Seminarberichte Nr. 15

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Mathematik und

Informatik

Seminarberichte aus dem Fachbereich Mathematik der FernUniversität

15 – 1982

Die Dozentinnen und Dozenten der Mathematik (Hrsg.)

Seminarbericht Nr. 15

Replaceability and µ-uniqueness - A unified viewpoint

von W. Beekmann und S.-C. Chang

Groupoids and the Mayer-Vietoris sequence

von R. Brown, P.R. Heath und K.H. Kamps The coshape category as an imprimitivity

algebra

von A. Frei und H. Kleisli

Homotopy pairs in Eckmann-Hilton duality von K.A. Hardie

Zur Stabilitat kompakter Blatterungen von H. Holmann

Radicals of Jordan algebras of degree 3

von H.P. Petersson und M.L. Racine Pro-categories and multiadjoint functors

von W. Tholen

Filtered colimits are directed colimits von W. Tholen

Seite

1 -

7 -

43 -

61 -

77 -

- 107 -

- 133 -

- 149 -

Wolfgang Beekmann and Shao-Chien Chang

In the last ten years the notions of replaceability and µ-uniqueness, in the theory of matrix methods of

summability, have been studied by several authors, see e.g. [1], [3] and [5]. It seems highly desirable to have characterizations of these notions in a more general setting, thus solving their naming problems, see [6]. In our discussion here, we shall use the notations in [3], [4] and [6].

Let A be an infinite complex matrix wi th <p c cA , i.e. cA includes the finite sequences. We recall that A is said tobe replaceable, if there exists a matrix D with CD= CA and lim

0x = O for all X E <p 1

and A is called µ-unique, if for given µ E

«::'

^{t E l}

und s E cß

A ^I the condition

µ limAx + t(Ax) + sx = ⁰ for all XE CA implies µ = O.

Since each f E cA has a representation (*) f(x) = µ limAx + t(Ax) + sx

wi th s ome µ E C , t E ,t and ß

s E cA , µ-uniqueness of A is equivalent to the fact that the nullfunctional does

not belong to

µ*:= {f E CA ß

·3µ=1=0, tE,t, sEcA s.t. (*) holdsL

- 1 -

DEFINITION: Let M be a'subset of

t

A is µ-unique over condition

• t

M, if för each the

f (x)

=

0 for all x EM

implies f

~

^µ

*.

We observe that µ-uniqueness over a subset of

an invariant property, since µ

*

is invariant (see is

[1], Satz 3) and that A is µ-unique, if and only if A is µ-unique over cA. Also,

A is non-replaceable, i f and onty i f A is µ-unique over <P·

(See [1], Folgerung 3). Thus the above definition

unifies the notions of µ-uniqueness and replaceability.

We first give a topological characterisation which can be considered as a step towards a solution of the naming problem and which extends a result due to Wilansky [ 6]

who considered µ-uniqueness (over cA). In the following theorem,

x(g) := g(e) - lkg(e ) k for g E c' ,

where e = ( 1 , 1 , • • • ) is the uni t sequence and c is the Banach-space of convergent sequences.

THEOREM 1. Let A be a matrix with <PC cA.

Then A is µ-unique over a subset M of cA, i f and only i f the functional X lies in the cr(c" ,c' )- closure of D[M]

=

^{Dx

matrix with cD

=

^{cA •}

x EM} whenever D is a

- 2 -

PROOF. Assume that A is non-µ-unique over M. Then, by our defini tion, there exists a functional f E µ

*

with f = O on M. By Zeller' s theorem (see e.g. [ 1], Folgerung 2) , f

=

limD for some D wi th cD

=

cA . Thus D(M] c c (the null-sequences) and x does not

0

lie in the cr(c",c')-closure of D[M]. This shows that the condition is sufficient.

Conversely, i f the condition fails, then there is a matrix

that

D

g=O

with and a functional on

g(y) = x(g) lim y + ty

x(g) :t=

o.

Now (y E c)

g E c' such

for some t E .f.. We define f := g ^O D; we then have f

=

0 on M and

f(x)

=

x(g) limDx + t(Dx)

Since x (g) :t=

o,

A is not µ-unique over M, hence our condition is nessesary.

