The Chromatic Splitting Conjecture at p n 2

Transcription

1 The at p n Agnès Beaudry University of Chicago AMS Joint Meetings in San Antonio, January 13, 015

2 Mathematics is not like a suspense novel. You have to start with the punchline. Peter May Theorem (B.) The chromatic splitting conjecture does not hold when n p.

3 Motivation Stable homotopy groups of spheres of the sphere spectrum: π S colim π n S n. HARD! Mahowald Uncertainty Principle (Non-technical formulation) If you have an easy algorithm to compute π S, it is most certainly wrong. General Philosophy Filter S by spectra whose homotopy are easier to compute.

4 Definition L n is Bousfield localization with respect to Morava E-theory E n. Theorem (Hopkins-Ravenel - Chromatic Convergence) Fix a prime p and let S ppq denote the p-local sphere. There is a tower of spectra such that... Ñ L n S ppq Ñ L n 1 S ppq Ñ... Ñ L 1 S ppq Ñ L 0 S ppq S ppq lim ÐÝn L n S ppq. π L n S ppq is relatively easier to compute than π S ppq Definition L n S ppq is the n th chromatic layer of S ppq at the prime p.

5 h0 h1 h h3 0 c0 Ph1 Ph d0 h 3 h 30h4 Pc0 h1h4 P h1 hh4 P h c1 g Pd0 h 0i h4c0 P c0 P 3 h1 P 3 h d 0 h 4 h 10 0 h5 n P 3 c0 q d1 h1h5 P 4 h1 p e 0 h0hh5 P 4 h t x h h5 h 0h3h5 P h 0i u h3d1 h5c0 h1h3h5 P 4 c0 g Ph1h5 f1 P 5 h1 z d 30 Phh5 P 5 h g h5d0 w h 3h5 d0l N B1 h 70Q e0r Ph5c0 Pu P 5 c0 d0e 0 B P 6 h1 C h5c1 P 6 h h3g gn e0m d1g d0u x e 0g h0h5i P 4 h 0i P 6 c0 P 7 h1 h1q P 7 h d0w B1 B3 d 0l g 3 d0e0r A D3 h1x1 d 0e0 E1 + C0 h5n h 5 R h 5 0 h6 X C h1h1 P 7 c0 q1 h3q h1h6 P 8 h1 gw B3 Ph5j k1 d0e0m B5 + D r1 P 8 h d 0u d0x e0gr C11 X3 Q3 h H1h3A d h0h4x h 30G1 p D 3 h h6 h1w1 h3(e1 + C0) d1e1 p1 h1h3h1 h3h6 hq3 E 8 -term of the Adams spectral sequence computing π S ^ pq (Isaksen): The E -page of the classical Adams spectral sequence

6 E 8 -term of the Adams spectral sequence computing π S ^ pq (Isaksen):

7 E 8 -term of the Adams spectral sequence computing π S ^ pq (Isaksen): General Philosophy The rays roughly correspond to the homotopy detected by π L n S.

8 Example The 0 th chromatic layer corresponds to rational homotopy theory, π L 0 S ppq r H ps, Qq Q The 1 st chromatic layer π L 1 S ppq is related to K-theory and the image of the j-homomorphism.

9 Current State of Computations S S () S (3) S (pk) S (pk+1) LnS () LnS (3) LnS (pk) LnS (pk+1) n> :??????? Behrens, Goerss, Lawson, n=, p= :??? B., Bobkova, Goerss, Henn, Mahowald, Ormsby, Rezk, Stapleton, Stojanoska, Shimomura, LS () LS (3) LS (pk) LS (pk+1) n=, p=3 : Goerss, Henn, Mahowald, Rezk, Shimomura, Wang, L1S () L1S (3) L1S (pk) L1S (pk+1) L0S = Q n=0 : Quillen, Sullivan, n=, p >3 : Behrens, Hopkins, Miller, Morava, Ravenel, Shimomura, Wilson, Yabe, n=1 : Adams, Mahowald, Miller,

