exotic spheres and topological modular formsmbehren1/presentations/exotic_spheres.pdfexotic spheres...
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Exotic spheres and topological modular forms
Mark Behrens (MIT)
(joint with Mike Hill, Mike Hopkins, and Mark Mahowald)
Fantastic survey of the subject:
Milnor, βDifferential topology: 46 years laterβ
(Notices of the AMS, June/July 2011)
http://www.ams.org/notices/201106/
PoincarΓ© Conjecture
Q: Is every homotopy n-sphere homeomorphic
to an n-sphere?
A: Yes!
β’ π = 2: easy.
β’ π β₯ 5: (Smale, 1961) h-cobordism theorem
β’ π = 4: (Freedman, 1982)
β’ π = 3: (Perelman, 2003)
Smooth PoincarΓ© Conjecture
Q: Is every homotopy n-sphere diffeomorphic to an n-sphere? A: Depends on n. β’ π = 2: True - easy. β’ π = 7: (Milnor, 1956) False β produced a smooth manifold
which was homeomorphic but not diffeomorphic to π7! [exotic sphere]
β’ π β₯ 5: (Kervaire-Milnor, 1963) β `oftenβ false. (true for π = 5,6). β’ π = 3: (Perelman, 2003) True. β’ π = 4: Unknown.
Smooth PoincarΓ© Conjecture
Q: Is every homotopy n-sphere diffeomorphic to an n-sphere? A: Depends on n. β’ π = 2: True - easy. β’ π = 7: (Milnor, 1956) False β produced a smooth manifold
which was homeomorphic but not diffeomorphic to π7! [exotic sphere]
β’ π β₯ 5: (Kervaire-Milnor, 1963) β `oftenβ false. (true for π = 5,6). β’ π = 3: (Perelman, 2003) True. β’ π = 4: Unknown.
Main Question
For which n do there exist exotic n-spheres?
Kervaire-Milnor
Ξπ β {oriented smooth homotopy n-spheres}/h-cobordism
(note: if π β 4, h-cobordant β oriented diffeomorphic)
For π β’ 2(4):
0 β Ξπππ
β Ξπ βππ
π
πΌπ π½β 0
Kervaire-Milnor
Ξπ β {smooth homotopy n-spheres}/h-cobordism
(note: if π β 4, h-cobordant β diffeomorphic)
For π β’ 2(4):
0 β Ξπππ
β Ξπ βππ
π
πΌπ π½β 0
Kervaire-Milnor
Ξπ β {smooth homotopy n-spheres}/h-cobordism
For π β’ 2(4):
0 β Ξπππ
β Ξπ βππ
π
πΌπ π½β 0
Ξπππ
= subgroup of those which bound a
parallelizable manifold
πππ = stable homotopy groups of spheres
= ππ+π ππ for π β« 0
π½: ππ ππ β πππ is the J-homomorphism.
Kervaire-Milnor
Ξπ β {smooth homotopy n-spheres}/h-cobordism
For π β’ 2(4):
0 β Ξπππ
β Ξπ βππ
π
πΌπ π½β 0
Ξπππ
= subgroup of those which bound a
parallelizable manifold
πππ = stable homotopy groups of spheres
= ππ+π ππ for π β« 0
π½: ππ ππ β πππ is the J-homomorphism.
Kervaire-Milnor
Ξπ β {smooth homotopy n-spheres}/h-cobordism
For π β’ 2(4):
0 β Ξπππ
β Ξπ βππ
π
πΌπ π½β 0
Ξπππ
= subgroup of those which bound a
parallelizable manifold
πππ = stable homotopy groups of spheres
= ππ+π ππ for π β« 0
π½: ππ ππ β πππ is the J-homomorphism.
