Cohomology: A Mirror of Homotopy
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1 Cohomology: A Mirror of Homotopy Agnès Beaudry University of Chicago September 19, 1
2 Spectra Definition Top is the category of based topological spaces with based continuous functions rx, Y s denotes the set of homotopy classes of continuous functions X Ñ Y π n X : rs n, X s the n th homotopy group of X.
3 Definition The smash product of X and Y is X ^ Y : X Y p Y q Y px q Definition The suspension is ΣX : S 1 ^ X Lemma ΣS n S n 1 Definition loooooomoooooon n Σ n X : S 1 ^... ^ S 1 ^ X S n ^ X
4 Definition A spectrum is a sequence E te n u 8 n of based topological spaces E n, with maps ε n : ΣE n Ñ E n 1. Definition If X P Top, then Σ 8 X is the spectrum with pσ 8 X q n Σ n X ε n : ΣΣ n X id ÝÑ Σ n 1 X Definition The sphere spectrum is S : Σ 8 S.
5 Definition The n-th homotopy group of E is π n E : lim ÝÑk π n k E k. Further, E : π E. Example The homotopy groups of the sphere spectrum are the stable homotopy groups of spheres: π n S lim ÝÑk π n k S k π s ns.
6 Definition E is a ring spectrum if E π E is a graded ring Example S is a ring spectrum.
7 Z-Modules S-Modules Z S M b N X ^ Y M b Z M X ^ S X Localizing set S Localizing spectrum E S 1 M L E X p-local module M ppq p-local spectrum X ppq L H Zppq X F p -module Mod-p spectrum M{ppq X {ppq p-complete module p-complete spectrum M p lim ÐÝ M{pp n q X p lim ÐÝ X {pp n q Hom Z pm, Nq F px, Y q Shift Suspension pσmq n M n 1 ΣX
8 Z-Modules S-Modules Z S M b N X ^ Y M b Z M X ^ S X p-local module M ppq p-local spectrum X ppq Localizing set S Localizing spectrum E S 1 M L E X F p -module Mod-p spectrum M{ppq X {ppq p-complete module p-complete spectrum M p lim ÐÝ M{pp n q X p lim ÐÝ X {pp n q Hom Z pm, Nq F px, Y q Shift Suspension pσmq n M n 1 ΣX
9 Z-Modules S-Modules Z S M b N X ^ Y M b Z M X ^ S X p-local module M ppq p-local spectrum X ppq Localizing set S Localizing spectrum E S 1 M L E X F p -module Mod-p spectrum M{ppq X {ppq p-complete module p-complete spectrum M p lim ÐÝ M{pp n q X p lim ÐÝ X {pp n q Hom Z pm, Nq F px, Y q Shift Suspension pσmq n M n 1 ΣX
10 Z-Modules S-Modules Z S M b N X ^ Y M b Z M X ^ S X p-local module M ppq p-local spectrum X ppq Localizing set S Localizing spectrum E S 1 M L E X F p -module Mod-p spectrum M{ppq X {ppq p-complete module p-complete spectrum M p lim ÐÝ M{pp n q X p lim ÐÝ X {pp n q Hom Z pm, Nq F px, Y q Shift Suspension pσmq n M n 1 ΣX
11 Z-Modules S-Modules Z S M b N X ^ Y M b Z M X ^ S X p-local module M ppq p-local spectrum X ppq Localizing set S Localizing spectrum E S 1 M L E X F p -module Mod-p spectrum M{ppq X {ppq p-complete module p-complete spectrum M p lim ÐÝ M{pp n q X p lim ÐÝ X {pp n q Hom Z pm, Nq F px, Y q Shift Suspension pσmq n M n 1 ΣX
12 Z-Modules S-Modules Z S M b N X ^ Y M b Z M X ^ S X p-local module M ppq p-local spectrum X ppq Localizing set S Localizing spectrum E S 1 M L E X F p -module Mod-p spectrum M{ppq X {ppq p-complete module p-complete spectrum M p lim ÐÝ M{pp n q X p lim ÐÝ X {pp n q Hom Z pm, Nq F px, Y q Shift Suspension pσmq n M n 1 ΣX
13 Z-Modules S-Modules Z S M b N X ^ Y M b Z M X ^ S X p-local module M ppq p-local spectrum X ppq Localizing set S Localizing spectrum E S 1 M L E X F p -module Mod-p spectrum M{ppq X {ppq p-complete module p-complete spectrum M p lim ÐÝ M{pp n q X p lim ÐÝ X {pp n q Hom Z pm, Nq F px, Y q Shift Suspension pσmq n M n 1 ΣX
14 Z-Modules S-Modules Z S M b N X ^ Y M b Z M X ^ S X p-local module M ppq p-local spectrum X ppq Localizing set S Localizing spectrum E S 1 M L E X F p -module Mod-p spectrum M{ppq X {ppq p-complete module p-complete spectrum M p lim ÐÝ M{pp n q X p lim ÐÝ X {pp n q Hom Z pm, Nq F px, Y q Shift Suspension pσmq n M n 1 ΣX
