The Adams Spectral Sequence

6/21 The Adams spectral sequence

References

• https://static1.squarespace.com/static/5aff705c5ffd207cc87a512d/t/5b0377abf950b75de22932e7/1526953900694/Homotopy+Theory.pdf
• https://ncatlab.org/nlab/show/Adams+spectral+sequence
• http://www.rrb.wayne.edu/papers/adams.pdf
• https://ncatlab.org/nlab/show/Introduction+to+the+Adams+Spectral+Sequence
• http://www.ms.uky.edu/~guillou/Aramian_ASS.pdf
• http://stanford.edu/~arpon/files/adams-ss.pdf
• https://ncatlab.org/nlab/show/Adams+resolution
• https://www.wikiwand.com/en/Adams_resolution
• https://www.wikiwand.com/en/Adams_spectral_sequence#/Ext_terms_from_the_resolution

Notes

Note from Paul: People describe it as Ext in \(\comod{\steenrod {}^{ \vee }}\) rather than (equivalently) in \(\operatorname{mod}{\steenrod}\). The difference: \(\steenrod\) is a free graded commutative algebra, which is easier to use than graded cocommutativity of \(\steenrod {}^{ \vee }\). Maybe say a bit about \(\comod{A}\) for \(A\in\HopfAlg\).

• Basic motivation: buff up the Serre spectral sequence, but only work stably.
• Big question: what is \([X, Y]\in {\mathsf{Ab}}\)? Start by understanding its \(p{\hbox{-}}\)torsion.
• Apply the functor \(H^*({-}) := H^*({-}; { \mathbf{F} }_p)\) to get \(\mathop{\mathrm{Hom}}_{{ \mathsf{Vect} }_{/{ \mathbf{F} }_p}}(H^* X, H^* Y)\).
• Find extra module structure on this Hom: module over \({\mathcal{A}}\) the Steenrod algebra
• Take derived functors to get \(\operatorname{Ext} _{{\mathcal{A}}}^*( H^*X, H^* Y)\).
• Applications
  • Compute \(\pi_* {\mathbb{S}}\otimes{ {\mathbf{Z}}_{\widehat{p}} }\)
  • Original use by Adams: Hopf Invariant One. Which \({\mathbf{R}}^n\) are division algebras?
  • Thom isomorphism theorem