Kervaire-Milnor
Ξπ β {smooth homotopy n-spheres}/h-cobordism
For π β’ 2(4):
0 β Ξπππ
β Ξπ βππ
π
πΌπ π½β 0
For π β‘ 2 4 :
0 β Ξπππ
β Ξπ βππ
π
πΌπ π½β β€
2 β Ξπβ1ππ
β 0
Ξπππ
β’ Trivial for n even
β’ Cyclic for n odd
Ξπππ
β’ Trivial for n even
β’ Cyclic for n odd β Generated by boundary of an explicit parallelizable
manifold given by plumbing construction
Ξπππ
β’ Trivial for n even
β’ Cyclic for n odd:
Ξπππ
=
22π 22π+1 β 1 ππ’π4π΅π+1
π + 1, π = 4π + 3
β€2 , π β‘ 1(4), βππ+1 π€ππ‘β Ξ¦πΎ = 1
0, π β‘ 1 4 , βππ+1 π€ππ‘β Ξ¦πΎ = 1
Upshot: n even β bp gives no exotic spheres
π β‘ 3 (4) β bp gives exotic spheres (π β₯ 7)
π β‘ 1 (4) β bp gives exotic sphere only if there are no ππ+1 with Ξ¦πΎ = 1
J-homomorphism
π½: ππππ β πππ β Ξ©π
ππ
Given πΌ: ππ β ππ, apply it pointwise to the standard stable framing of ππ to obtain a non-standard stable framing of ππ. Homotopy spheres are stably parallelizable, but not uniquely so β only get a well defined map
Ξπ βππ
π
πΌπ π½
J-homomorphism
π½: ππππ β πππ β Ξ©π
ππ
πβπ
Stable homotopy groups:
πππ β lim
πββππ+π(π
π) (finite abelian groups for π > 0)
Primary decomposition:
πππ = (ππ
π )(π)π πππππ e.g.: π3π = β€24 = β€8 β β€3
β’ Each dot represents a factor of 2, vertical lines indicate additive extensions
e.g.: (π3π )(2) = β€8, (π8
π )(2) = β€2 β€2
β’ Vertical arrangement of dots is arbitrary, but meant to suggest patterns
19
Computation: Mahowald-Tangora-Kochman Picture: A. Hatcher
β’ Each dot represents a factor of 2, vertical lines indicate additive extensions
e.g.: (π3π )(2) = β€8, (π8
π )(2) = β€2 β€2
β’ Vertical arrangement of dots is arbitrary, but meant to suggest patterns
20
21
Computation: Nakamura -Tangora Picture: A. Hatcher
(ns)(5)
n
22
Computation: D. Ravenel Picture: A. Hatcher
23
Adams spectral sequence
πΈπ₯π‘π΄π ,π‘(β€ π , β€ π) β ππ‘βπ
π π
ππ‘βπ π
24
Adams spectral sequence
πΈπ₯π‘π΄π ,π‘(β€ π , β€ π) β ππ‘βπ
π π
ππ‘βπ π
-Many differentials -ππ differentials go back by 1 and up by r
25
Adams spectral sequence
πΈπ₯π‘π΄π ,π‘(β€ π , β€ π) β ππ‘βπ
π π
ππ‘βπ π
26
Adams spectral sequence
πΈπ₯π‘π΄π ,π‘(β€ π , β€ π) β ππ‘βπ
π π
ππ‘βπ π
Kervaire Invariant
Ξ¦πΎ: πππ β β€ 2
Browder:
(Ξ¦πΎβ 0) β π = 2π β 2
27
Kervaire Invariant
Ξ¦πΎ: πππ β β€ 2
Browder:
(Ξ¦πΎ π₯ β 0) β π₯ πππ‘πππ‘ππ ππ¦ βπ2
ππ π΄ππ
28
Kervaire Invariant
Ξ¦πΎ: πππ β β€ 2
Browder:
(Ξ¦πΎ π₯ β 0) β π₯ πππ‘πππ‘ππ ππ¦ βπ2
ππ π΄ππ
Computation in ASS: Ξ¦πΎ β 0 for π β {2, 6, 14, 30, 62}
29
Kervaire Invariant
Ξ¦πΎ: πππ β β€ 2
Browder:
(Ξ¦πΎ π₯ β 0) β π₯ πππ‘πππ‘ππ ππ¦ βπ2
ππ π΄ππ
Computation in ASS: Ξ¦πΎ β 0 for π β {2, 6, 14, 30, 62}
Hill-Hopkins-Ravenel:
Ξ¦πΎ = 0 for all π β₯ 254
(Note: the case of π = 126 is still open) 30
Summary: Exotic spheres
Ξπ β 0 if:
β’ Ξπππ
β 0:
o π β‘ 3 (4) and π β₯ 7
o π β‘ 1 (4) and π β {1,5,13,29,61,125? } [Kervaire]
β’ Remains to check: is ππ
π
πΌπ π½β 0 for
o π even
oπ β {1,5,13,29,61,125? }
31
32
Summary: Exotic spheres
Ξπ β 0 if:
β’ Ξπππ
β 0: o π β‘ 3 (4) and π β₯ 7 o π β‘ 1 (4) and π β {1,5,13,29,61,125? }
β’ππ
π
πΌπ π½β 0 for π β‘ 2 (8)
β’ Remains to check: is ππ
π
πΌπ π½β 0 for
o π β‘ 0 (4) or π β‘ β2 (8)
oπ β {1,5,13,29,61,125? }
33
Low dimensional computations
β’ Limitation: only know πππ
2 for π β€ 63
β’ππ
π
πΌπ π½ π= 0 in this range for π β₯ 7.