In view of the remarks following our definition, we have

<WROLLARY 1. A is µ-unique, i f and onZy i f X Zies in the a(c",c'Y-cZosure of D[cA] far aZZ D with

CD= CA •

(See [6], 3.1 and 9.1.)

•

COROLLARY 2. A is non-repZaceabZe, i f and onZy i f

x

Zies in the a(c",c')-cZosure of

D[~]

for aZZ

D

- 3 -

The next two propositions are immediate consequences of our definition or of the remarks following the de- finition.

COROLLARY 3 • Let M 1 ^C M

2 ^C CA . If A is µ-unique over M

1 , then A is µ-unique over M 2 .

COROLLARY 4. If A is µ-unique over M, then A is µ-unique over N, whenever N c M and N :J M.

Finally, we consider the subset

P

=

{x E cA : (tA)x

=

t(Ax) for all t E l with tA E CA} . ß It is known ( see [ 2]) that P

=

^{<.p e} u for some u E cA \ ; or P

=

<.p , the last case being implied by non-replacea- bili ty. We now give a criterion in terms of the notion of µ-uniqueness over subsets to decide on when the first or second case occurs.

THEOREM 2. Let A be a matri·x wi th <.p C cA , and le t u E cA \ ip° • Then P

=

^<.p e u , i f and only i f A is non- µ-unique over <.p and is µ-unique over <.p U {u} •

PROOF. If P

=

^<.p e u , then A is replaceable, hence non-µ-unique over <.p. Assume that A be non-µ-unique over <.p u {u} • Then there exists f E µ* (and hence a matrix D wi th c

0

=

cA and lim0

=

O on <.p u {u} . Since

- 4 -

lim0

=

f) such that u E <.p there is a g

t

^c' _A

with g(u) =I= O and g

=

⁰ on (f) • From g (x) = µ lim

0x + t (Dx) + sx (x E c₀ = CA) we have (set X := ek)

0

=

tD +

.

s

^,

^{hence tD E c!}

_'

and

g(u)

=

0 + t(Du) - (tD)u

= o,

since u E P. This contradiction proves that the condi- tions are nessesary.

Conversely, if A is non-µ-unique over ^{(f) ,} then A is replaceable, and we may assume without lass of generali ty that limA ek

=

O for all k E JN" • Then, if A is µ-unique over ^(f) u { u} , we have limA u ^=I= 0 ,

since otherwiese f ^:= limA would be a functional in µ

*

which vanishes on ^(f) U { u} . Consequently,

u ~ qj c kernel of limA . Now, assume u Ej: P. Then there exists t

*

^E .e. such that

tA E c! and t*(Au) =I= (t*A)u . Define f E c' by

A

*1· *( ) ( * ) f(x) ^:= µ lIDAX + t Ax - t A X

where µ* ^{:= [} (t*A)u t* (Au)]/ limAu . Then on ^{(f) U {u}} and f E µ ^{=I= •} This means that A

f=O is non- µ-unique over ^(f) U {u} , contrary to the condition.

Hence u E P , and consequently P

=

^{(f) u {} u} •

- 5 -

•

References

[1] Beekmann, W.: Über einige limitierungstheoretische Invarianten. Math.

z.

150, 195-199 (1976).

[2] Beekmann, W.; Boas, J.; Zeller, K.: Der Teilraum P im Wirkfeld eines Limitierungsverfahrens ist invariant- Math. Z. 130, 287-290 (1973).

[3] Beekmann, W.; Chang, S.-C.: Same summability invariants.

manuscripta math. l.:!_, 363-378 (1980) [4] : An example in summability.

(to appear in Periodica Math. Hung.)

[5] Macphail, M. S.; Wilansky, A.: Linear functionals and summability invariants. Canad. Math. Bull .

.J..2,

233-242

(1974).

[6] Wilansky, A.: The µ-property of FK-spaces. Comment. Math.

~ , 3 71-3 80 ( 197 8) .

