@MikeMiller, I'm interested in a general 4-manifold, with a given handle decomposition. I'm less looking for an "efficient" way to calculate something for a given manifold, but more a constructive description that I could prove something about

In particular, there should be a natural map from homology of X to homology of π₁(X), and I want to know what the fundamental class of X is being sent to

What [X] is sent to can encode a lot of information, it seems! E.g. if X_2 is K(π₁(X),1), it must be sent to 0. And there is a result in 4-manifolds that says that a smooth 4-manifold up to ℂP² is given Euler characteristic, signature, π₁ and the image of [X].

This is not easy in general, and probably why that new paper by Teichner et al (apologies to the other authors I have forgotten) has many results stated for special classes of groups, where the cohomology is more tractable.

Ahh, that's a nice perspective!

@MikeMiller, since you seem to have read that new paper (I'm trying to read it currently, it's hard though well-written), do you understand from which theorem of Kreck in his "Surgery and duality" the result I stated follows? I can't that figure out at all.

I did not look back at the old Kreck paper, but if it's easily available I am glad to take a look now

It's this one: arxiv.org/abs/math/9905211

Or at least that's the one they cite

Hmmm, it might have to do with relative s-cobordisms...

The only thing that makes sense to me is as an application of Theorem 4, so there must be a standing assumption on the fundamental group

Iirc 'good' is pretty close but not quite the same as 'amenable'

I can't see it at all :/

But it seems like it is entirely possible that the image of [X] in H_4(π₁(X)) holds the information about the 2-type of X for simple π₁

My guess was that they arose as the "algebraic obstruction" that is supposed to be "elementary"

Alternatively I guess it could arise from the theorem after that, as $S^2 \times S^2$ becomes 2CP2 + CP2bar after connected sum with CP2

They seem to mean Theorem C. At the end of Page 9 they claim they sketch Kreck's argument

Oh yes

You were certainly right about the S² × S² part