15 Z-Chain Complexes S-Modules Z S M b N X ^ Y M b Z M X ^ S X p-local module M ppq p-local spectrum X ppq Localizing set S Localizing spectrum E S 1 M L E X F p -module Mod-p spectrum M{ppq X {ppq p-complete module p-complete spectrum M p lim ÐÝ M{pp n q X p lim ÐÝ X {pp n q Hom Z pm, Nq F px, Y q Shift Suspension pσmq n M n 1 ΣX
16 Z-Chain Complexes S-Modules Z S M b N X ^ Y M b Z M X ^ S X p-local module M ppq p-local spectrum X ppq Localizing set S Localizing spectrum E S 1 M L E X F p -module Mod-p spectrum M{ppq X {ppq p-complete module p-complete spectrum M p lim ÐÝ M{pp n q X p lim ÐÝ X {pp n q Hom Z pm, Nq F px, Y q Shift Suspension pσmq n M n 1 ΣX
17 Definition E-homology is the covariant functor: E : Spectra Ñ grpabq, E px q : π pe ^ X q P Z E-cohomology is the contravariant functor: E : Spectra Ñ grpabq, E px q : π F px, Eq P Z Lemma E π E E psq
18 Example An Eilenberg-MacLane space KpG, nq is a topological space such that π m KpG, nq #G m n m n. The Eilenberg MacLane spectrum HG satisfies phgq n KpG, nq. Lemma HG pσ 8 X q r H px, Gq.
19 General Philosophy The E-Adams spectral sequence One can do homological algebra in the category of spectra. Theorem Let E be a ring spectrum. The E-Adams spectral sequence is given by E s,t Ext s,t E pe E, E X q ùñ π t s L E X, where Ext s E E p, q are the derived functors of Hom E E p, q, and L E X is the E-localization of X. Mahowald Uncertainty Principle Any spectral sequence converging to the homotopy groups of sphere with an E -term that can be named using homological algebra will be infinitely far away from the actual answer.
20 Classical Adams Spectral Sequence: E H F Example Let E HF be the Eilenberg-MacLane spectrum, so that phf q px q H px, F q. The classical Adams spectral sequence is E s,t Ext s,t phf q HF pphf q, phf q Sq ùñ π t s L HF S π t s S
21 Classical Adams Spectral Sequence: E H F The E -page (Isaksen s chart): P h1 P h 8 Pc cohomological grading : s 6 3 Ph1 Ph d e f g c c1 1 h 3 h h1 h h3 h homotopy grading: t - s E s,t Ext s,t phf q H F pphf q, phf q q ùñ π t s S
22 h h1 h h3 Classical Adams Spectral Sequence: E H F The E 8 -page (Isaksen s chart): P h1 P h 8 s Pc 6 3 Ph1 Ph d h 3 h g c c1 1 h 3 h1h hh t - s E s,t Ext s,t phf q H F pphf q, phf q q ùñ π t s S
23 h h1 h h3 c Ph1 Ph d Pc P h1 h1h hh c1 The E 8 -page (Isaksen s chart): P h 8 s 6 3 h 3 h g 1 h t - s -component of π S (Hatcher s table) :
24 Adams-Novikov Spectral Sequence: E BP Definition BP is the Brown Peterson spectrum. where v n P BP pp n 1q. Definition BP Z ppq rv 1, v,...s The Adams-Novikov spectral sequence is E s,t Ext s,t BP BP pbp, BP Sq ùñ π t s L BP S π t s S pq
25 Adams-Novikov Spectral Sequence: E BP Definition BP is the Brown Peterson spectrum. where v n P BP pp n 1q. Definition BP Z ppq rv 1, v,...s The Adams-Novikov spectral sequence is E s,t Ext s,t BP BP pbp, BP Sq ùñ π t s L BP S π t s S pq
26 Figure..5. Ext(BP )forp =,t s 5. The E -page (Ravenel s chart): s 3 β 3 η β / β β / β /3 β / β t s
27 The E -page (Ravenel s chart): s 3 β 3 η β / β β / β /3 β / β t s
28 The E -page (Ravenel s chart): s 3 η β3 β / β β / β /3 β / β t s
29 The E -page (Ravenel s chart): s 3 η β3 β / β β / β /3 β / β t s
30 Some patterns of differentials:
31 Some patterns of differentials:
32 Good candidates for E: E s,t Ext s,t E E pe, E X q ùñ π t s L E X. Easier to compute the E -page Still detects enough homotopy
33 Morava K-Theories Fix a prime p, and an integer n 8. There is a ring spectrum Kpnq called the n-th Morava K-theory. Kpq H Q Kp1q K F p... Kp8q H F p
34 Chromatic tower: L n X : L K pq_..._k pnq X... Ñ L n X Ñ L n 1 X Ñ... Ñ L 1 X Ñ L X Theorem (Hopkins-Ravenel, Chromatic Convergence) Fix a prime p. If X is a finite spectrum, then there is a homotopy equivalence X ppq holimt... Ñ L n X Ñ L n 1 X Ñ... Ñ L X u.
