The bo-Adams

Mark Behrens (Notre Dame) Joint with Agnes Beaudry (Uchicago), Prasit Bhattacharya (Notre Dame), Dominic Culver (Notre Dame), Zhouli Xu (Uchicago) Generalized Adams spectral sequences

• 푅 = ( theory with cup products) • 푅∗ ∶= 푅∗ 푝푡 = 휋∗푅

• 푅∗푅 = ring of “R-cooperations”. • Dual to 푅∗푅 = ring of R-operations (natural transformations 푅∗ − → 푅∗) • E.g. 푅 = 퐻픽푝 ordinary mod 2 cohomology :

푅∗푅 = 퐴 = 푅∗푅 = 퐴∗ = dual Steenrod algebra

• R-Adams-spectral sequence (ASS): 푅 푠,푡 퐸1 = 푅푡 푅 ∧ ⋯ ∧ 푅 ∧ 푋 ⇒ 휋푡−푠(푋푅) 푠 Generalized Adams spectral sequences

R-ASS: 푅 푠,푡 퐸1 = 푅푡 푅 ∧ ⋯ ∧ 푅 ∧ 푋 ⇒ 휋푡−푠(푋푅) 푠 • 푋 = space or spectrum

• 휋∗ 푋푅 = stable homotopy groups of the ``R-localization’’ of 푋 E.g. 푅 = 퐻픽푝: ∧ 휋∗푋푅 = 휋∗푋 푝

• If 푅∗푅 is flat over 푅∗, then 푅∗푅 is a Hopf algebra, and

푅 푠,푡 퐸2 = 퐸푥푡푅∗푅(푅∗, 푅∗푋) WARNING: this is Ext of comodules over the coalgebra 푅∗푅 Adams spectral sequence 퐸푥푡푠,푡(픽 , 픽 ) ⇒ 휋푠 퐴∗ 푝 푝 푡−푠 푝

4 Adams spectral sequence 퐸푥푡푠,푡(픽 , 픽 ) ⇒ 휋푠 퐴∗ 푝 푝 푡−푠 푝

-Many differentials -푑 differentials go back by 1 and up by r 푟 5 Adams spectral sequence 퐸푥푡푠,푡(픽 , 픽 ) ⇒ 휋푠 퐴∗ 푝 푝 푡−푠 푝

-Many differentials -푑 differentials go back by 1 and up by r 푟 6 Adams spectral sequence 퐸푥푡푠,푡(픽 , 픽 ) ⇒ 휋푠 퐴∗ 푝 푝 푡−푠 푝

7 Adams spectral sequence 퐸푥푡푠,푡(픽 , 픽 ) ⇒ 휋푠 퐴∗ 푝 푝 푡−푠 푝

8 Adams-Novikov spectral sequence (p=2) Adams-Novikov spectral sequence (p=2) Adams-Novikov spectral sequence (p=2) 12 13 푏표 푠,푡 ∧푠 bo-ASS: 퐸1 = 푏표푡 푏표 ⇒ 휋푡−푠(푆)

• Motivation: bo is 푣1-periodic (aka Bott periodic) – good at detecting 푣1-periodic homotopy.

−1 • Used by Mahowald to compute 푣1 휋∗푆 at 푝 = 2. (proving the 2-primary 푣1-periodic telescope conjecture)

• Lellmann-Mahowald computed bo-ASS for sphere through dimension 20

• GOOD NEWS: no differentials through this range! 푏표 푠,푡 • BAD NEWS: 푏표∗푏표 is NOT flat over 푏표∗ --- thus hard to compute 퐸2

푏표 푠,푡 • TODAY: method to compute 퐸2 , computes bo-ASS through dimension 40 “with ease” The real motivation: tmf-ASS

• tmf = topological modular forms (sees 푣1 and 푣2-periodic homotopy)

• Much more powerful than bo 16 Adams spectral sequence 퐸푥푡푠,푡(픽 , 픽 ) ⇒ 휋푠 퐴∗ 푝 푝 푡−푠 푝

17 Adams spectral sequence 퐸푥푡푠,푡(픽 , 픽 ) ⇒ 휋푠 퐴∗ 푝 푝 푡−푠 푝

18 The real motivation: tmf-ASS

• tmf = topological modular forms (sees 푣1 and 푣2-periodic homotopy)

• Much more powerful than bo

• Bad news: 푡푚푓∗푡푚푓 not flat over 푡푚푓∗

• Good news: getting a fairly good understanding of 푡푚푓∗푡푚푓 (B-Ormsby-Stapleton-Stojanoska --- aka B.O.S.S.)