34
Low dimensional computations
35
Stem p = 2 p = 3 p = 5
4 0 0 0
8 e 0 0
12 0 0 0
16 h4 0 0
20 ΞΊbar Ξ²1^2 0
24 h4 Ξ΅ Ξ· 0 0
28 Ξ΅ ΞΊbar 0 0
32 q 0 0
36 t Ξ²2 Ξ²1 0
40 ΞΊbar^2 Ξ²1^4 0
44 g2 0 0
48 e0 r 0 0
52 ΞΊbar q Ξ²2^2 0
56 ΞΊbar t 0 0
60 kbar^3 0 0
Non-trivial elements in πΆππππ π½: π β‘ 0 (4)
Low dimensional computations
36
Stem p = 2 p = 3 p = 5
6 Ξ½^2 0 0
14 k 0 0
22 Ξ΅ k 0 0
30 ΞΈ4 Ξ²1^3 0
38 y Ξ²3/2 Ξ²1
46 w Ξ· Ξ²2 Ξ²1^2 0
54 v2^8 Ξ½^2 0 0
62 h5 n Ξ²2^2 Ξ²1 0
Non-trivial elements in πΆππππ π½: π β‘ β2 (8)
Low dimensional computations
37
Stem p = 2 p = 3 p = 5
1 0 0 0
5 0 0 0
13 0 Ξ²1 a1 0
29 0 Ξ²2 a1 0
61 0 Ξ²4 a1 0
Non-trivial elements in πΆππππ π½:
π β {1,5,13,29,61} [where Ξπππ
= 0 because of Kervaire classes]
Low dimensional computations
Conclusion
For π β€ 63, the only π for which Ξπ = 0 are: 1,2,3,4,5,6,12,61
38
Beyond low dimensionsβ¦
Strategy: try to demonstrate Coker J is non-zero in certain dimensions by producing infinite periodic families such as the one above.
Need to study periodicity in πβπ
39
Periodicity in πβπ
40
Periodicity in πβπ
41
Periodicity in πβπ
42
Periodicity in πβπ
43
Periodicity in πβπ
44
Periodicity in πβπ
45
Periodicity in πβπ
46
Periodicity in πβπ
47
Periodicity in πβπ
48
Periodicity in πβπ
49
Periodicity in πβπ
50
Periodicity in πβπ
51
Periodicity in πβπ
52
Periodicity in πβπ
53
Periodicity in πβπ
54
Periodicity in πβπ
55
Periodicity in πβπ
56
(ns)(5)
v1 - periodic layer consists solely of a-family
period = 2(p-1) = 8
57
(ns)(5)
v1 - periodic layer consists solely of a-family
period = 2(p-1) = 8
58
Greek letter notation: the πΌ-family
59
(ns)(5)
v2 - periodic layer = b-family
period = 2(p2 - 1) = 48
60
(ns)(5)
61
π£1-torsion in the π£2-family
62
Greek Letter Names (Miller-Ravenel-Wilson) 63
(ns)(5)
period = 2(p3 - 1) = 248
v3 - periodic layer = g-family
64
β’ π£1-periodicity β completely understood
β’ π£2-periodicity β know a lot for π β₯ 5
β Knowledge for π = 2,3 is subject of current research.
β For Ξπ, we will see π = 2 dominates the discussion
β’ π£3-periodicity β know next to nothing!
