@Drew wait do you mean Lie Knive?

This might be a very silly question:

Does a spectral sequence collapse at the r page if all differentials on that page vanish? The converse is obvious but i'm reading something now that suggests that this is an iff statement.

is it even true for the E2 page?

no. think about the serre spectral sequence for $\Omega S^n$ for instance

by which I mean the Serre spectral sequence for the fiber sequence $\Omega S^n \to PS^n \to S^n$ that you use to compute homology or cohomology of $\Omega S^n$. It only has differentials on one page which is ~n (I forget which exactly). So there aren't any differentials on the low pages but it doesn't collapse there.

That's what I thought. Thanks.

Where can I read about transgressions in spectral sequences (say in the leray spectral sequence for starters) and how to compute them?

@SaalHardali I found some interesting explanation in Kirk & davis book

it's not complicated, very clear and concise. If you need a more "deep" reference, I believe you can read something on Switzer's book

I have another old-facts-references question: What's the canonical citation for the fact that Adams' old stable category is indeed equivalent to any of its more modern incarnations?

@Riccardo thanks i'll check it out

A different matter: Is there a p-local hurevicz theorem for every p? what about a K(n)-local hurevicz?

@SaalHardali yes, there is a p-local version for every p. i'm not sure what a K(n)-local version would mean; there's no meaningful notion of 'n-connected' in the K(n)-local category

(alternatively, i think there's a reasonable interpretation of the theorem for connective morava k-theory but it is false.)

p-local here means that i'm inverting all primes except p right?

Or does it mean completion?

I'm always confused about whether p-local means localize w.r.t. S_(p) (invert all primes) or localize w.r.t. the moore spectrum (completion).

While i'm at it is there a hurevicz theorem for both of these?

i meant inversion, but it's up to you. there's a version of hurewicz for p-localization and H_*(-; Z_(p)), and there's also a version of hurewicz for p-completion and H_*(-; Fp)

the former i think you can prove by hand without any substantial deviation from integral hurewicz. the second one is maybe a little more clever, but you can read about it in mosher & tangora('s section on mod-C theory)

Thanks!

@EricPeterson is there any kind of ordering on the K(n)-local picard group?