Wolfgang Beekmann

Fachbereich Mathematik und Informatik

Fernuniversität -Gesamthochschule- Postfach 940

5800 Hagen

Federal Republic of Germany

- 6 -

Shao-Chien Chang

Department of Mathematics Brack University

St. Catharines 0ntario

Canada L2S 3 A1

by

Ronald BROWN, Philip R. HEATH and Klaus Heiner KAMPS

1. Introduction

It was shown in [ 4] that a fibration p : E - B of groupoids gives rise to a family of exact sequences

( 1 • 1 ) 1 - Fpx {x} - E{x}

ax

- B{px} - 'IT _o F px - 'IT o E - 'IT o B of groups and pointed sets, where x E Ob (E) , Fpx denotes the fibre ^{p -1 p (x)} over p (x) , ^'IT

0 is the set of cornponents and for example E{x} is the vertex group of E at x .

The advantage of using fibrations of groupoids is to give a simple and clear derivation of the operation of B on the family of sets {TT _{o y y} F } _{E Ob (B)} and of ext;i:-a properties of exactness at 'IT F

o px The operation includes rnany operations in hornotopy theory, involving change of base points. Similarly, rnany of the exact sequences in hornotopy theory are easily derived

frorn (1.1) (see [3], [4], [6], [19] - [22], [13]). Algebraic applications to exact sequences in non-abelian cohornology are given in [4], [5] and, rnore recently, to exact orbit sequences in group theory in [17]. These latter sequences also find

applications in the Nielsen fixed point theory for fibre spaces ( [ 1 4]) .

The object of this paper, which goes back to an earlier version by the first author (1972), is to generalize the

- 7 -

sequence (1.1) and the operations to a situation of a Mayer- Vietoris type. The prototype of such a sequence in a non- abelian situation was given somewhat obscurely in [1 ], a

paper which was in effect a footnote to a paper [23] of Olum, which itself showed how non-abelian cohomology could be used

to prove the van Kampen theorem on the fundamental group of a union of spaces.

Here we will show how a sequence of Mayer-Vietoris type arises from a pullback square in which two (opposite) maps are fibrations of groupoids. We also establish a weak form of operation and a type of 5-lemma result.

We give some applications to homotopy theory, andin the last section relate the methods of non-abelian cohomology to the groupoid version of the van Kampen theorem given in [2]. I t is hoped that the relative forms of non-abelian

cohomology given here will prove a pointer for higher dimensionaJ analogues of those methods.

- 8 -

2. The Mayer - Vietoris sequence

Let p : D - C be a morphism of groupoids. We say p is a fibration, if p is star surjective; that is for each object d E Ob(D) each element in C with initial point p(d) can be

'lifted' to an element in D with initial point d (see [4], 1, 2.1).

We adopt two conventions. (i) The component of an object x of a groupoid is written x or x~ - i t will be clear from the context in which groupoid this component is taken. (ii) The maps induced by a morphism f will usually be written f , as the interpretation will be clear from the context.

Throughout this section we consider a pullback square of groupoids

B

-

f ( 2. 1 ) p

!

_A

_-

_C

f

so that B is the subgroupoid of A x D whose elements are pairs (a.,ö) such that f(a.)

=

p(ö) , and p, f are given by (a.,ö) ~ a., (a.,ö) ~ ö respectively.

Let b = (a ,d )

0 0 0 be an object of B

- 9 -

2.2 Theorem. I f p is a fibration, then there is a function

- 1T g

0 ·which fits into a diagram, called the Mayer- Vietoris sequence

( 2. 3)

TTOD

ß

1/ "'z

C{c } - n B ~ TI C

0 0 _ / 0

p

/ f

1T A

0

with the following 'exactness' properties:

( i) TI A,

0

~ d &n 1T D satisfy

0

there is a b &n ^TI" B such that

0

(ii) (iii)

f(b)

=a.,

p(b)

=a.

Im(!::,)

t:,(1)=b •

0

f(a) =p(d)

and

then

(iv) y 1, y

2 in C{c

0 } satisfy /::,(y

1) =6(y

2) i f and only i f there are elements a in A{a

0}, ö in D{d

0 } such that

Proof. We first prove (i), which does not involve t:, • Since f(a)=p(d), thereis anelement y in C(p(d),f(a)).