10 Theorem (Ravenel) Fix a prime p. There are spectrum Kpmq called the m th Morava K-theory spectrum such that L n S L K p0q K pnq S. S (p) L ns L K(n) S L n 1S L n 1L K(n) S L 1S L 0S

11 Theorem (Ravenel) Fix a prime p. There are spectrum Kpmq called the m th Morava K-theory spectrum such that L n S L K p0q K pnq S. S (p) L ns L K(n) S L n 1S L n 1L K(n) S L 1S L 0S

12 Hopkins s Conjecture (CSC-Strong form) The map i : L n 1 S Ñ L n 1 L K pnq S in L n S L K pnq S L n 1 S i L n 1 L K pnq S is a split monomorphism and L n 1 L K pnq S L n 1 S _ ª 1 i 1 i j n L n max ik S p i k q j.

13 Conjecture (CSC at n ) For S the p-local sphere, L 1 L K pq S L 1 S _ L 1 S 1 _ L 0 S 3 _ L 0 S 4. L n S L K pnq S L n 1 S i L n 1 L K pnq S

14 Remark The chromatic splitting conjecture as stated above is true when: n 1 at all primes (Adams, Mahowald) n for p 5 (Hopkins, Shimomura) n for p 3 (Goerss, Henn, Mahowald) It does not hold at n, p.

15 Theorem (B.) The strong form of the chromatic splitting conjecture for n p does not hold. Mahowald Uncertainty Principle If you have an easy algorithm to compute π S, it is most certainly wrong. Remark Method of proof was suggested by Paul Goerss. Shimomura and Wang s computations also hint at it.

16 Conjecture (CSC) L 1 L K pq S L 1 S _ L 1 S 1 _ L 0 S 3 _ L 0 S 4. The Moore spectrum V p0q is the cofiber S ÝÑ S Ñ V p0q. Then π L 0 V p0q r HpV p0q, Qq 0. Conjecture (CSC) L 1 L K pq V p0q L 1 V p0q _ L 1 Σ 1 V p0q.

17 Conjecture (CSC) π L 1 L K pq V p0q π L 1 V p0q ` π 1 L 1 V p0q Lemma If the chromatic splitting conjecture is true in its strongest form, then π 3 L 1 L K pq V p0q 0.

18 Lemma If the chromatic splitting conjecture is true, then π 3 L 1 L K pq V p0q 0. Proposition (B.) π 3 L 1 L K pq V p0q 0

19 Ingredients There is an equivalence, L K pq S E h G. For G G 1 Z, Call E h G1 is the 1 -sphere. There is a duality resolution, L K pq S Ñ E h G1 Ñ E h G1. E h G1 Ñ X 0 Ñ X 1 Ñ X Ñ X 3.

20 Proof Sketch. E h G1 Ñ X 0 Ñ X 1 Ñ X Ñ X 3 gives a spectral sequence E p,t 1 π t L 1 px p ^ V p0qq π t p L 1 pe h G1 ^ V p0qq. L K pq S Ñ E h G1 Ñ E h G1 gives a map F : π 3 L 1 L K pq V p0q Ñ π 3 L 1 pe h G1 ^ V p0qq. There is e P E 3,0 1 detecting a class in π 3 L 1 pe h G1 ^ V p0qq. e is in the image of F. 6 π 3 L 1 L K pq V p0q 0.

21 Remark At p 3, the class analogous to e gives rise to the blue summand: L 1 L K pq S L 1 S _ L 1 S 1 _ L 0 S 3 _ L 0 S 4. At odd primes, e is an L 0 -local but at p, e is an L 1 -local class. Questions What does the rational part of L 1 L K pq S look like at p? What is L 1 L K pq S at n and p? Can we describe of L n 1 L K pnq S in terms of a L k S i for 0 k n?

22 Thank you!