COHOMOLOGYOFTOPOLOGICALGRAEMEGROUPS (*)

SEGAL

The following work was done jointly by G. J. Mitchison and me.
Its purpose is to define cohomology groups of a topological group G
withcoefficientsin a topologicalabeliangroupA(on whichG

actscontinuously) by a method analogous to defining the cohomology of adiscrete group as a derived functor. I shall begin by introducing acategory of topological abelian groups.

§ 1. Let Topab denote the category of compactly generated andlocally contractible (*) hausdorff topological abelian groups and continuousiphomomorphisms. A sequence A’——> A ——> A” in Topab will be calleda short exact sequence ifi) i is an imbedding of A’ as a closed subgroup of A;ii) p induces a topological isomorphism A JA! —sA”;iii) A’ has a local cross-section in A (i.e. the fibration A"—>A-—>A"
is locally trivial topologically).

From short exact sequences one can build up longer exact sequencesin the usual way.

(*) I risultati

contenutiin questo

lavorosono

in thecategory

stati esposti

nefla

conferenza

tenuta il 26 marzo 1969.
() I meanan abelian

group

A

of

k-spaces

[2].

Thus

the

composition A X A — A is supposed to be continuous when the product is understood in the sense of k-spaces. The condition of local contractibility is inserted
only to ensure that each object has a contractible resolution ((1.1) below); in fact
Topab can be embedded in a larger abelian category with enough injectives, andthe restriction is unnecessary.

25

378

GraemeSegal

ProrosiTION 1.1. Any object A of Topab has a contractible resolution, i.e. there is an exact sequence
0—->A—>A"—
Alo A2— ...
such that each A’ is contractible as a topological space.
PROOF.

A suitable resolution is, in the notation of the appendix:
A—>EA— EBA — EBA — ...

A left-exact

additivefunctor

F: Topab— Ab, whereAb

is thecategory of abelian groups, will be called derivable if, for each short

exact sequence A'—>A—A”
the sequencein Topabfor which A" is contractible,

0— F(A") => F(A)—> F(A")— 0is exact. If F is such a functor one can define its derived functors R°F
for p=0 by RPF(A)=H"(F(A")),lution of A. This is justified by

ProrosiTiON

1.2.

where

A— A" is a coniractible reso-

If F is derivable then R’F(A) does not dependon A".
Proor:Let E'4 be the canonical contractible resolution constructedabove. Consider the double complex
Ae

A —

A! —

E)—>

Ep

EY, >

——>

vey

in which both the rows and the columns are exact (because the functors

E and B are exact, as explained in the appendix), and all the objects
except those in the first row are contractible. When F is applied to thiscomplex the rows and the columns, except for the first in each case,remain acyclic because F is derivable, so the cohomology of the firstrow is isomorphic to that of the first column, i.e. R?F(A) defined using
A—>A"—> Al'— ... is the same as R?F(A) defined using the canonicalresolution

A => E

4.

Given the exactness of A — Ey, it is clear that we have

Cohomology

of topological groups

379

ProrosITION 1.3. If F: Topab —> Ab is derivable,and A"—>A— A"
is a short exact sequence in Topab, then there is an infinite exact sequence

0— F(A") — F(A)—> F(A") —>—> R'F(A") — R'F(A) — RIF(A")—> R* F(A") — ...

ExaMprLE1.4.

Let X be a paracompact space. Then the functor

A> Map (X; A), where Map (X; A) denotes the space of continuous
maps of X into A, is derivable, and R? Map (X; A)=H"(X, A), thecohomology of X with coefficients in the sheaf of germs of continuousfunctions from X into A.
Proor:
If A"—>A—>A" is a short exact sequence in Topabwith A’ contractible then A==A’ > A” as topological space, for a locallytrivial principal fibration with a contractible fibre is trivial (*). So
Map (X; A) — Map (X; A”) is surjective, and the functor is derivable.
From the short exact sequence A — EA — BA one obtains an exact

sequence
Map (X; EA) — Map (X; BA)— R!' Map (X; A)— 0.
This shows that R! Map (X; A)=[X;BA], the set of homotopy-classesof maps from X to BA, for a map X —> BA is null-homotopic if and only
if it lifts to EA. But BA is a classifying-space for locally trivial principal
A-bundles on paracompact spaces, so[X; BA]=H'(X; A),andR'Map (X;

A)=H (X; A).

