Exotic spheres and topological modular forms Mark Behrens
Exotic spheres and topological modular forms Mark Behrens (MIT) (joint with Mike Hill, Mike Hopkins, and Mark Mahowald)
Fantastic survey of the subject: Milnor, “Differential topology: 46 years later” (Notices of the AMS, June/July 2011) http: //www. ams. org/notices/201106/
Poincaré Conjecture •
Smooth Poincaré Conjecture •
Smooth Poincaré Conjecture •
Main Question For which n do there exist exotic n-spheres?
Kervaire-Milnor •
Kervaire-Milnor •
Kervaire-Milnor •
Kervaire-Milnor •
Kervaire-Milnor •
Kervaire-Milnor •
• Trivial for n even • Cyclic for n odd
• Trivial for n even • Cyclic for n odd – Generated by boundary of an explicit parallelizable manifold given by plumbing construction
J-homomorphism •
J-homomorphism •
Computation: Mahowald-Tangora-Kochman Picture: A. Hatcher 19
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Computation: Nakamura -Tangora Picture: A. Hatcher 21
( ns)(5) Computation: D. Ravenel Picture: A. Hatcher n 22
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Kervaire Invariant • 27
Kervaire Invariant • 28
Kervaire Invariant • 29
Kervaire Invariant • 30
Summary: Exotic spheres • 31
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Summary: Exotic spheres • 33
Low dimensional computations • 34
Low dimensional computations 35
Low dimensional computations 36
Low dimensional computations 37
Low dimensional computations • 38
Beyond low dimensions… • 39
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( n s) (5) v 1 - periodic layer consists solely of a-family period = 2(p-1) = 8 57
( n s) (5) v 1 - periodic layer consists solely of a-family period = 2(p-1) = 8 58
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( n s) (5) v 2 - periodic layer = b-family period = 2(p 2 - 1) = 48 60
( ns)(5) 61
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Greek Letter Names (Miller-Ravenel-Wilson) 63
( n s) (5) v 3 - periodic layer = g-family period = 2(p 3 - 1) = 248 64
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Cohomology theories • 69
Cohomology theories • 70
Cohomology theories • 71
Hurewicz Homomorphism • 72
73 Example: KO (real K-theory)
Topological Modular Forms KO TMF • • 75
Topological Modular Forms • 76
The decent spectral sequence for TMF (p=2) 77
Hurewicz image of TMF (p = 3) 80
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The decent spectral sequence for TMF (p=2) Thm: (B-Mahowald) The complete Hurewicz image: 84
85 Hurewicz image of TMF (p = 2)
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- Slides: 86