Dedicated to the Memory of J. H. C. Whitehead

Theory and Applications of Categories Vol No pp

ON FINITE INDUCED CROSSED MODULES

AND THE HOMOTOPY TYPE OF MAPPING CONES

Dedicated to the memory of J H C Whitehead

RONALD BROWN AND CHRISTOPHER D WENSLEY

Transmitted byLawrence Breen

ABSTRACT Results on the niteness of induced crossed mo dules are proved b oth

algebraically and top ologically Using the Van Kamp en typ e theorem for the fundamen

tal crossed mo dule applications are given to the typ es of mapping cones of classifying

spaces of groups Calculations of the cohomology classes of some nite crossed mo dules

are given using crossed complex metho ds

Intro duction

Crossed mo dules were intro duced by JHC Whitehead in They form a part of what

can b e seen as his programme of testing the idea of extending to higher dimensions the

metho ds of combinatorial group theory of the s and of determining some of the

extra structure that was necessary to mo del the geometry Other pap ers of Whitehead of

this era show this extension of combinatorial group theory tested in dierent directions

In this case he was concerned with the algebraic prop erties satised by the b oundary

map

X A A

of the second relative homotopy group together with the standard action on it of the

fundamental group A This is the fundamental crossedmodule X Aofthepair

X A In order to determine the second homotopy group of a CW complex he formu

lated and proved the following theorem for this structure

g be obtainedfrom the connectedspace A by attaching Theorem W Let X A fe

cel ls Then the second relative homotopy group X A may bedescribed as the free

crossed Amodule on the cel ls

The pro of in uses transversality and knot theory ideas from the previous pap ers

See for an exp osition of this pro of Several other pro ofs are available The

The rst author was supp orted by EPSRC grant GRJ for a visit to Zaragoza in November

and is with Dr T Porter supp orted for equipment and software by EPSRCGrant GRJ

Non ab elian homological algebra

Received by the editors March and in revised form April

Published on May

Mathematics Sub ject Classication G F P Q

Key words and phrases crossed mo dules homotopytyp es generalised van Kamp en Theo

remcrossed resolution Postnikovinvariant

Ronald Brown and Christopher D Wensley Permission to copy for private use granted

Theory and Applications of Categories Vol No

survey byBrown and Huebschmann and the b o ok edited by HogAngeloni Metzler

and Sieradski give wider applications

The pap er of Mac Lane and Whitehead uses Theorem W to show that the

dimensional homotopy theory of p ointed connected CW complexes is completely mo d

elled by the theory of crossed mo dules This is an extra argument for regarding crossed

mo dules as dimensional versions of groups

One of our aims is the explicit calculation of examples of the crossed mo dule

A V A A

of a mapping cone when V A are nite The key to this is the dimensional

Van Kamp en Theorem VKT proved by Brown and Higgins in This implies a

generalisation of Theorem W namely that the crossed mo dule is induced from the

identity crossed mo dule V V by the morphism V A

Presentations of induced crossed mo dules are given in and from these we prove

a principal theorem Theorem that crossed mo dules induced from nite crossed

mo dules by morphisms of nite groups are nite We also use top ological metho ds to

prove a similar result for nite pgroups Corollary These results give a new range

of nite crossed mo dules

Sequels to this pap er discuss crossed mo dules induced by a normal inclusion and

calculations obtained using a group theory package

The origin of the VKT was the idea of extending to higher dimensions the notion

of the fundamental group oid as suggested in in This led to the discovery of

the relationship of dimensional group oids to crossed mo dules in work with Sp encer

This relationship reinforces the idea of higher dimensional group theory and was

essential for the pro of of the VKT for the fundamental crossed mo dule In view

of the results of Mac Lane and Whitehead and of metho ds of classifying spaces of

crossed mo dules byLoday and Brown and Higgins see section the VKT

allows for the explicit computation of some homotopy typ es in the form of the crossed

mo dules which mo del them

In some cases the Postnikovinvariant of these typ es can b e calculated as the

following example shows

CorollaryLet C denote the cyclic group of order n and let BC denote its

n n

classifying space The second homotopy mo dule of the mapping cone X BC BC

is a particular cyclic C mo dule A say The cohomology group H C A is a cyclic

n n n n

group of order n and the rst Postnikovinvariantof X is a generator of this group

The metho d used for the calculation of the cohomology class here is also of interest

