<k (S k ) k (S k ) >k (S k ) Infinite subgroups completely - PowerPoint PPT Presentation

 <k (S k )  k (S k )  >k (S k )

Infinite subgroups completely understood

Values stabilize along diagonals:  n+k (S k ) =  n+k+1 (S k+1 ) for k >> 0

Stable homotopy groups: s := lim  n+k (S k ) k   n Primary decomposition: s =  n (  n  3 s ) (p) s = Z 24 = z 8 + Z 3 e.g.: p prime

Computation: Mahowald-Tangora-Kochman Picture: A. Hatcher • Each dot represents a factor of 2, vertical lines indicate additive extensions 𝑡 ) (2) = ℤ 8 , 𝑡 ) (2) = ℤ 2 ⨁ℤ 2 (𝜌 3 (𝜌 8 e.g.: • Vertical arrangement of dots is arbitrary, but meant to suggest patterns 5

Computation: Nakamura -Tangora Picture: A. Hatcher 6

(  n Computation: D. Ravenel s ) (5) Picture: A. Hatcher n • Each dot represents a factor of p (  39 s ) (5) = Z 25 e.g.: • Vertical arrangement of dots is arbitrary, but meant to suggest patterns

• Each dot represents a factor of 2, vertical lines indicate additive extensions 𝑡 ) (2) = ℤ 8 , 𝑡 ) (2) = ℤ 2 ⨁ℤ 2 (𝜌 3 (𝜌 8 e.g.: • Vertical arrangement of dots is arbitrary, but meant to suggest patterns 8

Computation: Nakamura -Tangora Picture: A. Hatcher 9

(  n Computation: D. Ravenel s ) (5) Picture: A. Hatcher n 10

(  n s ) (5) v 1 - periodic layer period = 2(p-1) = 8 11

(  n s ) (5) v 2 - periodic layer period = 2(p 2 - 1) = 48 12

(  n s ) (5) v 3 - periodic layer period = 2(p 3 - 1) = 248 13

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15 Example: KO (real K-theory)

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Hurewicz image of TMF (p = 2) 17