K-theory

K-theory

Topological K-theory is a generalised

cohomology

theory on the

category

of

compact

Hausdorff spaces

.
It classifies the

vector bundles

over a space

X

up to

stable equivalences

.
Equivalently, via the

Serre-Swan theorem

, it classifies the

finitely generated projective modules

over the

C

*

-

algebra

C

⁢

(

X

)

.

Let

A

be a unital

C

*

-algebra over

ℂ

and denote by

M

∞

⁢

(

A

)

the algebraic

direct limit

of matrix algebras

M

n

⁢

(

A

)

under the

embeddings

M

n

⁢

(

A

)

→

M

n

+

1

⁢

(

A

)

:

a

↦

(

a

0

0

0

)

.
Identify the

completion

of

M

∞

⁢

(

A

)

with the stable algebra

A

⊗

𝕂

(where

𝕂

is the

compact operators

on

l

2

⁢

(

ℕ

)

),
which we will continue to denote by

M

∞

⁢

(

A

)

.
The

K

0

⁢

(

A

)

group is the

Grothendieck group

(

abelian group

of formal

differences

) of the

homotopy classes

of the

projections

in

M

∞

⁢

(

A

)

.
Two projections

p

and

q

are

homotopic

if there exists a norm

continuous

path of projections from

p

to

q

.
Let

p

∈

M

m

⁢

(

A

)

and

q

∈

M

n

⁢

(

A

)

be two projections.
The sum of their homotopy classes

[

p

]

and

[

q

]

is the homotopy class of their

direct sum

:

[

p

]

+

[

q

]

=

[

p

⊕

q

]

where

p

⊕

q

=

diag

⁢

(

p

,

q

)

∈

M

m

+

n

⁢

(

A

)

.
Alternatively, one can consider

equivalence classes

of projections up to

unitary transformations

.

Unitary

equivalence

coincides with

homotopy equivalence

in

M

∞

⁢

(

A

)

(or

M

n

⁢

(

A

)

for

n

large enough).

Denote by

U

∞

⁢

(

A

)

the direct limit of

unitary groups

U

n

⁢

(

A

)

under the embeddings

U

n

⁢

(

A

)

→

U

n

+

1

⁢

(

A

)

:

u

↦

(

u

0

0

1

)

.
Give

U

∞

⁢

(

A

)

the direct limit

topology

, i.e. a subset

U

of

U

∞

⁢

(

A

)

is open if and only if

U

∩

U

n

⁢

(

A

)

is an

open subset

of

U

n

⁢

(

A

)

, for all

n

.
The

K

1

⁢

(

A

)

group is the Grothendieck group (abelian group of formal differences) of the homotopy classes of the unitaries in

U

∞

⁢

(

A

)

.
Two unitaries

u

and

v

are homotopic if there exists a norm continuous path of unitaries from

u

to

v

.
Let

u

∈

U

m

⁢

(

A

)

and

v

∈

U

n

⁢

(

A

)

be two unitaries.
The sum of their homotopy classes

[

u

]

and

[

v

]

is the homotopy class of their direct sum:

[

u

]

+

[

v

]

=

[

u

⊕

v

]

where

u

⊕

v

=

diag

⁢

(

u

,

v

)

∈

U

m

+

n

⁢

(

A

)

.
Equivalently, one can work with

invertibles

in

GL

∞

⁢

(

A

)

(an invertible

g

is

connected

to the unitary

u

=

g

⁢

|

g

|

-

1

via the

homotopy

t

→

g

⁢

|

g

|

-

t

).

Higher K-groups can be defined through repeated

suspensions

,

K

n

⁢

(

A

)

=

K

0

⁢

(

S

n

⁢

A

)

.

(1)

But, the Bott periodicity theorem means that

K

1

⁢

(

S

⁢

A

)

≅

K

0

⁢

(

A

)

.

(2)

The main properties of

K

i

are:

K

i

⁢

(

A

⊕

B

)

=

K

i

⁢

(

A

)

⊕

K

i

⁢

(

B

)

,

(3)

K

i

⁢

(

M

n

⁢

(

A

)

)

=

K

i

⁢

(

A

)

(Morita invariance)

,

(4)

K

i

⁢

(

A

⊗

𝕂

)

=

K

i

⁢

(

A

)

(stability)

,

(5)

K

i

+

2

⁢

(

A

)

=

K

i

⁢

(

A

)

(Bott periodicity)

.

(6)

There are three flavours of topological K-theory to handle the cases
of

A

being complex (over

ℂ

), real (over

ℝ

) or Real
(with a given real structure).

K

i

⁢

(

C

⁢

(

X

,

ℂ

)

)

=

𝐾𝑈

-

i

⁢

(

X

)

(complex/unitary)

,

(7)

K

i

⁢

(

C

⁢

(

X

,

ℝ

)

)

=

𝐾𝑂

-

i

⁢

(

X

)

(real/orthogonal)

,

(8)

𝐾𝑅

i

⁢

(

C

⁢

(

X

)

,

J

)

=

𝐾𝑅

-

i

⁢

(

X

,

J

)

(Real)

.

(9)

Real

K-theory

has a Bott period of 8, rather than 2.

References

1

N. E. Wegge-Olsen,

K-theory and

C

*

-algebras

.

Oxford science publications. Oxford University Press, 1993.

2

B. Blackadar,

K-Theory for Operator Algebras

.

Cambridge University Press, 2nd ed., 1998.

3

M. Rørdam, F. Larsen and N. J. Laustsen,

An

Introduction

to K-Theory for

C

*

-Algebras

.

Cambridge University Press, 2000.

Title

K-theory

Canonical name

Ktheory

Date of creation

2013-03-22 12:58:06

Last modified on

2013-03-22 12:58:06

Owner

mhale (572)

Last modified by

mhale (572)

Numerical id

17

Author

mhale (572)

Entry type

Topic

Classification

msc 19-00

Synonym

Topological K-theory

Related topic

KHomology

Related topic

AlgebraicKTheory

Related topic

GrothendieckGroup

Generated on Fri Feb 9 19:14:35 2018 by

LaTeXML