It intro duces a new small free crossed resolution of the cyclic group of order n in order to

construct an explicit co cycle corresp onding to the ab ove crossed mo dule This indicates

a wider p ossibility of using crossed resolutions for explicit calculations It is also related

to Whiteheads use of what he called in homotopy systems and which are simply

free crossed complexes

The initial motivation of this pap er was a conversation with Rafael Sivera in Zaragoza

in Novemb er which suggested the lack of explicit calculations of induced crossed

Theory and Applications of Categories Vol No

mo dules This led to discussions at Bangor on the use of computational group theory

packages which culminated in a GAP program and to the development of general

theory

Crossed mo dules and induced crossed mo dules

In this section we recall the denition of induced crossed mo dules and of results of

on presentations of induced crossed mo dules Wethengive some basic examples of these

Recall that a crossedmodule M M P is a morphism of groups M P

together with an action m p m of P on M satisfying the two axioms

CM m p mp

CM m n mn

for all m n M and p P Wesaythat M is nite when M is nite

The category XM of crossed mo dules has as ob jects all crossed mo dules Morphisms

in XM are pairs g f forming commutative diagrams

M P

in which the horizontal maps are crossed mo dules and g f preserve the action in the sense

p fp

that for all m M p P wehave g m gm If P is a group then the category

XMP of crossed P mo dules is the sub category of XM whose ob jects are the crossed

P mo dules and in whicha morphism M P N P of crossed P mo dules

p p

is a morphism g M N of groups such that g preserves the action g m gm

for all m M p P and g

Standard algebraic examples of crossed mo dules are

i an inclusion of a normal subgroup with action given by conjugation

ii an inner automorphism crossed mo dule M Aut M inwhich m is the

automorphism n m nm

iii a zero crossed mo dule M P where M is a P mo dule

iv an epimorphism M P with kernel contained in the centre of M

Examples of nite crossed mo dules may b e found among those ab ove the induced

crossed mo dules of this pap er and its sequels and copro ducts and tensor

pro ducts of nite crossed P mo dules

Further imp ortant examples of crossed mo dules are the free crossed mo dules referred

to in the Intro duction which are rarely nite They arise algebraically in considering

identities among relations which are nonab elian forms of syzygies

We next dene pul lback crossedmodules Let P Q b e a morphism of groups

and let N Q b e a crossed Qmo dule Let N P b e the pullbackof N by

Theory and Applications of Categories Vol No

so that N fp n P N j p ngand p n p Let P act on N by

p p

p n p p p n The verication of the axiom CM is immediate while CM is

proved as follows

Let p n p n N Then

p n p n p n p p p n n n

p p p n

p n

Proposition The functor XMQ XMP has a right adjoint

Proof This follows from general considerations on Kan extensions

The universal prop erty of induced crossed mo dules is the following Let M P

C Q b e crossed mo dules In the diagram

M P

the pair is a morphism of crossed mo dules suchthatforany crossed Qmo dule

C Q and morphism of crossed mo dules f there is a unique morphism

g M C of crossed Qmo dules suchthat g f

It is a consequence of this universal prop ertythatifM P F R the free group

on a set R and if w R Q is the restriction of to the set R then F Risthefree

crossed mo dule on w in the sense of Whitehead see also Constructions

of this free crossed mo dule are given in these pap ers

A presentation for induced crossed mo dules for a general morphism is given in Prop o

sition of We will need two more particular results The rst is Prop osition of that

pap er and the second is a direct deduction from Prop osition

Proposition If P Q is a surjection and M P is a crossedP

MM K where K Ker and M K denotes the subgroup of M module then M

generatedbyallm m for al l m M k K

The following term and notation will b e used frequentlyLetP b e a group and let

T be a set We dene the copower P T to b e the free pro duct of groups P t T

each with elements p tp P and isomorphic to P under the map p t p If Q is a

group then P Q will denote the cop ower of P with the underlying set of the group Q

