4 added 21 characters in body

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 $u$?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}$, but I am unsure where the other relations are coming from. 3 added 109 characters in body I am trying to 'get my hands dirty', so to speak, with some of the calculations with the Adams spectral sequence in Ravenel's Complex Cobordism book, and I have a few questions (I hope it is OK to ask them in the one question). In great generality for a Hopf algebroid$(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$\pi_*(bu)$, \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 this $\overline{\xi}_2 \otimes \overline{\xi}_2 + \xi_1 \otimes \xi_1^2 \overline{\xi}_2 + \xi_1 \overline{\xi}_2\otimes \xi_1^2$ represent $u = h_{11}^2$?u$? 3) How can I calculate this differential? For example how do we calculate$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.

2 spelling in title

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