Finally, one sees that R? Map(X; A)==H"(X; A) for all A by
induction on p. For R" Map (X; A)=R"~!Map (X; BA) if p=2; andH?(X; A)=H""!(X; BA) if p=2 because EA is a soft sheaf [3].

§ 2. Let G be a compactly generated topological group, and let
G-Topab be the category of compactly generated locally contractible
topological abelian groups A on which G acts continuously. A sequence(3) See Appendix

B.

380

GraemeSegal

A’—>A—>A” in G-Topab will be a short exact sequence if it is one
in Topab when the G-actions are neglected.
Call an object of G-Topab

soft if it is of the form Map (G; A),

where A is a contractible group and Map denotes the set of continuous
maps with the compact-open topology. (Thus a soft group is contractible).
And call a left-exact additive functor F: Topab — Ab derivable if foreach short exact sequence A'—>A—>A" with A” soft the sequence
0— F(A") —> F(A) —> F(A’")—> 0 is exact. As before one defines thederived functors RF by RPF(A)=H"(F(A")) were A—>A’ is a softresolution of A. This is justified in virtue of the following two propositions, analogous to (1.1) and (1.2).

ProrosrtioN 2.1.
Proor:

Each object A of G-Topab has a soft resolution.

Let E()yA= Map (G; EA). Define a G-embedding

A — E)A

by composing the embedding of A in EA with that of EA in Map (G; EA)
as the constant functions. Then A —> E()A

has a local cross-section, so

it leads to a short exact sequence A — EgA — BgA in G-Topab. A suitable resolution of A is

ProrosITION 2.2.

A — E()A— E(¢)B()A—> E()BigyA — ...

If F: G-Topab— Ab is derivable then RPF(A)is independent of the soft resolution of A used to define it.
Proor:To see that the proof of (1.2)

suffices to show that the functors Es)

and Bs)appliesin this case it

are exact. In the caseof E(w), this is true because if A’—>
A — A” is a short exact sequencein G-Topab then in the induced sequence EA" —> EA — EA” there is a
global cross-section of the fibration EA — EA” (the group EA being
contractible), and so there is an induced global cross-section of the
fibration E(¢yA= Map (G; EA) — E)A” = Map(G; EA"). In the caseof Bs), a local cross-section of B(eyA —> B(e)A” is given by the composi-tion p-s2-81, where s; is a local cross-section of E(gyA”—> B(s)A”, sz isa cross-section
projection.

of

EA—>Ew)A”,

and

p: EA

—> Bi)AAb

derivable,

1s

the

And of course we have
ProrositioN

A"—>A—

2.3.

If

F: G-Topab—

is

and

A” is a short exact sequence in G-Topab, then there is an

infinite exact sequence
0—> F(A")= F(A) F(A”) => R'F(A)— ....

Cohomology of topological groups

381

ExamrLE 2.4. The functor A > I'®4 which associates to an objectof G-Topab its G-invariant subgroup is derivable.
Proor:Let A”—> A—> A” be a short exact sequence in G-Topabwith A’= Map (G; As). We must show that TA — T°A” is surjective.
Let yeT€A" cA”, and let xe A be such that x+>y. Then, identifying
A’ with a subgroup of A, g-x—x is an element ©(g) of A” for each geG.
It is casily seen that ¢ : G— A’ is a crossed homomorphism G):

let its

adjoint map be ¢: GX G—> Ao. Let € be the element of A’ definedby g—>4Y(g™!, 1). Then (g-£E—&E)eA’ is the map
Y => (yg,1)—d(y—t,

1).

But

U(y~1g, =r!

D=d(y~, D+y; Wg, D—-d(y~!, D=U(g, v)

because ¢ is a crossed homomorphism. That is, g-£—E=g-x—x for all
ge€@G, and so x—£ is an element of A which maps to y.

