group cohomology (topological definition)

group cohomology (topological definition)

Let

G

be a

topological group

. Suppose some contractible space

X

admits a

fixed point

free action

of

G

, so that the

quotient map

p

:

X

→

X

/

G

is a

fibre map

. Then

X

/

G

, denoted

B

⁢

G

is called the

classifying space

of

G

. Classifying spaces always exist and are unique up to

homotopy

. Further, if

G

has the structure of a CW- complex, we can choose

B

⁢

G

to have one too.

The group (co)

homology

of

G

is defined to be the (co)homology of

B

⁢

G

. From the long-exact

sequence

associated to the fibre map,

p

, we know that

π

n

⁢

(

G

)

=

π

n

+

1

⁢

(

B

⁢

G

)

for

n

≥

0

. In particular the

fundamental group

of

B

⁢

G

is

π

0

⁢

(

G

)

, which inherits a group structure as a

quotient

of

G

. Let

H

denote

π

0

⁢

(

G

)

. Then

H

acts freely on the cells of

B

⁢

G

*

, the

universal

over of

B

⁢

G

. Hence the cellular resolution for

B

⁢

G

*

, denoted,

C

*

⁢

(

B

⁢

G

*

)

, is a sequence of free

Z

⁢

H

- modules and

Z

⁢

H

- linear maps. Taking coefficients in some

Z

⁢

H

- module

A

, we have

H

n

⁢

(

G

;

A

)

=

H

n

⁢

(

C

*

⁢

(

B

⁢

G

*

)

;

A

)

⁢

and

⁢

H

n

⁢

(

G

;

A

)

=

H

n

⁢

(

C

*

⁢

(

BG

*

)

;

A

)

In particular, when

G

is

discrete

,

p

must be the covering map associated to a

universal cover

. Hence

X

=

B

⁢

G

*

and

C

*

⁢

(

B

⁢

G

*

)

is exact, as

X

is

contractible

and hence has trivial homology. Note in this case

H

=

G

. So for a

discrete group

G

, we have,

H

n

⁢

(

G

;

A

)

=

E

⁢

x

⁢

t

Z

⁢

G

n

⁢

(

Z

,

A

)

⁢

and

⁢

H

n

⁢

(

G

;

A

)

=

Tor

ZG

n

⁢

(

Z

,

A

)

Also, as passing to the universal cover

preserves

π

n

for

n

>

1

, we know that

π

n

⁢

(

B

⁢

G

)

=

0

for

n

>

1

.

B

⁢

G

is always

connected

and for a discrete group

π

0

⁢

(

G

)

=

G

so we have

B

⁢

G

=

k

⁢

(

G

,

1

)

, the Eilenberg - Maclane space.

As an example take

G

=

S

⁢

U

1

. Note topologically,

S

⁢

U

1

=

S

1

=

k

⁢

(

Z

,

1

)

. As

π

n

⁢

(

G

)

=

π

n

+

1

⁢

(

B

⁢

G

)

for

n

≥

0

, we know that

B

⁢

S

⁢

U

1

=

k

⁢

(

Z

,

2

)

=

C

⁢

P

∞

.

More explicitly, we may identify

S

⁢

U

1

with the unit

complex numbers

. This acts freely on the infinite complex sphere (which is contractible) leaving a quotient of

C

⁢

P

∞

.

Hence

H

n

⁢

(

S

⁢

U

1

,

Z

)

=

Z

if

2

divides

n

and

0

otherwise.

Similiarly

B

⁢

C

2

=

R

⁢

P

∞

and

B

⁢

S

⁢

U

2

=

H

⁢

P

∞

, as

C

2

and

S

⁢

U

2

are

isomorphic

to U(R) and U(H) respectively. So

H

n

⁢

(

C

2

,

Z

2

)

=

Z

2

for all

n

and

H

n

⁢

(

S

⁢

U

2

,

Z

)

=

Z

if

4

divides

n

and

0

otherwise.

Title

group cohomology

(topological

definition

)

Canonical

name

GroupCohomologytopologicalDefinition

Date of creation

2013-03-22 14:32:24

Last modified on

2013-03-22 14:32:24

Owner

whm22 (2009)

Last modified by

whm22 (2009)

Numerical id

18

Author

whm22 (2009)

Entry type

Definition

Classification

msc 55N25

Related topic

CohomologyGroupTheorem

Defines

group cohomology

Defines

classifying spaces

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