exotic spheres and topological modular formsmbehren1/presentations/exotic_spheres.pdfexotic spheres...
TRANSCRIPT
![Page 1: Exotic spheres and topological modular formsmbehren1/presentations/Exotic_spheres.pdfExotic spheres and topological modular forms Mark Behrens (MIT) (joint with Mike Hill, Mike Hopkins,](https://reader035.vdocuments.us/reader035/viewer/2022071010/5fc89007b9fd9939a53aa813/html5/thumbnails/1.jpg)
Exotic spheres and topological modular forms
Mark Behrens (MIT)
(joint with Mike Hill, Mike Hopkins, and Mark Mahowald)
![Page 2: Exotic spheres and topological modular formsmbehren1/presentations/Exotic_spheres.pdfExotic spheres and topological modular forms Mark Behrens (MIT) (joint with Mike Hill, Mike Hopkins,](https://reader035.vdocuments.us/reader035/viewer/2022071010/5fc89007b9fd9939a53aa813/html5/thumbnails/2.jpg)
Fantastic survey of the subject:
Milnor, βDifferential topology: 46 years laterβ
(Notices of the AMS, June/July 2011)
http://www.ams.org/notices/201106/
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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)
![Page 4: Exotic spheres and topological modular formsmbehren1/presentations/Exotic_spheres.pdfExotic spheres and topological modular forms Mark Behrens (MIT) (joint with Mike Hill, Mike Hopkins,](https://reader035.vdocuments.us/reader035/viewer/2022071010/5fc89007b9fd9939a53aa813/html5/thumbnails/4.jpg)
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.
![Page 5: Exotic spheres and topological modular formsmbehren1/presentations/Exotic_spheres.pdfExotic spheres and topological modular forms Mark Behrens (MIT) (joint with Mike Hill, Mike Hopkins,](https://reader035.vdocuments.us/reader035/viewer/2022071010/5fc89007b9fd9939a53aa813/html5/thumbnails/5.jpg)
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.
![Page 6: Exotic spheres and topological modular formsmbehren1/presentations/Exotic_spheres.pdfExotic spheres and topological modular forms Mark Behrens (MIT) (joint with Mike Hill, Mike Hopkins,](https://reader035.vdocuments.us/reader035/viewer/2022071010/5fc89007b9fd9939a53aa813/html5/thumbnails/6.jpg)
Main Question
For which n do there exist exotic n-spheres?
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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
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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
![Page 9: Exotic spheres and topological modular formsmbehren1/presentations/Exotic_spheres.pdfExotic spheres and topological modular forms Mark Behrens (MIT) (joint with Mike Hill, Mike Hopkins,](https://reader035.vdocuments.us/reader035/viewer/2022071010/5fc89007b9fd9939a53aa813/html5/thumbnails/9.jpg)
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.
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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.
![Page 11: Exotic spheres and topological modular formsmbehren1/presentations/Exotic_spheres.pdfExotic spheres and topological modular forms Mark Behrens (MIT) (joint with Mike Hill, Mike Hopkins,](https://reader035.vdocuments.us/reader035/viewer/2022071010/5fc89007b9fd9939a53aa813/html5/thumbnails/11.jpg)
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.
![Page 12: Exotic spheres and topological modular formsmbehren1/presentations/Exotic_spheres.pdfExotic spheres and topological modular forms Mark Behrens (MIT) (joint with Mike Hill, Mike Hopkins,](https://reader035.vdocuments.us/reader035/viewer/2022071010/5fc89007b9fd9939a53aa813/html5/thumbnails/12.jpg)
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
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Ξπππ
β’ Trivial for n even
β’ Cyclic for n odd
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Ξπππ
β’ Trivial for n even
β’ Cyclic for n odd β Generated by boundary of an explicit parallelizable
manifold given by plumbing construction
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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
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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
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J-homomorphism
π½: ππππ β πππ β Ξ©π
ππ
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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
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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
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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
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21
Computation: Nakamura -Tangora Picture: A. Hatcher
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(ns)(5)
n
22
Computation: D. Ravenel Picture: A. Hatcher
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23
Adams spectral sequence
πΈπ₯π‘π΄π ,π‘(β€ π , β€ π) β ππ‘βπ
π π
ππ‘βπ π
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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
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25
Adams spectral sequence
πΈπ₯π‘π΄π ,π‘(β€ π , β€ π) β ππ‘βπ
π π
ππ‘βπ π
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26
Adams spectral sequence
πΈπ₯π‘π΄π ,π‘(β€ π , β€ π) β ππ‘βπ
π π
ππ‘βπ π
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Kervaire Invariant
Ξ¦πΎ: πππ β β€ 2
Browder:
(Ξ¦πΎβ 0) β π = 2π β 2
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Kervaire Invariant
Ξ¦πΎ: πππ β β€ 2
Browder:
(Ξ¦πΎ π₯ β 0) β π₯ πππ‘πππ‘ππ ππ¦ βπ2
ππ π΄ππ
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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}
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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
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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? }
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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? }
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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.
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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)
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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)
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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]
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Low dimensional computations
Conclusion
For π β€ 63, the only π for which Ξπ = 0 are: 1,2,3,4,5,6,12,61
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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
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Periodicity in πβπ
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Periodicity in πβπ
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Periodicity in πβπ
42
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Periodicity in πβπ
43
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Periodicity in πβπ
44
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Periodicity in πβπ
45
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Periodicity in πβπ
46
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Periodicity in πβπ
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Periodicity in πβπ
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Periodicity in πβπ
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Periodicity in πβπ
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Periodicity in πβπ
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Periodicity in πβπ
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Periodicity in πβπ
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Periodicity in πβπ
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Periodicity in πβπ
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Periodicity in πβπ
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(ns)(5)
v1 - periodic layer consists solely of a-family
period = 2(p-1) = 8
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(ns)(5)
v1 - periodic layer consists solely of a-family
period = 2(p-1) = 8
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Greek letter notation: the πΌ-family
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(ns)(5)
v2 - periodic layer = b-family
period = 2(p2 - 1) = 48
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(ns)(5)
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π£1-torsion in the π£2-family
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Greek Letter Names (Miller-Ravenel-Wilson) 63
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(ns)(5)
period = 2(p3 - 1) = 248
v3 - periodic layer = g-family
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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!
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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)
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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
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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
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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)
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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¦ )
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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€.
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Example: KO (real K-theory)
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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!
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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
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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.
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π»π β³πππ; πβπ‘ β π2π‘βπ πππΉ The decent spectral sequence for TMF
(p=2)
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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
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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
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Hurewicz image of TMF (p = 3)
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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
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π£2-periodicity at the prime 2
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π£2-periodicity at the prime 2
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Thm: (B-Mahowald)
The complete Hurewicz image:
The decent spectral sequence for TMF
(p=2) 84
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Hurewicz image of TMF (p = 2)
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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