Topological K-theory ( $K^{*} (X)$ )

is free abelian, with one generator for each cell. This follows by induction from the six-term exact sequence for adjoining a wedge of even-dimensional cells.

Künneth theorem for complex K-theory with an even-cell factor

 0  0

Y

is a finite even-cell complex, the exterior product is an isomorphism

K^{0} (Y) \otimes K^{i} (X) ≅ K^{i} (Y \times X) .

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Cellular induction proves this because

K^{0} (Y)

is free and

K^{- 1} (Y) = 0

Complex K-theory of complex projective space ( $K^{0} (CP^{n})$ )

 0  0

t = 1 - [\overline{γ}]

for the tautological complex line bundle, then

K^{0} (CP^{n}) ≅ Z [t] / (t^{n + 1}), K^{- 1} (CP^{n}) = 0.

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K-theory Wang sequence of a mapping torus

 0  0

For a self-map

f : Z \to Z

, the mapping torus has an exact sequence containing

K^{- 1} (Z) 1 - f^{*} K^{- 1} (Z) ⟶ K^{0} (T_{f}) ⟶ K^{0} (Z) 1 - f^{*} K^{0} (Z) .

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K^{- 1} (Z) = 0

, then

K^{0} (T_{f}) ≅ ker (1 - f^{*} : K^{0} (Z) \to K^{0} (Z))

K-theory of the mapping torus of the factor swap on two complex projective planes ( $K^{0} (T_{swap})$ )

 0  0

For the factor swap on

CP^{2} \times CP^{2}

, the invariant subgroup of

Z [x, y] / (x^{3}, y^{3})

(1)

has basis

1, x y, x^{2} y^{2}, x + y, x^{2} + y^{2}, x y^{2} + x^{2} y

. Hence the mapping torus has

K^{0} ≅ Z^{6}

Splitting principle for complex vector bundles

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For a complex vector bundle

E \to X

, there is a map

p : F (E) \to X

such that

p^{*}

is injective on cohomology and

p^{*} E

splits as a sum of line bundles. Symmetric identities in the Chern roots can therefore be proved after this pullback.

Chern root ( $x_{i}$ )

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If a pulled-back complex vector bundle splits as

L_{1} \oplus \dots \oplus L_{r}

, its formal Chern roots are

x_{i} = c_{1} (L_{i})

. The Chern classes are the elementary symmetric polynomials in these roots.

Chern character ( $ch : K^{0} (X) \to H^{ev} (X; Q)$ )

 0  0

If the formal Chern roots of

E

are

x_{1}, \dots, x_{r}

, then

ch (E) = \sum_{i = 1}^{r} e^{x_{i}} .

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It extends to virtual bundles and is a ring homomorphism because direct sum joins root lists while tensor product replaces them by all sums

x_{i} + y_{j}

Chern character on an even-dimensional sphere is integral

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Under the natural identification

H^{ev} (S^{2 n}; Z) \subset H^{ev} (S^{2 n}; Q),

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the Chern character maps

K^{0} (S^{2 n})

into integral cohomology. A Bott generator is an exterior product of

n

degree-two Bott elements, whose Chern characters multiply to an integral top-dimensional generator.

Divisibility of the top Chern number on an even-dimensional sphere

 0  0

For a complex vector bundle

E \to S^{2 n}

⟨ c_{n} (E), [S^{2 n}] ⟩

(1)

is divisible by

(n - 1)!

. Since the lower Chern classes vanish, the Newton identity gives

ch_{n} (E) = (- 1)^{n + 1} \frac{c _{n} ( E )}{( n - 1 )!},

(2)

and the Chern character on an even-dimensional sphere is integral.

K-theory Thom class ( $λ_{E}$ )

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For a complex vector bundle

E \to X

, the K-theory Thom class

λ_{E} \in K^{0} (Th (E))

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generates the Thom isomorphism

K^{i} (X) ≅ K^{i} (Th (E))

K-theory Euler class ( $e^{K} (E) = Λ_{- 1} (\overline{E})$ )

 0  0

Pulling the K-theory Thom class back along the zero section gives

e^{K} (E) = Λ_{- 1} (\overline{E}) = \sum_{j} (- 1)^{j} [Λ^{j} \overline{E}] .

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K-theory Gysin sequence of a sphere bundle

 0  0

The cofibration

S (E) \to D (E) \to Th (E)

and the Thom isomorphism give

\dots \to K^{i} (X) \cdot e^{K} (E) K^{i} (X) p^{*} K^{i} (S (E)) p_{!} K^{i + 1} (X) \to \dots .

(1)

Odd K-theory of the sphere bundle of two tautological lines ( $K^{- 1} (S (γ \oplus γ))$ )

 0  0

For

E = γ \oplus γ

over

CP^{n}

and

t = 1 - [\overline{γ}]

, the K-theory Euler class is

t^{2}

. Hence

K^{- 1} (S (E)) ≅ ker (t^{2} : Z [t] / (t^{n + 1}) \to Z [t] / (t^{n + 1})) = Z {t^{n - 1}, t^{n}}

(1)

for

n \geq 1

For

n = 0

, the base is a point, the sphere bundle is

S^{3}

, and its odd K-theory is

Z

Adams operation ( $ψ^{k}$ )

 1  0

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

ψ^{k} ([L]) = [L]^{k} .

(1)

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)

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Topological K-theory by

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Topological K-theory is a branch of mathematics that studies vector bundles over topological spaces through the lens of homotopy theory. It arises in both algebraic topology and functional analysis and is a fundamental concept in modern mathematics, bridging several areas, including geometry, representation theory, and mathematical physics. The main idea behind K-theory is to classify vector bundles (or more generally, modules over topological spaces) up to stable isomorphism.

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