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Differentials in the Adams-Novikov spectral sequence and the geometric boundary theorem

2012-09-18T06:17:59Z

$\newcommand\Ext{\mathrm{Ext}} \newcommand\Z{\mathbb{Z}} \newcommand\G{\mathbb{G}}$ The reference for this question will be the paper by Henn, Karamanov and Mahowald - "The homotopy of the $K(2)$-local Moore spectrum at the prime 3 revisted" (so everything is at $n=2,p=3$).

Let $\G_2$ be the extended Morava stabiliser group, and let $\G_2^1$ be the kernel of the reduced norm $\G_2 \to \Z_3$. Then Henn,Karamanov and Mahowald use a permutation resolution of $\mathbb{G}_2^1$ to construct a spectral sequence (the algebraic spectral sequence)

$$ E_1^{p,q,t}=\Ext_{\mathbb Z_3[[\G_2^1]]}^q(C_p,M_t)\Longrightarrow H^{p+q}(\G_2^1,M_t) $$ for $M=(E_2)_*(V(0)) = (E_2)_*/(3)$ and then use this to calculate $H^*(\G_2,(E_2)_*/(3))$ , which in turn is used to calculate $\pi_*(L_{K(2)}V(0))$ (via the Adams-Novikov spectral sequence)

It's a very technical paper, and suffice to say I can't define all the terms I'm going to use in an introduction. But I am specifically interested in some of the differentials in the Adams-Novikov spectral sequence. In particular, let's just start with the first differential $d_5(\Delta_k \tilde{\alpha} \beta) = \pm \Delta_{k-1}\beta^4 v_1$. (The calculation is on pp. 32-33)

Consider the short exact sequence

$$ 0 \to \Sigma^4 (E_2)_*/(3) \stackrel{v_1}{\to} (E_2)_*/(3) \to (E_2)_*(3,u_1) \to 0$$

It turns out that $\Delta_k \tilde{\alpha} \beta$ is in the kernel of multiplication by $v_1$ and so must be in the image of the Bockstein $\delta_{\G_2}^1$ in $H^*(\G_2,-)$ and $\delta_{\G_2^1}^1$ in $H^*(\G_2^1,-)$ associated to the above short exact sequence (after appropriate suspension). So I go and look up the calculation of $H^*(\G_2^1,(E_2)_*/(3,u_1))$ and work out (by degree) what possible classes it can be to get that $$\delta_{\G_2^1}^1((\omega^2 u^{-4})^{3k+2}\beta) = \pm \Sigma^4 \Delta_k \tilde{\alpha}\beta,$$

and in fact the same is true for $\delta^1_{\G_2^1}$. What I am interested in is how the geometric boundary theorem is then used to get the result. From Ravenel (2.3.4) this applies to a cofiber sequence $$W \stackrel{f}{\to} X \stackrel{g}{\to} Y \stackrel{h}{\to} \Sigma W$$ with $E_*(h)=0$. If such a situation applies then we have maps $\delta_r$ for $2 \le r \le \infty$ such that $\delta_2 = \delta^1$ in our language (I think), but more importantly $\delta_r d_r = d_r \delta_r$.

Thus I think the calculation of $d_5(\Delta_k \tilde{\alpha} \beta)$ should basically be something akin to $d_5(\Delta_k \tilde{\alpha} \beta) = d_5\delta(-) = \delta d_5(-)$, where $d_5(-)$ is something we can calculate via knowledge of differentials in $H^*(\G_2,(E_*)/(3,u_1)).$

Is this the right idea? I am slightly confused by the $\Sigma^4$ floating around and more confused by the fact that HKM seem to be doing some additional calculations (e.g. why do I care about $\delta^1((\omega^2 u^{-4})^{3k+2} \beta)$?)

Some calculations with the Adams spectral sequence and the cobar complex

2012-05-23T10:59:36Z

I am trying to 'get my hands dirty', so to speak, with some of the calculations with the Adams spectral sequence in Ravenel's Complex Cobordism book, and I have a few questions (I hope it is OK to ask them in the one question).

In great generality for a Hopf algebroid $(A,\Gamma)$ we can define the cobar complex $C_\Gamma^*(M)$ by $C_\Gamma^*(M)=\overline{\Gamma}^{\otimes s} \otimes M$ where $\overline{\Gamma}=\text{ker} \epsilon: \Gamma \to A$ with coboundary map $$d_s(\gamma_1 \otimes \cdots \otimes \gamma_s \otimes m) = \cdots + \sum_{i=1}^s \gamma_1 \otimes \dots \otimes \gamma_{i-1} \otimes \psi(\gamma_i) \otimes \cdots \otimes \gamma_s \otimes m + \cdots$$

(where $\psi(\gamma_i)$ is the coproduct and I have omitted the first and last term for brevity) and then $\text{Ext}_{\Gamma}(A,M)$ is the cohomology of this cobar complex.