Since p is a fibration, y lifts to an element ö in D ( d,

a

₁) , say, where p (

a

₁) = f ( a) so that the component of (a,d1) in B maps to

ä, d

by p,

f

respectively.

This proves (i).

- 10 -

' 1 ' ^{,1 1}

We now define ^{t.1 .1 i 1 1} Let y E C{c

0 } Since an element ö in ^{D (} d , d)

0

'

p is say,

a fibration, and we define

y lifts to

t. (y) to be the component of (a

0 ,d) in B. If ö' in D(d ,d')

0

lifts y , and 1 is the identity at a

0 , then f(1) =p(ö'ö- 1 ) and so (1,ö'ö- 1 ) joins (a

0 ,d) to

(a0 ,d') in B. Thus t.(y) is well defined. Further p(a ,d)=a, f(a ,d)=d and d=d

0 0 0 0 This proves that

p ( ~m ( t.) ) = { ä.

0 } ,

f (

Im ( t.) ) = {

d

0 } •

also

To prove the remaini.ng part of (ii), suppose b in 1T 0 B satisfies

p

^(b) = ao, f (b) = do . Let b = (a,d) ; then there are elements a,ö, say,in A(a,a

0 ) , D(d

0 ,d) respectively.

The element y = f (a)p(ö) is defined and belongs to C{c

0 } •

We prove that t.(y) =b .

say, be a lift of y . Then

p(ö1) =y=f(a)p(ö) (a,ö -1

1ö ) is an element of and i t follows that

B((a,d), (a

0 ,d

1) ) . which proves (ii).

Hence 6 (y) = (a

0 ,d

1)

~

= (a,d)

~ = b ,

The proof of (iii) is simple; the identity at d

0 lifts the identity of C{c

0 } , so that t. (1) = b

0 •

To prove (iv), suppose for E

=

1, 2 that belongs to C{c

0} and that ÖE in D(d

0 ,dE) lifts yE , so that t.(yE) is the component of (a

0 ,dE) in B.

Suppose there are elements a in A{a

0} , ö in D{d

0}

such that

- 11 -

Then

Thus if then p ( K ) = f ( a) and

(a,K) joins (a

0,d

2) to (a

0,d

1) in B • Hence b. (y

1)

=

^{6. (y}₂) Suppose converse ly tha t b. ( y

1 )

=

^{b. (} y

2) . Then we can find an element (a,K) in

a E A { ao} , K E D ( d2 , d1 ) in D {d

0 }

B joining (a

0 ,d

2) to and f ( a)

=

p ( K) • Let

(a0 ,d

1) • Then

-1 -1

0

=o

₁

Ko

₂

Finally, the proof of (v) is easy, using the definition of B . D

~e remark that if A is a trivial groupoid then 2.2 specializes to the exact sequence of a fibration of groupoids

([4], 4.3(b)) except for the operation of c{c

0 } on the set of components of the fibre. We ~onsider operations in section 3.

The special case of 2.2 where A is connected, that is

1T 0A

=

0 , has already been used in [ 1 6 ] , dealing wi th problems of duality in homotopy theory.

Theorem 2.2 is used in [25] to deduce a Mayer-Vietoris sequence for homotopy pullbacks of groupoids. Analogous results have also been stated in [18].

Note that part of Theorem 2.2, namely (ii) - (v) could have been obtained from [4], (4.2) by an alternative approach using the methods of [9] for groupoids instead of pointed topological spaces.

- 12 -

From Theorem 2. 2 we can deduce a sequence o'f, a more familiar

t

form. !

Let 1T

0A n 1T

0D be the pullback of

1T A

0 f

2.4 Corollary. The following sequence ~s exact 1-B{b

},Cp,f),

A{a }xD{d} f-1

p C{c }~TI B_t..1T Art1T D - 1 ,

0 0 0 0 0 0 0

The sequence in 2.4 contains less information than (2.3) as can be seen by comparing the exactness at

with 2.2(iv).

C{c}

0 of 2.4

One part of Corollary 2.4 is that ~ is surjective.