Theory and Applications of Categories Vol No

Proposition If P Q is an injection and M P is a crossedP

module let T be a right transversal of P in Q Let Q act on the copower M T by

q p

the rule m t m u where p P u T and tq pu Let M T Q

be denedbym t t mt Let S be a set of generators of M asagroup and let

P p

S fx x S p P gThen

M M T R

where R is the normal closureinM T of the elements

rt P

hrt s ui rt s u rts u rs S tu T

Proof Let N M T Prop osition of yields that M is the quotientof N by

the subgroup hN N i generated by hn n i n n nn nn N and which is called

P p

in the Peier subgroup of N Now N is generated bythesetS Tfs ts

p q pp

S p P t T g and this set is Qinvariant since s t s uwhereu T p P

satisfy tq p u It follows from Prop osition of that hN N i is the normal closure

P P

of the set hS T S Ti of basic Peier commutators

Example The dihedral crossedmodule Weshowhow this works out in the follow

ing case which exhibits a number of typical features WeletQ b e the dihedral group D

with presentation hx y x y xy xy i and let M P b e the cyclic subgroup

C of order generated by y Let C f n g b e the cyclic group of order n

A right transversal T of C in D is given by the elements x i C Hence C has a

n n

presentation with generators a y x i C and relations given by a i C

i n n

i i i

together with the Peier relations Now a x yx yx Further the action is given

x y a

bya a a a Hence a a so that the Peier relations are

i i i ni i j i

a a a a It is well known that wenowhave a presentation of the dihedral group

j i j j i

D from whichwe recover the standard presentation hu v u v uv uv i by

setting u a a v a so that u a a Then

u x v y

so that y acts on C by conjugation by v However x acts by

x x

u u v vu

Note that this is consistent with the crossed mo dule axiom CM since

x x

v vu vuu u vu

We call D D D the dihedral crossedmoduleItfollows from these formulae

n n n

that is an isomorphism if n is o dd and has kernel and cokernel isomorphic to C if n

is even In particular if n is even then by results of section BD BC canbe

regarded as having one nontrivial element represented by u

Theory and Applications of Categories Vol No

Corollary Assume P Q is injective If M has a presentation as a group

with g generators and r relations the set of generators of M is Pinvariant and n Q

M then M has a presentation with gn generators and rn g nn relations

Another corollary determines induced crossed mo dules under some ab elian conditions

This result has useful applications If M is an ab elian group or P mo dule and T is a

set we dene the copower of M with T written M T to b e the sum of copies of M

one for each elementof T

Corollary Let M P beacrossed P module and P Q a monomor

phism of groups such that M is abelian and M is normal in Q Then M is abelian

and as a Qmodule is just the induced Qmodule in the usual sense

Proof We use the result and notation of Prop osition Note that if u t T and

r S then u rt ut r t mut t mu for some m M by the

normality condition The Peier commutator given in Prop osition can therefore b e

rewritten as

rt m

rt s u rts u r ts u rts u

Since M is ab elian s s Thus the basic Peier commutators reduce to ordinary

commutators Hence M is the cop ower M T and this with the given action is the

usual presentation of the induced Qmo dule

Example Let M P Q b e the innite cyclic group C let b e the identity

C C and the action of a generator of and let be multiplication by Then M

Q on M is to switchthetwo copies of C This result could also b e deduced from well

known results on free crossed mo dules However our results showthatwe get a similar

conclusion simply by replacing each C in the ab oveby a nite cyclic group C and

this fact is new

On the niteness of induced crossed mo dules

In this section we give an algebraic pro of that a crossed mo dule induced from a nite

crossed mo dule by a morphism with nite cokernel is also nite In a later section we will

prove a slightly less general result but by top ological metho ds which will also yield results

on the preservation of the Serre class of a crossed mo dule under the inducing pro cess