[Current collaboration does not have same ring to it: “BBBCX”] The real motivation: tmf-ASS

• Hope: tmf-ASS should get up through dimension 60 or 70 “with ease”.

• Wang-Xu recently showed 휋61푆 has no 2-torsion CONSEQUENCE: no exotic spheres in dimension 61! (super-hard mod 2 ASS computation)

• If we could push our understanding of 휋∗푆 into the 90’s, we would have a shot at resolving the outstanding Kervaire invariant question in dimension 126 The real motivation: tmf-ASS

−1 • Potential to understand 푣2 휋∗푆 at 푝 = 2 (we currently only have good understanding at odd primes)

• Telescope conjecture??? (don’t know this for 푣2-periodic homotopy at any prime – folks believe it’s false…) Connective K-theory cooperations Brown-Gitler spectrum perspective 푡ℎ (𝑖 퐵𝑖 (퐵𝑖 = 푖 integral Brown-Gitler spectrumڂ = 퐻ℤ •

2 • 퐻∗ 퐵𝑖 ⊂ 퐻∗ 퐻ℤ = 픽2 휁1 , 휁2, … 𝑖−1 subspace spanned by monomials of weight ≤ 2푖 (weight(휁𝑖) = 2 ) 휁𝑖 = 푐(휉𝑖)

4𝑖 [𝑖 Σ 푏표 ∧ 퐵𝑖 [Mahowald-Milgramڀ ⋍ 푏표 ∧ 푏표 • Connective K-theory cooperations Brown-Gitler spectrum perspective

• 푏표 ∧ 푏표푠 ≃ Σ4 퐼 푏표 ∧ 퐵 퐼= 𝑖1,…,𝑖푠 퐼ڀ

• 퐵퐼 = 퐵𝑖1 ∧ ⋯ ∧ 퐵𝑖푠 • 퐼 = 푖1 + ⋯ + 푖푠

푏표<2 퐼 −훼 퐼 >, 퐼 푒푣푒푛 • 푏표 ∧ 퐵 ≃ 퐻푉 ∨ ൝ 퐼 퐼 푏푠푝<2 퐼 −훼 퐼 −1>, 퐼 표푑푑

• 푉퐼 = graded 픽2-vector space • 훼 푖 = number of 1’s in dyatic expansion of i • 훼 퐼 = 훼 푖1 + ⋯ + 훼(푖푠) • 푏표<푛> = 푛th Adams cover of bo (bsp = symplectic K-theory)

Adams spectral sequence for “good” part of 푏표 ∧ 푏표 bo-ASS: 퐸1 − 푡푒푟푚

4 퐼 <2 퐼 −훼 퐼 > 푏표 푠,푡 푠 휋푡Σ 푏표 , 퐼 푒푣푒푛 퐸1 = 푏표푡푏표 ≃ ໄ 푉퐼 ⊕ ൝ 4 퐼 <2 퐼 −훼 퐼 −1> 휋푡Σ 푏푠푝 , 퐼 표푑푑 퐼= 𝑖1,…,𝑖푠 bo-ASS: 퐸1 − 푡푒푟푚

4 퐼 <2 퐼 −훼 퐼 > 푏표 푠,푡 푠 휋푡Σ 푏표 , 퐼 푒푣푒푛 퐸1 = 푏표푡푏표 ≃ ໄ 푉퐼 ⊕ ൝ 4 퐼 <2 퐼 −훼 퐼 −1> 휋푡Σ 푏푠푝 , 퐼 표푑푑 퐼= 𝑖1,…,𝑖푠 bo-ass: 퐸2-term bo-ass: 퐸2-term Our contribution: can use 퐸푥푡퐴∗ 픽2, 픽2 to 푒푣𝑖푙 ∗,∗ compute 퐸2 directly! Our contribution: can use 퐸푥푡퐴∗ 픽2, 픽2 to 푒푣𝑖푙 ∗,∗ compute 퐸2 directly! Our contribution: can use 퐸푥푡퐴∗ 픽2, 픽2 to 푒푣𝑖푙 ∗,∗ compute 퐸2 directly!