SLIDE 1
<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 - - 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
SLIDE 2
SLIDE 3
Values stabilize along diagonals: n+k(Sk) = n+k+1(Sk+1) for k >> 0
SLIDE 4
Stable homotopy groups: n
s := lim n+k(Sk) k
Primary decomposition: n
s =
(n
s)(p)
e.g.: 3
s = Z24 = z8 + Z3
p prime
SLIDE 5
- Each dot represents a factor of 2, vertical lines indicate additive extensions
e.g.: (𝜌3
𝑡)(2) = ℤ8,
(𝜌8
𝑡)(2) = ℤ2⨁ℤ2
- Vertical arrangement of dots is arbitrary, but meant to suggest patterns
5
Computation: Mahowald-Tangora-Kochman Picture: A. Hatcher
SLIDE 6
6
Computation: Nakamura -Tangora Picture: A. Hatcher
SLIDE 7
(n
s)(5)
n Computation: D. Ravenel Picture: A. Hatcher
- Each dot represents a factor of p
e.g.: (39
s)(5) = Z25
- Vertical arrangement of dots is arbitrary, but meant to suggest patterns
SLIDE 8
- Each dot represents a factor of 2, vertical lines indicate additive extensions
e.g.: (𝜌3
𝑡)(2) = ℤ8,
(𝜌8
𝑡)(2) = ℤ2⨁ℤ2
- Vertical arrangement of dots is arbitrary, but meant to suggest patterns
8
SLIDE 9
9
Computation: Nakamura -Tangora Picture: A. Hatcher
SLIDE 10
(n
s)(5)
n
10
Computation: D. Ravenel Picture: A. Hatcher
SLIDE 11
(n
s)(5)
v1 - periodic layer
period = 2(p-1) = 8
11
SLIDE 12
(n
s)(5)
v2 - periodic layer
period = 2(p2 - 1) = 48
12
SLIDE 13
(n
s)(5)
period = 2(p3 - 1) = 248
v3 - periodic layer
13
SLIDE 14
14
SLIDE 15
Example: KO (real K-theory)
15
SLIDE 16
16
SLIDE 17
Hurewicz image of TMF (p = 2)
17