Theorem Let M P bea crossedmodule and let P Q be a morphism

of groups Suppose that M and the index of P in Q are nite Then the inducedcrossed

module M Q is nite

Proof Factor the morphism P Q as where is injective and is surjective

Then M is isomorphic to M It is immediate from Prop osition that if M is nite

then so also is M So it is enough to assume that is injective and in fact we assume

it is an inclusion

Theory and Applications of Categories Vol No

Let T b e a right transversal of P in Q Let Y M T be the copower of M and

T and let Y Q and the action of Q on Y b e as in Prop osition The equations

tq pu which determine this action in fact provide a function

T Q P T t q p u

A basic Peier relation is then of the form

tv mv p

m tn v n v m t v mv n v m u

where m n M t u T and q v mv

Wenow assume that the nite set T has l elements and has b een given the total order

t t t An elementof Y may b e represented as a word

m u m u m u

e e

Suchawordissaidtobereduced when u u i e and to b e ordered if

i i

u u u in the given order on T Thisyeilds a partial ordering of M T where

m u m u whenever u u

i i j j i j

A twist uses the Peier relation to replace a reduced word w w m tn v w

with vtby w w n v m uw If the resulting word is not reduced multiplication

in M and M may b e used to reduce it In order to showthatanyword may b e ordered

v u

by a nite sequence of twists and reductions we dene an integer weight function on the

set W of nonemptywords of length at most n by

ni e

l j W Z m t m t m t l

i n n j j e j

It is easy to see that w w whenw w is a reduction Similarlyfora twist

n n

w w m t m t w w w m t n t w

i j i j i j k

i i i

the weight reduction is

nei nei

w w l l j j j j l

n n i i i k

so the pro cess terminates in a nite number of moves

This ordering pro cess is a sp ecial case of a purely combinatorial sorting algorithm

discussed in

Wenow sp ecify an algorithm for converting a reduced word to an ordered word

Various algorithms are p ossible some more ecient than others but wearenotinterested

in eciency here We call a reduced word k ordered if the subword consisting of the rst

k elements is ordered and the remaining elements are greater than these Every reduced

word is at least ordered Given a k ordered reduced word nd the rightmost minimal

element to the rightofthek th p osition Move this element one place to the left with

Theory and Applications of Categories Vol No

atwist and reduce if necessary The resulting word may only b e j ordered with jk

but its weight will b e less than that of the original word Rep eat until an ordered word

is obtained

Let Z M M M b e the pro duct of the sets M M ft g Then the

t t t t i

algorithm yeilds a function Y Z such that the quotient morphism Y M factors

through Since Z is nite it follows that M is nite

Remark In this last pro of it is in general not p ossible to give a group structure

on the set Z such that the quotient morphism Y M factors through a morphism to

Z For example in the dihedral crossed mo dule of example with n the set Z

will have elements and so has no group structure admitting a morphism onto D This

explains why the ab ove metho d do es not give an algebraic pro of of Corollary which

gives conditions for M to b e a nite pgroup However in we will give an algebraic

pro of for the case P is normal in Q

Top ological applications

As explained in the Intro duction the fundamental crossed mo dule functor assigns a

crossed mo dule X A A to any base p ointed pair of spaces X A We will

use the following consequence of Theorem C of which is a dimensional Van Kamp en

typ e theorem for this functor

Theorem Theorem D Let B V beacobredpair of spaces let f V A

be a map and let X A B Suppose that A B V arepathconnected and the pair

B V is connected Then the pair X A is connected and the diagram

V B V

X A A

presents X A as the crossed Amodule B V inducedfrom the crossed

V module B V by the group morphism V A inducedby f

As p ointed out earlier in the case P is a free group on a set Rand is the identitythen

the induced crossed mo dule P is the free crossed Qmo dule on the function jR R Q