65
66
Exotic spheres from π½-family
β’ π½π = π½π/1,1 exists for π β₯ 5 and π β₯ 1
[Smith-Toda]
Ξπ β 0 for π β‘ β2 π β 1 β 2 πππ 2(π2 β 1)
67
68
Coker J
n = 0 mod 4 n = -2 mod 8 (including Kervaire Inv 1) n = 2^k - 3 (where Ξ_n^bp = 0 because of Kervaire class)
Stem p = 2 p = 3 p = 5 Stem p = 2 p = 3 p = 5 Stem p = 2 p = 3 p = 5
4 0 0 0 6 Ξ½^2 0 0 1 0 0 0
8 e 0 0 14 k 0 0 5 0 0 0
12 0 0 0 22 Ξ΅ k 0 0 13 0 Ξ²1 a1 0
16 h4 0 0 30 ΞΈ4 Ξ²1^3 0 29 0 Ξ²2 a1 0
20 ΞΊbar Ξ²1^2 0 38 y Ξ²3/2 Ξ²1 61 0 Ξ²4 a1 0
24 h4 Ξ΅ Ξ· 0 0 46 w Ξ· Ξ²2 Ξ²1^2 0 125? 0
28 Ξ΅ ΞΊbar 0 0 54 v2^8 Ξ½^2 0 0
32 q 0 0 62 h5 n Ξ²2^2 Ξ²1 0
36 t Ξ²2 Ξ²1 0 70 0 0
40 ΞΊbar^2 Ξ²1^4 0 78 Ξ²2^3 0
44 g2 0 0 86 Ξ²6/2 Ξ²2
48 e0 r 0 0 94 Ξ²5 0
52 ΞΊbar q Ξ²2^2 0 102 Ξ²6/3 Ξ²1^2 0
56 ΞΊbar t 0 0 110 0
60 kbar^3 0 0 118 0
64 0 0 126 0
68 <a1,Ξ²3/2,Ξ²2> 0 134 Ξ²3
72 Ξ²2^2 Ξ²1^2 0 142 0
76 0 Ξ²1^2 150 0
80 0 0 158 0
84 Ξ²5 Ξ²1 0 166 0
88 0 0 174 0
92 Ξ²6/3 Ξ²1 0 182 Ξ²4
96 0 0 190 Ξ²1^5
100 Ξ²2 Ξ²5 0 198 0
104 0 206 Ξ²5/4
108 0 214 Ξ²5/3
112 0 222 Ξ²5/2
116 0 230 Ξ²5
120 0 238 Ξ²2 Ξ²1^4
124 Ξ²2 Ξ²1 246 0
128 0 254 0
132 0 262 0
136 0 270 0
140 0 278 Ξ²1
144 0 286 Ξ²3 Ξ²1^4
148 0 294 0
152 Ξ²1^4 302 0
156 0 310 0
160 0 318 0
Cohomology theories
β’ Use homology/cohomology to study homotopy
β’ A cohomology theory is a contravariant functor
πΈ: {Topological spaces} {graded ab groups}
π πΈβ(π)
β’ Homotopy invariant:π β π β πΈ π = πΈ(π)
β’ Excision: π = π βͺ π (CW complexes)
β― β πΈβ π β πΈβ π β πΈβ π β πΈβ π β© π β
69
Cohomology theories
β’ Use homology/cohomology to study homotopy
β’ A cohomology theory is a contravariant functor
πΈ: {Topological spaces} {graded ab groups}
π πΈβ(π)
β’ Homotopy groups:
ππ πΈ β πΈβπ(ππ‘)
(Note, in the above, n may be negative)
70
Cohomology theories
β’ Example: singular cohomology β πΈπ π = π»π(π)
β ππ π» = β€, π = 0,0, else.
β’ Example: Real K-theory β πΎπ0 π = πΎπ π = Grothendieck group of β-vector bundles over π.
β πβπΎπ = (β€, β€ 2 , β€ 2 , 0, β€, 0, 0, 0, β€, β€ 2 , β€ 2 , 0, β€, 0, 0, 0β¦ )
71
Hurewicz Homomorphism
β’ A cohomology theory E is a (commutative) ring theory if
its associated cohomology theory has βcup productsβ
πΈβ(π) is a graded commutative ring
β’ Such cohomology theories have a Hurewicz
homomorphism:
βπΈ: πβπ β πβπΈ
Example: π» detects π0π = β€.
72
Example: KO (real K-theory)
73
β’ To get more elements of Ξπ, need to start
looking at π£2-periodic homotopy.
β’ Need a cohomology theory which sees a
bunch of π£2-periodic classes in its
Hurewicz homomorphism
β’ π‘ππβ π - topological modular forms!