35 The stable p-local homotopy category of finite spectra is filtered: X () X (3) X... X (pk) X (pk+1) L n X L n X... L n X L n X L n 1 X L n 1 X... L n 1 X L n 1 X L X L X... L X L X L 1 X L 1 X... L 1 X L 1 X L H Q X
36 L n X L K pq_..._k pnq X X (p)... F n X L n X L K(n) X F n X L n 1 X L n 1 L K(n) X... L 1 X L X
37 L n X L K pq_..._k pnq X X (p)... F n X L n X l n X F n X L n 1 X L n 1 L K(n) X... L 1 X L X
38 L n X L K pq_..._k pnq X X (p)... F n X L n X L K(n) X F n X L n 1 X L n 1 L K(n) X... L 1 X L X
39 L n X L K pq_..._k pnq X X (p)... L n X L K(n) X L n 1 X L n 1 L K(n) X... L 1 X L X
40 L n X L K pq_..._k pnq X X (p)... F n X L n X L K(n) X F n X L n 1 X L n 1 L K(n) X... L 1 X L X
41 Theorem There is a ring spectrum E n (called Morava E-theory) and a group G n (called the Morava Stablizer group) acting on E n such that L K pnq S E h Gn n. The E-Adams-Novikov spectral sequence has a K pnq-local analogue. Theorem Ext s,t E E pe, E X q ùñ π t s L E X The Kpnq-local E n -Adams-Novikov spectral sequence can be described as H pg n, pe n q q ùñ π L K pnq S.
42 Chromatic Level n 1 Example Kp1q K F G 1 Z, pe 1q Z rv 1 1 s λ pv 1 q λ 1 v 1, λ P G 1 H pg 1 ; pe 1 q q H pz ; Z rv 1 1 sq
43 E -page : H pg 1 ; pe 1 q q H pz ; Z rv 1 1 sq
44 E -page : H pg 1 ; pe 1 q q H pz ; Z rv 1 1 sq
45 Compare with Adams-Novikov Spectral Sequence for BP η 3 s β3 β / β β / β /3 β / β t s
46 General Philosophy Chromatic level n phenomena correspond to the n-line of the Adams-Novikov spectral sequence for BP s 3 β 3 η β / β β / β /3 β / β t s Figure..5. Ext(BP )forp =,t s 5.
47 Chromatic Level n H pg ; pe q q ùñ π L K pq S Remark π L K pq S has been computed for p 3. The remaining case is p.
48 Chromatic Level n, p H pg ; pe q q ùñ π L K pq S Facts H pg ; pe q q is hard to compute. G G 1 Z There is a fiber sequence L K pq S Ñ E h G1 Ñ E h G1 E h G1 1/-sphere H pg 1 ; pe q q ùñ π E h G1
49 Still too hard. Remark H pg 1 ; pe q q ùñ π E h G1 S ÝÑ S Ñ S{ loooooooomoooooooon H pg 1 ; pe q {q ñ π pe h G1 ^ S{q 1 L K pqs{ I have computed the E -page of this spectral sequence, H pg 1 ; pe q {q.
50 L S{ L K pq S{ L 1 S{ L 1 L K pq S{ Theorem (Mahowald - Telescope Conjecture) This implies: π L 1 L K pq S{ v 1 1 π L K pq S{ and v 1 v 1 1 H pg ; pe q {q ùñ π L 1 L K pq S{ loooooooooomoooooooooon 1 H pg 1 ; pe q {q ùñ π L 1 pe h G1 ^ S{q 1 π L 1L K pq S{
51 v 1 1 H pg 1 ; pe q S{q ùñ π L 1 pe h G1 ^ S{q Figure: π L 1 pe h G1 ^ S{q. Classes in gray are shifts of the generators under multiplication by rv 1 s
52 L S{ L K pq S{ L 1 S{ L 1 L K pq S{ Have : L 1 L K pq pe h G1 ^ S{q 1 L 1L K pq S{ There is a fiber sequence: L 1 L K pq S{ Ñ L 1 pe h G1 ^ S{q Ñ L 1 pe h G1 ^ S{q We are one computation away from π L 1 L K pq S{.
53 Thank you!
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