Thus Theorem implies Whiteheads Theorem W of the Intro duction A considerable

amountofwork has b een done on this case b ecause of the connections with identities

among relations and metho ds such as transversality theory and pictures have proved

successful particularly in the homotopy theory of dimensional complexes

However the only route so far available to the wider geometric applications of induced

crossed mo dules is Theorem We also note that this Theorem includes the relative

Hurewicz Theorem in this dimension on putting A V and f V V the inclusion

We will apply this Theorem to the classifying spaceofacrossedmodule as dened

byLodayinorBrown and Higgins in This classifying space is a functor B

Theory and Applications of Categories Vol No

assigning to a crossed mo dule M M P apointed CW space B M with the

following prop erties

The homotopy groups of the classifying space of the crossedmodule M M

P are given by

Coker for i

B M Ker for i

for i

The classifying space B P is the usual classifying space BP of the group

P and BP is a subcomplex of B M Further there is a natural isomorphism of crossed

modules

B MBP M

If X is a reduced CW complex with skeleton X then there is a map

X B X X

inducing an isomorphism of and

It is in these senses that it is reasonable to sayasintheIntro duction that crossed

mo dules mo del all p ointed homotopy typ es

Wenowgivetwo direct applications of Theorem

Corollary Let M M P beacrossedmodule and let P Q bea

morphism of groups Let BP B M be the inclusion Consider the pushout

B M BP

Then the fundamental crossedmodule of the pair X B Q is isomorphic to the induced

crossedmodule M Q and this crossedmodule determines the typeofX

Proof The rst statement is immediate from Theorem The nal statement follows

from results of since the morphism Q X is surjective

Remark An interesting sp ecial case of the last corollary is when M is an inclusion

of a normal subgroup since then B M is of the homotopytyp e of B PM So wehave

determined the typ e of a homotopy pushout

BR BP

Theory and Applications of Categories Vol No

in which p P R is surjective

We write V for the cone on a space V

Corollary Let P Q be a morphism of groups Then the fundamental crossed

module BQ BP BQ is isomorphic to the inducedcrossedmodule P Q

Finiteness theorems by top ological metho ds

The aim of this section is to show that the prop erty of b eing a nite pgroup is preserved

by the pro cess of induced crossed mo dules We use top ological metho ds

An outline of the metho d is as follows Supp ose that Q is a nite pgroup To prove

that M is a nite pgroup it is enough to prove that Ker M Qisanitepgroup

But this kernel is the second homotopy group of the space X of the pushout and so

is also isomorphic to the second homology group of the universal cover X of X In order

to apply the homology MayerVietoris sequence to this universal cover weneedtoshow

that it may b e represented as a pushout and we need information on the homology of the

spaces determining this pushout So we start with the necessary information on covering

spaces

Wework in the convenient category TOP of weakly Hausdor k spaces Let

X X b e a map of spaces In the examples we will use will b e a covering map

Then induces a functor

TOPX TOPX

It is known that has a right adjoint and so preserves colimits

For regular spaces the pullbackofa covering space in the ab ove category is again a

covering space These results enable us to identify a covering space of an adjunction space

as an adjunction space obtained from the induced covering spaces

If further is a covering map and X is a CW complex then X maybegiven the

structure of a CW complex

We also need a sp ecial case of the basic facts on the path comp onents and fundamental

group of induced covering maps Given the following pullback

f b

X A

A X

and p oints a A x X suchthatfa x in which is a universal covering map and

X A X are path connected then there is a sequence

b b

A a x A a X f a A

This sequence is exact in the sense of sequences arising from brations of group oids

whichinvolves an op eration of the fundamental group X f a on the set Aofpath

Theory and Applications of Categories Vol No

b b

comp onents of A It follows that the fundamental group of A is isomorphic to Ker f

and that A is bijective with the set of cosets X f af A a It is also clear

b b

that the covering A A is regular and that all the comp onents of A are homeomorphic

Let M b e the crossed mo dule M P and let P Q b e a morphism of

groups Let

X BQ B M

f d d

as in diagram Let X X b e the universal covering map and let BQ B M BP

b e the pullbacks of X under the maps BQ X B M X B P X Then wemay

write

f d

X BQ B M

by the results of section

From the exact sequence we obtain the following exact sequences in which X

Q PM

BQ Q Q PM

B M PM Q PM B M

d d

BP P Q PM BP

d d d

Proposition Under the above situation let the groups BP B M BQ

be denotedby P M Q respectively and let B M denote a component of B M Then

there is an exact sequence

d d

B M H Q X BP H B M H P

d d

H P BP H M B M H Q

Proof This is immediate from the MayerVietoris sequence for the pushout and the

X fact that H X

Corollary If P Q is the inclusion of a normal subgroup and X BQ

BP then X is isomorphic to H P I QP where I G denotes the augmentation

ideal of a group G

This result agrees with Corollary of in which the induced crossed mo dule itself

is computed in the case P is normal in Q via the use of copro ducts of crossed P mo dules