§ 3. The cohomology of a discrete group can be calculated froma well-known cochain complex. One can define an analogous complex ofcontinuouscochains

C'¢(A)

in the topological case:one sets Cef'(A)== Map (GX... X G; A), and defines d : Cs”(A)—> C5*Y(A) by the
Cpls]

classical formula
pL1
df(g

,

vaey

grs)=

XZ

.
(—=1)fe0

sony

Bly

wey

Sps1).

A is embedded in C¢’(A)= Map (G; A) as the constant maps. Then
A—> Cg(A) is a resolution of A in G-Topab, for there are (non-equivariant) contracting homotopies £2: C4"1(A) — C6”(A) defined by
hf(go

,

teoy

gp) =f(1,

80s

«np

gp).

When A is a contractible group the groups Cg¢?(A) are soft, so wehave() Le. 9(g1g,) = olg,) +g;+ 9(g,) for all g;, g,eG.

382

GraemeProrositioN

3.1.

Segal

If A is contractible then R’T'6(A)can be cal-culated from the complex of continuous cochains.
In general this will not be true, but there is a spectral sequence
relating the cohomologyof the continuouscochains to the cohomologyof G. In fact if A— A’ is a soft resolution of A then from the double
complex T°C'c(A’) we obtain
ProrosiTioN=HY(G X

3.2.

There

is

a

spectralsequence

with

FE{?=

ho X G; A) converging to RTCA).

Proor.

For each gq the complex T'°A?— I'°C'¢(A?) is acyclic, so

the total cohomology of T°C'¢(A") is R'T°(A). On the other hand for anyp>p the complex T°C¢”(A") can be identified with Map (G X

and

we

have

seen

in

(1.4)

that

the

. X

cohomologyofG; A),

this

is

Cp

HYG X ... X Gj A).
REMARKS.
i) Notice that EP=C¢"(A), so that the edgehomomorphism
EP — RTA) is a natural transformation from the cohomology of thecontinuous cochains to the cohomology of G.
il) If, for example, A is discrete and

H(G X

...

XG; A)= Hom (H(G)Q

H, (G) is torsion-free then... QH(G);A)

and one has the spectral sequence of Eilenberg-Moore-etc.
Extn?) (£4;A) pel RTCA).

To an object A of G-Topab one can associate a sheaf of abeliangroups oA on the space BG by defining cA(U)= Map® (p~U; A), where
U is an open set of BG and p: EG — BG is the projection. This gives
a functor ¢: G-Topab — Sh(BG), where Sh(BG) is the abelian category ofsheaves on BG. It is easy to see that o takes exact sequences to exact

sequences, and soft objects to soft sheaves. So from the natural transformation

IA — HBG;

R’T°A— HP(BG;oA)

oneoA) for all p=0.

derivesnaturaltransformations

One has a transformation

spectral sequence of (3.2) into that of the double complex

Ex*=HYBG;o Map (G"'!; AY).

of theCohomology

of topological groups

383

But

HBG;

o Map (G**;A7))= Map® (EG;Map (G**';A9))== Map (EG X G*; A9),oF =HYEG X G?; A) by (2.4). If A is discrete this spectral sequence
coincides at the Ej-level with that of (3.2), because EG is contractible;
and one deduces

ProrosITION 3.3.
§ 4.
dimensions

If A is discrete then RT9(A)=H*(BG; cA).

To conclude, I shall show
1 and 2 havethat the cohomology

their usual interpretations.

groups in

Let Hom (G;A)

denote the abelian group of crossed homomorphisms from G to A modulo
principal crossed homomorphisms (i.e. ones of the form g+> g-¢—a forsome aA). And let Ext (G; A) denote the set of isomorphism-classesof extensions A — E —> G which have a local cross-section. Ext (G; A)
is a contravariant functor in G, because G'— G induces E = E X ¢G’;and a covariant functor in A, because A — A’ induces E> (E X A") /A.

The composition-law
Ext (G;A) X Ext (G;A) —>Ext(G;given by (Ei, E))—>E; X gE., together
Ext (G; A) into an abelian group.

