a finite CW complex $X$?

I'm referring to a pairing analogue to the one explained in Kochman page 129 Proposition $4.2.10$. In particular I'd like to have some kind of relationship between the differential in the homology version and in the cohomology version, since my idea is to try figure out the cohomological one (maybe using some kind of multiplicative structure) and infer something about the homological ones.

Thanks in advance for your help.

Answer 1

Answer to Q2: By the Thom isomorphism, $E_{n,0}=H_n(B;\mathbb Z)$. The bordism group you are interested in is isomorphic to a twisted spin bordism group, i.e. elements are represented by $f:M^n\to B$ together with a spin structure on $TM\oplus f^*\eta$, up to bordism. This follows by a Pontryagin-Thom construction. The edge homomorphism sends such an element to $f_*([M])$, which follows from the Thom isomorphism.

For the study of the edge homomorphism, you may want to use surgery below the middle dimension, which reduces to the case of manifolds $M$ of a particular form. Maybe you can use some classification result of manifolds in dimensions 4 and 5 with given fundamental group.

More could be said if you would specify the stable bundle.

Another thought: If one is interested in calculating the low dimensional spin bordism groups of $M\eta$, then an Adams spectral sequence might help. Note that $MSpin$ is isomorphic to $ko$ in dimensions up to 7, so that at the prime 2 one will only need to know $Sq^1$ and $Sq^2$ on $M\eta$.