The first calculation (pp. 64-66) is the $E_2$ term for the calculation of $\pi_*(bo)$, which is equal to $\text{Ext}_{\mathscr{A}(1)_*}(\mathbb{F}_2,\mathbb{F}_2)$ . This is abutted to by a Cartan-Eilenberg Spectral sequence which has $E_2$ term equal to $\mathbb{F}_2[h_{10},h_{11},h_{20}]$ , where $h_{i,j}$ corresponds to the class $[\overline{\xi}_i^{2^j}]$ in the cobar complex. The first claim is that $d_2(h_{20}) = h_{10}h_{11}$, and this follows from the fact that in the cobar complex $d(\xi_2) = \xi_1 \otimes \xi_1^2$, which in turn follows from the coproduct of the mod 2 Steenrod algebra. This gives $E_3$ term $\mathbb{F}_2(u,h_{10},h_{11})/(h_{10}h_{11})$ where $u$ corresponds to $h_{20}^2$. Again we can calculate $d(\overline{\xi}_2 \otimes \overline{\xi}_2) = \overline{\xi}_2 \otimes \xi_1 \otimes \xi_1^2 + \xi_1 \otimes \xi_1^2 \otimes \overline{\xi}_2 $ in the cobar complex. Ravenel then states

...the cobar complex is not commutative and when we add correcting terms to $\overline{\xi}_2 \otimes \overline{\xi}_2$ in the hope of getting a cycle we get instead $d(\overline{\xi}_2 \otimes \overline{\xi}_2 + \xi_1 \otimes \xi_1^2 \overline{\xi}_2 + \xi_1 \overline{\xi}_2\otimes \xi_1^2) = \xi_1^2 \otimes \xi_1^2 \otimes \xi_1^2$

which is used to conclude $d_3(u)=h_{11}^3$

Finally my questions:

1) Why are the correcting terms $\xi_1 \otimes \xi_1^2 \overline{\xi}_2$ and $\xi_1 \overline{\xi}_2\otimes \xi_1^2$?

2) (This may be answered by 1) Why does $\overline{\xi}_2 \otimes \overline{\xi}_2 + \xi_1 \otimes \xi_1^2 \overline{\xi}_2 + \xi_1 \overline{\xi}_2\otimes \xi_1^2$ represent $u$ in the cobar complex?

3) How can I calculate this differential? For example how do we calculate $d(\xi_1 \otimes \xi_1^2 \overline{\xi}_2$)?

That part 2 of this question concerns the May spectral sequence for calculating $\text{Ext}_\mathscr{A}(\mathbb{F}_2,\mathbb{F}_2)$. One can compute the $E_2$ term of the May spectral sequence to have generators (in the region $t-s \le 13$) $h_j = h_{1,j}$, $b_{i,j} = h_{i,j}^2$ and $x_7 = h_{20}h_{21} + h_{11}h_{30}$. There are some relations given without proof; $h_jh_{j+1} = 0, h_2b_{20} = h_0x_7$ and $h_2x_7 = h_0b_{21}$. I think that the relation $h_jh_{j+1}=0$ comes from the fact that $d_1(h_{2,j}) = h_jh_{j+1}$, but I am unsure where the other relations are coming from.

Milnor exact sequence in $K(n)$ local Morava $E$-theory

2012-03-30T00:05:34Z

Let $L_E$ denote Bousfield localisation with repsect to the cohomology theory $E$. I am trying to follow through some calculations in Hovey-Strickland's paper Morava $K$-theories and localisation

Claim 7.10(e) is that

$$L_{K(n)}X = \underset{\leftarrow}{\text{holim}}_I L_{E(n)}X \wedge S/I$$

where the homotopy limit is over a tower of generalised Moore spectra (the Moore spectra $S/I$ is defined in 4.12 and the tower in 4.22)

Define $$E_*^{\vee}:=\pi_{*} (L_{K(n)} (E \wedge X))$$

the $K(n)$-local version of Morava $E$-theory (where, I believe, Morava $E$-theory here is what I might call a completed Johnson-Wilson theory, but I don't believe it really matters).

The claim (8.04) is then that we can extract a Milnor exact sequence from this:

$$0 \to \varprojlim_I {}^1 (E/I)_{\ast+1}(X) \to E_*^\vee X \to \varprojlim_I (E/I)_*X \to 0 $$

I'm not sure how to show this. I would like to think that we can get a sequence

$$0 \to \varprojlim_I {}^1 \pi_{\ast+1}(L_{E(n)}(E \wedge X) \wedge S/I)\to E_{\ast}^\vee X \to \varprojlim_I \hspace{1mm} \pi_* (L_{E(n)}(E \wedge X) \wedge S/I) \to 0 $$

and then if you drop the $E(n)$-localisation it seems to work, but I'm not really sure about this.