A X A)withA X A— A,

makes

ProrosiTioN 4.1. If A"—>A—>A” is a short exact sequence in
G-Topab then there is an exact sequence
0—->TCA" >T%4 ->T%4” —- Hom (G; A) —> Hom (G; A)——> Hom (G;Proor.
g+>g-x—x,A")=> Ext (G; A)— Ext (G;A)
— Ext (G; A”).

The first d assigns to yeI'®A” the crossed homomorphism
where

x€éA

has image y in A”.

a crossed homomorphism

G +> A”

The

the extension

second d assigns to

G X aA.

To verify

exactness Is trivial,
ProrosiTiON 4.3.

If A is soft then Hom (G; A)= Ext (G; A)=0.

Proor.
Let A= Map (G; Ao). Given a crossed homomorphism
f:G— A, let its adjointbe f: G X G—> Ap. Then a trivial calculation

384

GraemeSegal(done in the proof of (2.4)) shows that f(g)=g-p—o, where ¢(g)==f(g~!, 1). Thus f is principal and Hom (G; A)=0.
Let A— E— G be an extension. Because A is contractible there
is a continuous cross-section ¢ : G—> E. Define a«: G X G~> A so that

P(81, B2)=0(g1, 8)p(g1)p(g),and let its adjoint be a: GX GX G—>A, Then if B: G—>A has
adjoint B: GX

G— A, where

Ble, g)=g-alg’ g, 1),a calculation shows that g> 3(g)o(g) is a splitting of the extension E.
Hence Ext (G; A)=0.
PROPOSITION 4.3.

RT¢(A)=Hom(G;A), and RT°(A)=Ext(G; A).

Proor.
Because of the exact sequence A —> E(g)A — B(s)A4, bothRTCA) and Hom (G; A) can be identified with the cokernel ofT'°(E(6)A) => TBA),
so they are isomorphic. And similarly
RTC(A)=R'T%B(xA),
while
Ext (G;

Appendix

A)=

Hom

(G;

B(nA).

(A).

In [6] I explained how one can associate naturally to a topologicalspace A a contractible space EA obtained from the semi-simplicial space“naples{An}, where An=A X ... X A, by the usual realization process; i.e. EA
is obtained from the topological sum IIA. >X A" where A" is the stan
dard m-simplex, by making
simplexes.

identifications

Another way of describing EA

along the boundaties

is as follows:of theit is the space ofstep-functions on the unit interval I with values in A, where by stepfunction one means a function constant on each of the half-open intervals

Cohomology of topological groups(t;, fi1]

for some

partition

O=f<ti<..

385<tp.=1of I. But fromthispoint of view the topology seems rather obscure.
If A and B are two spaces then E(A X B)=EA X EB in the categoryof k-spaces, sc that EA is a group in the category if A is one. When

EA is thought of as a function-space the composition-law is the obvious
one, and EA admits a continuous monomorphism on to a dense subgroupof L'(I : A). The group A is embedded in EA as a closed subgroup (theconstant functions) which is normal if and only if A is abelian. One
defines BA=EA/A. The following proposition replaces the vague remark
on the same subject in [6].

ProrosiTioN (A.1).

If A is a locally contractible group (not nec-

essarily abelian) then it has a local cross-stection in EA, so that thefibration A —> EA —> BA is locally trivial. (Note: « Locally contractible »means that for each neighbourhood U of each point x there is a neighbourhood V of x contained in U which is contractible in U).

Proor.

BA

is the realization

where By=A,/A=A

which

>

of the semi-simplicial.. X A. Let B"A be the n-skeleton

space({B.},[6]of BA,

is obtained by attaching B. X A" to B*~!'A by a certain map(B. X AM U (Bd X A") — B*=14,
where B,* is the degenerate part of B,. Suppose that a cross-section ofE"-14 — B"'A has been found in a neighbourhood of the base-pointin BA. When the bundle EA — BA is pulled back to B, X A”* itbecomes trivial, so the problem of extending the partial section reduces
to that of extending a map V — A, where V is an open set of (IB, X A") U
UB, X AP), to a neighbourhood of V in By X A". That can be done
because (B, X AM U(B,4 X A?) is a neighbourhood deformation retractin B, XX A®; which is true in turn because
Bl={(a;,..., adehA KX ..X A: a=1

for somei)

is a neighbourhood deformation retract in B,, if A is locally contractible.
ProposiTiON (A.2).
If A is a locally contractible group
and BA are locally contractible.

then EA

Proor. EA is locally contractible because any contractible topological group is locally contractible. The local contractibility of BA
follows from that of EA in view of the local cross-section found in (A.1).