Corollary Let M M P beacrossedmodule and let P Q bea

morphism of groups If M P and Q are nite pgroups then so also is M

Theory and Applications of Categories Vol No

Proof It is standard that the reduced homology groups of a nite pgroup are nite

pgroups The same applies to the reduced homology of the classifying space of a crossed

mo dule of nite pgroups The latter may b e proved using the sp ectral sequence of a

covering and Serre C theory as in Chapters IX and X of In the present case we

need information only on H B M and some of its connected covering spaces and this

may b e deduced from the exact sequence due to Hopf

H K H G K Z H K H G

for any connected space K with fundamental group G see for example Exercise on

p of Prop osition shows that Ker M Q X is a nite pgroup

Since Q is a nite pgroup it follows that M is a nite pgroup

Note that these metho ds extend also to results on the Serre class of an induced crossed

mo dule whichweleave the reader to formulate

Cohomology classes

Recall that if G is a group and A is a Gmo dule then elements of H G Amay

b e represented by equivalence classes of crossedsequences

A M P G

namely exact sequences as ab ovesuch that M P is a crossed mo dule The

equivalence relation b etween such crossed sequences is generated by the basic equivalences

namely the existence of a commutative diagram of morphisms of groups as follows

A M P G

A G

M P

such that f g is a morphism of crossed mo dules Such a diagram is called a morphism

of crossed sequences

The zero cohomology class is represented by the crossed sequence

A A G G

whichwe sometimes abbreviate to

A G

In a similar spirit wesay that a crossed mo dule M P represents a cohomology

class namely an elementof H Coker Ker

Theory and Applications of Categories Vol No

Example Let C denote the cyclic group of order n written multiplicatively

with generator u Let C C b e given by u u This denes a crossed mo dule

n n

with trivial op erations This crossed mo dule represents the trivial cohomology class in

H C C in view of the morphism of crossed sequences

n n

C C C C

n n n n

C C C C

n n

where if t is the generator of the top C then tu

Example Weshow that the dihedral crossed mo dule D of Example represents

the trivial cohomology class This is clear for n o dd since then is an isomorphism For

n even we simply construct a morphism of crossed sequences as in the following diagram

C C C C

f f

C D D C

n n

where if t denotes the non trivial elementof C then f tx f t u Just for

interest we leave it to the reader to prove that there is no morphism in the other direction

between these crossed sequences

A crossed mo dule M M P determines a cohomology class

k H Coker Ker

If X is a connected p ointed CW complex with skeleton X then the class

k H X X

of the crossed mo dule X X is called the rst Postnikov invariant of X This class

is also represented by X A for any connected sub complex A of X such that X A

is connected and A It may b e quite dicult to determine this Postnikov

invariant from a presentation of this last crossed mo dule and even the meaning of the

word determine in this case is not so clear There are practical advantages in working

directly with the crossed mo dule since it is an algebraic ob ject and so it or families

of such ob jects may b e manipulated in many convenient and useful ways Thus the

advantages of crossed mo dules over the corresp onding co cycles are analogous to some

of the advantages of homology groups over Betti numb ers and torsion co ecients

However in work with crossed mo dules and in applications to homotopy theory in

formation on the corresp onding cohomology classes such as their nontriviality or their

Theory and Applications of Categories Vol No

order is also of interest The aim of this section is to givebackground to sucha de

termination and to givetwo example of nite crossed mo dules representing nontrivial

elements of the corresp onding cohomology groups

The following general problem remains If G A are nite where A is a Gmo dule

how can one characterise the subset of H G A of elements represented by nite crossed

mo dules This subset is a subgroup since the addition maybedenedby a sum of crossed

sequences of the Baer typ e An exp osition of this is given by Danas in It might

always b e the whole group

The natural context in whichtoshowhow a crossed sequence gives rise to a co cycle

is not the traditional chain complexes with op erators but that of crossed complexes