Answer by Drew Heard for Definition of CW complexes

2012-03-21T10:29:37Z

It basically says that a CW complex has the coherent topology from its closed cells. This should be wrapped up into any definition of a CW complex that you see.

For example in Massey's book it is equivalent to statement (iv):

A subset $A$ is closed if and only if $A \cap \bar{e}$ is closed for all $n$-cells, $e^n,n=0,1,2,\ldots$

(You can see this equivalence at the wikipedia page).

Comment by Drew Heard

2013-04-05T14:18:03Z

I guess you get the same extension problem if you just try to use the AHSS?

Comment by Drew Heard

2012-12-19T10:36:40Z

There are a couple more references, also related to the Picard group - the dual of the question mark complex is referenced in Goerss-Henn-Mahowald-Rezk's "Picard groups at chromatic level 2 for $p = 3$" paper - they at least tell you a bit about its $K$ theory and $KO$ theory. There are also some (hard!) calculations in Ichigi-Shimomura's "$E(2)_*$-invertible spectra smashing with the Smith-Toda spectrum $V(1)$ at the prime 3" (see section 3 in particular)

Comment by Drew Heard

2012-11-17T00:00:03Z

If you specifically want to see these in action, 'Bordism, Stable Homotopy, and Adams Spectral Sequences' by Kochman, does some serious $p=2$ calculations of the stable homotopy groups of spheres

Comment by Drew Heard

2012-11-16T09:49:45Z

Also, if you check out the work of Shimomura and colleagues, who use the chromatic spectral sequence in a fairly heavy way, I'm sure you'll find Massey products a plenty!

Comment by Drew Heard

2012-11-16T09:41:31Z

But how many people are really doing ASS calculations? If you look at, say, the calculation of the homotopy of tmf at $p=2,3$ then it makes heavy use of Massey products.

Comment by Drew Heard

2012-10-07T05:00:46Z

In particular, I hadn't realised that there was a spectral sequence $E_1^{n,k} = \bigoplus_{n=0}^2 \pi_k M_n(S) \Rightarrow \pi_k L_{E(2)} S$ for example

Comment by Drew Heard

2012-10-07T04:57:12Z

Good question, something I've been wondering about. Have you read Behren's "The homotopy groups of the $E(2)$-local sphere at $p > 3$, revisited"? There is a lot of good stuff in there! You might be interested in section 7 where he uses the calculations of $H^*(M_1^1)$ and $H^*(M_0^2)$ to calculate the homotopy of the $K(2)$ local sphere, the $E(2)$ local sphere and the $K(2)/E(2)$ local mod $p$ Moore spectrum.

Comment by Drew Heard

2012-08-21T22:57:00Z

Perhaps "Commutative Algebra: with a View Toward Algebraic Geometry" by Eisenbud is what you are after?

Comment by Drew Heard

2012-07-17T15:21:17Z

Another option is Kochman's book "Bordism, Stable Homotopy and Adams Spectral Sequences". If I recall correctly the differentials are fully calculated in this range, using Massey products.

Comment by Drew Heard

2012-07-07T08:58:33Z

You can find a proof in Hatcher's Algebraic Topology book - Proposition 4.56 on pp. 385

Comment by Drew Heard

2012-06-29T09:44:35Z

Not necessarily related to infinite families but in Hopkins ICM address he discusses what is known geometrically about the first 16 stable homotopy groups (arxiv.org/abs/math/0212397)

Comment by Drew Heard

2012-05-24T03:06:15Z

@Tyler: Thanks for that. So are you saying, in the example I have in mind, $d[\xi_1 \vert \xi_1^2\xi_2 ]=[\xi_1 \vert \xi_1^2 \vert \xi_2] + [\xi_1 \vert \xi_2 \vert \xi_1^2]+\cdots$?

Comment by Drew Heard

2012-05-24T01:45:06Z

@John - thanks that is pretty neat!

Comment by Drew Heard

2012-05-24T00:46:39Z

Ok, I've verified the relations in 2 now - thank you again. So it is really 'grunt work' - trying to find boundaries who have the right summand to produce a relation?

Comment by Drew Heard

2012-05-23T23:42:57Z

Thank you very much! I've finally managed to locate your thesis, so I'll sit down this morning and try and work through the calculations. (Hopefully I've fixed in the typo's in the first question now. I guess my problem in part 3 is - how does one do the coproduct $\psi(\xi_1^2 \overline{\xi}_2)$?)