74
Topological Modular Forms
KO
β’ π£1-periodic β 8-periodic
β’ Multiplicative group
β’ Bernoulli numbers
TMF
β’ π£2-periodic β 576-periodic
β’ Elliptic curves
β’ Eisenstein series
(modular forms)
192-periodic at π = 2
144-periodic at π = 3
75
Topological Modular Forms
β’ There is a descent spectral sequence:
π»π β³πππ; πβπ‘ β π2π‘βπ πππΉ
β’ Edge homomorphism:
π2ππππΉ β Ring of integral modular forms
(rationally this is an iso)
β’ πβπππΉ has a bunch of 2 and 3-torsion, and the descent spectral sequence is highly non-trivial at these primes.
76
π»π β³πππ; πβπ‘ β π2π‘βπ πππΉ The decent spectral sequence for TMF
(p=2)
77
Exotic spheres from π½-family
β’ π½π = π½π/1,1 exists for π β₯ 5 and π β₯ 1
[Smith-Toda]
Ξπ β 0 for π β‘ β2 π β 1 β 2 πππ 2(π2 β 1)
β’ π½π exists for π = 3 and π β‘ 0,1,2,3,5,6 9
[B-Pemmaraju]
Ξπ β 0 for π β‘ β6, 10, 26, 42, 74, 90 mod 144
78
Exotic spheres from π½-family
β’ π½π = π½π/1,1 exists for π β₯ 5 and π β₯ 1
[Smith-Toda]
Ξπ β 0 for π β‘ β2 π β 1 β 2 πππ 2(π2 β 1)
β’ π½π exists for π = 3 and π β‘ 0,1,2,3,5,6 9
[B-Pemmaraju]
Ξπ β 0 for π β‘ β6, 10, 26, 42, 74, 90 mod 144
79
Hurewicz image of TMF (p = 3)
80
81
Coker J
n = 0 mod 4 n = -2 mod 8 (including Kervaire Inv 1) n = 2^k - 3 (where Ξ_n^bp = 0 because of Kervaire class)
Stem p = 2 p = 3 p = 5 Stem p = 2 p = 3 p = 5 Stem p = 2 p = 3 p = 5
4 0 0 0 6 Ξ½^2 0 0 1 0 0 0
8 e 0 0 14 k 0 0 5 0 0 0
12 0 0 0 22 Ξ΅ k 0 0 13 0 Ξ²1 a1 0
16 h4 0 0 30 ΞΈ4 Ξ²1^3 0 29 0 Ξ²2 a1 0
20 ΞΊbar Ξ²1^2 0 38 y Ξ²3/2 Ξ²1 61 0 Ξ²4 a1 0
24 h4 Ξ΅ Ξ· 0 0 46 w Ξ· Ξ²2 Ξ²1^2 0 125? 0
28 Ξ΅ ΞΊbar 0 0 54 v2^8 Ξ½^2 0 0
32 q 0 0 62 h5 n Ξ²2^2 Ξ²1 0
36 t Ξ²2 Ξ²1 0 70 0 0
40 ΞΊbar^2 Ξ²1^4 0 78 Ξ²2^3 0
44 g2 0 0 86 Ξ²6/2 Ξ²2
48 e0 r 0 0 94 Ξ²5 0
52 ΞΊbar q Ξ²2^2 0 102 Ξ²6/3 Ξ²1^2 0
56 ΞΊbar t 0 0 110 0
60 kbar^3 0 0 118 0
64 0 0 126 0
68 <a1,Ξ²3/2,Ξ²2> 0 134 Ξ²3
72 Ξ²2^2 Ξ²1^2 0 142 0
76 0 Ξ²1^2 150 0
80 0 0 158 0
84 Ξ²5 Ξ²1 0 166 0
88 0 0 174 Ξ²1^3 0
92 Ξ²6/3 Ξ²1 0 182 Ξ²3/2 Ξ²4
96 0 0 190 Ξ²2 Ξ²1^2 Ξ²1^5
100 Ξ²2 Ξ²5 0 198 0