We explain how this works here For more information on the relations b etween crossed

complexes and the traditional chain complexes with op erators see

Recall that a free crossed resolution of the group G is a free aspherical crossed complex

F together with an epimorphism F G with kernel F

Example The cyclic group C of order n is written multiplicatively with generator

t We give for it a free crossed resolution F as follows Set F C with generator w

and for r set F C Here for r F is regarded as the free C mo dule on

r r n

one generator w and we set w w The morphism C C sends w to t and

i n

the op eration of F on F for r is via The b oundaries are given by

w w

for r odd w w w

r i i

for r even and greater than w w w w

r i n

Previous calculations show that is the free crossed C mo dule on the element

w C Thus F is a free crossed complex It is easily checked to b e aspherical and so

is with a crossed resolution of C

Let A be a Gmo dule Let C G A denote the crossed complex C whichis G in

dimension A in dimension with the given action of G on A and which is elsewhere

as in the following diagram

A G

Let F b e a free crossed resolution of G It follows from the discussions in

that a co cycle of G with co ecients in A can b e represented as a morphism of crossed

complexes f F C G A over This co cycle is a cob oundary if there is an op erator

morphism l F A over F G such that l f

F F F F

A G

Theory and Applications of Categories Vol No

To construct a co cycle on F from the crossed sequence rst construct a mor

phism of crossed complexes as in the diagram

F F F F

f f f

A M P G

using the freeness of F and the exactness of the b ottom row Then comp ose this with

the morphism of crossed sequences

A M P G

A G G

Hence it is reasonable to say that the morphism f of diagram is a co cycle corre

sp onding to the crossed sequence

Wenow use these metho ds in an example

denote the injection sending a generator Theorem Let n and let C C

t of C to u where u denotes a generator of C Let A denote the C module which

n n n

is the kernel of the inducedcrossedmodule N C C Then H C A is

n n n

cyclic of order n and has as generator the class of this inducedcrossedmodule

Proof By Corollary the ab elian group C is the pro duct V C AsaC

n n n

mo dule it is cyclic with generator v sayWrite v v i n Then each

v is a generator of a C factor of V Thekernel A of is a cyclic C mo dule on the

i n n n

Write a a v v As an ab elian group A has generators generator a v v

i i n

a a a with relations a aa a

n n

We dene a morphism f from F to the crossed sequence containing N as in diagram

where

f maps w to u

f maps the mo dule generator w of F to v v

f maps the mo dule generator w of F to a

n n n

C C C

C C

f f f

A C C

n n

Theory and Applications of Categories Vol No

The op erator morphisms f over f are dened completely by these conditions

The group of op erator morphisms g C A over f maybeidentied with

A under g g w Under this identication the b oundaries are transformed

resp ectively to and to a a a So the dimensional cohomology group is the

i i

group A with a identied with a i n This cohomology group is therefore

n i i

isomorphic to C and a generator is the class of the ab ove co cycle f

Corollary The mapping cone X BC BC satises X C and X

B n n

is the C module A of Theorem The rst Postnikov invariant of X is a generator

n n

of the cohomology group H X X which is a cyclic group of order n

The following is another example of a determination of a nontrivial cohomology class

by a crossed mo dule The metho d of pro of is similar to that of Theorem and is left

to the reader

Example Let n be even Let C denote the C mo dule whichis C as an ab elian

n n

group but in which the generator t of the group C acts on the generator t of C by sending

C and a generator of this group is represented it to its inverse Then H C C

by the crossed mo dule C C C with generators t t u say and where

n n n

t t u Here u C op erates by switching t t It is not clear if this crossed

n n

mo dule can b e an induced crossed mo dule for n However n gives the case n

of Theorem

Remark The crossed mo dule C C C also app ears as an example in

pp The pro of given there that its corresp onding cohomology class is nontrivial

is obtained by relating this class to the obstruction to a certain kind of extension

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