386

GraemeProrosiTiON (A.3).

Segal

If A"—>A—A” is a short exact sequence inTopab then the induced sequences EA" — EA —> EA”

andBA’ —> BA ——> BA” are exact.
Proor. The algebraic exactness of the sequences is trivial. The
local cross-section of A —> A” induces a local cross-section of EA — EA”,
because E is a functor on topological spaces. A local cross-section ofBA— BA” can be obtained from local sections of EA”— BA” andEA
— EA”.
Appendix(B).

I shall prove the following theorem, which is true either in the
category of topological spaces or in the category of k-spaces.

THEOREM (B.1).

If A” is a contractible closed subgroup of a topo-

logical group A which has a local cross-section then it has a global
cross-section, ie. A=A" X (A/A’) as topological space. (Notice thatA” and A are not assumed abelian),

The proof depends on the following lemma inspired by [4].
ProrosiTiON (B.2).
If X is a uniform space, and UcX X X isan entourage of the diagonal, then the covering {U.}icx of X is numerable [1], where U.={yex: (x, y)eU}.

ExampLE,
If G is a topological group and V is a neighbourhoodof the identity in G then the covering {gV}.cc of G is numerable,
Proor or (B.2).
Mather ([4] Thm. 1) has shown that is enough tofind functions ¢.: X — [0, 1] for each xeX such that Z ¢(y)=1 for

all yeXand ¢«(y)=0 when y¢U..

veU,f(x, y)<1,

Choose an écart f on X such thatand let 0. : X — [0, 1] be defined by

0:(y)= sup (1—f(x, y), 0).
Well-order the points of X and defineOy) =0:(y) — sup 8:(y).
Then ¢. is continuous because the functions {0} are uniformly equicontinuous and hence have a continuous supremum. Clearly Xo.=1,and ¢:(y)=0 if yeU..

Cohomology

PRroOF OF (B.1).

of topologicalgroups387

Because of (B.2) the fibration A"—>A-—>A"=A/A’is trivial over each of the sets of a numerable covering of A”, so by
a well-known

argumentof Milnor(cf. [5], 3.6) one can suppose

it istrivial over the sets {Uren of a locally finite countable covering. Let
Wi=UsUU

U ... UUs.

Suppose one has found a cross-section wi: Wir—>A, and has a crosssection ug + Ur—>A. Then one can define weer: Weer
— A byWr =wi(x)for xeWr-—Uy=ur(x)

for xeUr—Wq= w(x) - clup(x) wrx)",0(x)), for xeW, NU;

where ¢: A’ > [0, 1]—>A’ is a contraction and 0: X—[0, 1] is afunction equal to 1 on Wiy—U,. Finally, define the desired cross-section

wi: A”=> A by w(x)=wr(x) for large k: that is permissible because
wi(x)=wg.1(x) if x¢Ur, and the covering {U:} is locally finite.
Testo pervenuto il 3 giugno

1969,

Bozze

1970,

licenziate

il 10 luglio

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Partitions of unity

in the theoryof fibrations,Ann,of Math.,78

223-255.

GaBriEL, P. and Zisman, M.: Calculus of fractions and homotopy theory,
Ergebnisse der Math. u. ihrer Grenzgebiete, 35, Springer, Berlin, 1967.

[3] Huser,
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P. J.:(1961),[4] MATHER,

Homolopicalcohomology

and

Cech

coliomology,

Math.

Ann,

73-76.

M.:

Paracompactness

and partitionsof unity.

[5] Pavrais, R. S.: Slices and equivariant embeddings, in A. BOREL, et al.: Seminar[6]

on transformation groups, Ann.

of Math. Studies, 46 (1960).

SecAL,

classifying spaces,G.

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B.:

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andPubl. Math,

LH.E.S.,

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