104 0 206 Ξ²2^2 Ξ²1 Ξ²5/4
108 0 214 Ξ²5/3
112 0 222 Ξ²2^3 Ξ²5/2
116 0 230 Ξ²6/2 Ξ²5
120 0 238 Ξ²5 Ξ²2 Ξ²1^4
124 Ξ²2 Ξ²1 246 Ξ²6/3 Ξ²1^2 0
128 0 254 0
132 0 262 0
136 0 270 0
140 0 278 Ξ²1
144 0 286 Ξ²3 Ξ²1^4
148 0 294 0
152 Ξ²1^4 302 0
156 0 310 0
160 0 318 Ξ²1^3 0
π£2-periodicity at the prime 2
82
π£2-periodicity at the prime 2
83
Thm: (B-Mahowald)
The complete Hurewicz image:
The decent spectral sequence for TMF
(p=2) 84
Hurewicz image of TMF (p = 2)
85
86
Coker J
n = 0 mod 4 n = -2 mod 8 (including Kervaire Inv 1) n = 2^k - 3 (where Ξ_n^bp = 0 because of Kervaire class)
Stem p = 2 p = 3 p = 5 Stem p = 2 p = 3 p = 5 Stem p = 2 p = 3 p = 5
4 0 0 0 6 Ξ½^2 0 0 1 0 0 0
8 e 0 0 14 k 0 0 5 0 0 0
12 0 0 0 22 Ξ΅ k 0 0 13 0 Ξ²1 a1 0
16 h4 0 0 30 ΞΈ4 Ξ²1^3 0 29 0 Ξ²2 a1 0
20 ΞΊbar Ξ²1^2 0 38 y Ξ²3/2 Ξ²1 61 0 Ξ²4 a1 0
24 h4 Ξ΅ Ξ· 0 0 46 w Ξ· Ξ²2 Ξ²1^2 0 125? w kbar^4 0
28 Ξ΅ ΞΊbar 0 0 54 v2^8 Ξ½^2 0 0 = in tmf
32 q 0 0 62 h5 n Ξ²2^2 Ξ²1 0 = not in tmf, not known to be v2-periodic
36 t Ξ²2 Ξ²1 0 70 <kbar w,Ξ½,Ξ·> 0 0 = not in tmf, but v2-periodic
40 ΞΊbar^2 Ξ²1^4 0 78 Ξ²2^3 0 = Kervaire
44 g2 0 0 86 Ξ²6/2 Ξ²2 = trivial
48 e0 r 0 0 94 Ξ²5 0
52 ΞΊbar q Ξ²2^2 0 102 v2^16 Ξ½^2 Ξ²6/3 Ξ²1^2 0
56 ΞΊbar t 0 0 110 v2^16 k 0
60 kbar^3 0 0 118 v2^16 Ξ·^2 kbar 0
64 0 0 126 0
68 v2^8 k Ξ½^2 <a1,Ξ²3/2,Ξ²2> 0 134 Ξ²3
72 Ξ²2^2 Ξ²1^2 0 142 v2^16 Ξ· w 0
76 0 Ξ²1^2 150 (v2^16 Ξ΅ kbar)Ξ·^2 v2^9 0
80 kbar^4 0 0 158 0
84 Ξ²5 Ξ²1 0 166 0
88 0 0 174 beta32/8 Ξ²1^3 0
92 Ξ²6/3 Ξ²1 0 182 beta32/4 Ξ²3/2 Ξ²4
96 0 0 190 Ξ²2 Ξ²1^2 Ξ²1^5
100 kbar^5 Ξ²2 Ξ²5 0 198 v2^32 Ξ½^2 0
104 v2^16 Ξ΅ 0 206 k Ξ²2^2 Ξ²1 Ξ²5/4
108 0 214 Ξ΅ k Ξ²5/3
112 0 222 Ξ²2^3 Ξ²5/2
116 2v2^16 kbar 0 230 Ξ²6/2 Ξ²5
120 0 238 w Ξ· Ξ²5 Ξ²2 Ξ²1^4
124 v2^16 k^2 Ξ²2 Ξ²1 246 v2^8 Ξ½^2 Ξ²6/3 Ξ²1^2 0
128 v2^16 q 0 254 0
132 0 262 <kbar w,Ξ½,Ξ·> 0
136 <v2^16 k kbar,2,Ξ½^2> 0 270 0
140 0 278 Ξ²1
144 v2^9 0 286 Ξ²3 Ξ²1^4
148 v2^16 Ξ΅ kbar 0 294 v2^16 Ξ½^2 v2^18 0
152 Ξ²1^4 302 v2^16 k 0
156 <Ξ^6 Ξ½^2,2Ξ½,Ξ·^2> 0 310 v2^16 Ξ·^2 kbar 0
160 0 318 Ξ²1^3 0