Adams operation ( $ψ^{k}$ )

The Adams operations are natural ring endomorphisms of complex K-theory characterized on line bundles by

ψ^{k} ([L]) = [L]^{k} .

(1)

Table of contents
- Adams operation on a Bott class Adams operation
  - Adams-operation obstruction to retracting a truncated complex projective space Adams operation on a Bott class
- Cannibalistic class Adams operation
  - Adams operation and the boundary pushforward of a sphere bundle Cannibalistic class
    - Second Adams operation on the odd K-theory of the sphere bundle of two tautological lines Adams operation and the boundary pushforward of a sphere bundle

Adams operation on a Bott class

 0  0

u \in K^{0} (S^{2 m})

is a Bott element generator, then

ψ^{q} (u) = q^{m} u .

(1)

Adams-operation obstruction to retracting a truncated complex projective space

 0  0

Let

CP_{k}^{k + n} = CP^{k + n} / CP^{k - 1}

. If the bottom-cell inclusion

S^{2 k} = CP_{k}^{k} \to CP_{k}^{k + 2}

admits a retraction, then

24

divides

k

. Writing

x = [\overline{γ}] - 1

, a retracted Bott generator has the form

x^{k} + a x^{k + 1} + b x^{k + 2}

. Comparing it with its image under

ψ^{2} (x) = 2 x + x^{2}

gives

a = - k /2

and

b = k (3 k + 5) /24

; integrality forces both

3

and

8

to divide

k

Cannibalistic class ( $ρ^{k} (E)$ )

 0  0

The cannibalistic class is defined by

ψ^{k} (λ_{E}) = ρ^{k} (E) λ_{E} .

(1)

It satisfies

ρ^{k} (E \oplus F) = ρ^{k} (E) ρ^{k} (F)

. For a line bundle,

ρ^{k} (L) = \frac{1 - L ^{k}}{1 - L} = 1 + \overline{L} + \dots + \overline{L}^{k - 1} .

(2)

Adams operation and the boundary pushforward of a sphere bundle

 0  0

For

p_{!} : K^{- 1} (S (E)) \to K^{0} (X)

in the K-theory Gysin sequence,

p_{!} (ψ^{k} x) = ρ^{k} (E) ψ^{k} (p_{!} x) .

(1)

This follows by applying

ψ^{k}

δ x = λ_{E} p_{!} (x)

Second Adams operation on the odd K-theory of the sphere bundle of two tautological lines ( $ψ^{2} : K^{- 1} (S (γ \oplus γ)) \to K^{- 1} (S (γ \oplus γ))$ )

 0  0

Choose

a, b

with

p_{!} (a) = t^{n - 1}

and

p_{!} (b) = t^{n}

, where

t = 1 - [\overline{γ}]

. Then

ψ^{2} (a) = 2^{n + 1} a - (n + 1) 2^{n} b, ψ^{2} (b) = 2^{n + 2} b .

(1)

 Ancestors (6)

 Incoming links (2)

 View article source

 Discussion (0)

New discussion

There are no discussions about this article yet.

 Articles by others on the same topic (1)

Adams operation by

Wikipedia Bot  1

 View more

Adams operation is a concept from the field of algebra, specifically in the context of homotopy theory and stable homotopy theory. It is named after the mathematician Frank Adams, who introduced it while studying stable homotopy groups of spheres. In more detail, Adams operations are a family of operations on the ring of stable homotopy groups of spheres, which can be linked to the concept of formal group laws.

 Read the full article

  See all articles in the same topic Create my own version

 Discussion (0)  Subscribe (1)

 Discussion (0)