We are interested in conditions for ~ to·be. bijective, i.e. for ^1r

0 to preserve the pullback (2.1) ~

Recall that a groupoid C is simply connected if C{c0 } consists of a single element for all in Ob ( ^{C) ;} C is 1-connected, if C is connected and simply connected.

2.5 Corollary. The function ~ of 2.4 is a bijection i f and only i f for all choices of (a

0 ,d

0 ) E Ob (B) , the group C{c0 } ( where c

0 = f (a

0 ) = p (d

0 )) is the product se t

- 13 -

In partiauZar, ~ is bijeative i f C ~s simpZy aonneated,

01' i f ^{f : A {} a } - C { c } or

0 0 ~s surjeative.

Proof. Suppose that ~ is bijective. Then for any (a0 ,d

0 ) E Ob(B) there is a sequence (2.3) with base points a , d etc. For this sequence Ker ~

=

1

0 0 by hypothesis,and

so f-¹p is surjective in 2.4.

Conversely, let

~ ~

suppose ~ (b

0 )

=

~ (b) .

b

= (

a , d ) , b

= (

a, d) E Ob (B) ' and

0 0 0

Form (2.3) with the base points a ,d 0 0

etc. Then b E Ker ~

=

Im t::,. by 2. 4. So there is a y E C{c } such

0

that t::,.(y)=b. ~ Then y

=

f (a)p(ö) for some ^{a E A{a}

0 } , ö E D{d

0 }

by hypothesis, so ~ ~

b=6(y) =1i(1) =b

0 by 2.2(iv). o

It will be seen in section 5 that Corollary 2.5 is the essence of the method of Olum in [23]. The extension of Olum's method discussed in [1] comes from the following simple type of corollary.

2.6 Corollary. I f both A and D are 1-aonneated, then

1,, :C{c }---'IT B is bijeative.

•

0 0

We have a complete description of 'IT

0B which generalizes 4.4 of [4]. The simple proof is left to the reader.

2.7 Corollary. Let XCOb(B) be a aompZete set of representatives of the fibres of ~:'ITB-'ITAn'ITD.

0 0 0

There is a bijeation of 'IT

0B with the disjoint union over X of double aosets of fA{a

0}, pD{d

0}

(ao,do) EX • D

- 14 -

in C{fa}

0 foT' aii

The next proposition deals with the naturality of the

function ß . Suppose given a commutative diagram of rnorphisms of groupoids

B D

~ ~

B'

-

^D'

( 2. 8)

1 ^{lp• p}

A' C'

y

A

f'

~

_C

f

in which the inner and outer squares are pullbacks and p and p' are fibrations. Let b

0 = (a

0 ,d

0 ) be an object of B , let ^C

=

^{f (a ) ,}

0 0 and let

a'=)..(a) d'=p(d) b'=µ(b) c ' = v ( c ) .

Q O ¹ 0 0 f O O I O 0

2.9 Proposition. The induaed diagram

is aommutative.

•

C' {c'}

0 ß'

1T B

0

The amount of structure discussed so far enables us to prove a 4-lemma type result, in fact that part of the 5-lemrna which deals with injectivity.

We suppose given the situation of (2.8) and the induced diagram

- 15 -

TT B

0

(2.10)

~1

TT B'

<P ' 0

2. 11 Theorem. I f

1T An 1T D

0 0

1

^{AM p}

1T AI M 1T D'

0

A' {a'}

0 0

and

are surjective and \J : C{c } - C' {c'} is injective„ then µ

0 0

-1 ~ ~

isinjectiveon Ker<P=<P (a

0 ,d

0 ) .

~ ~

Proof. Let

S

₁

,.5

₂ E TT

0B satisfy

<P(b1) = </l(b2) = (ao,do) = </l(bo) .

µ (b

1) = µ (b

2) , and Then by 2.2 (ii) we have

~ ~

6'v(y

1) =µ6(y

1) =µ(b

1) =µ(b

2) =µ6(y

2) =6'v(y 2) . Hence by 2.2 (iv) there are a' EA'{a'} o' ED'{d'}

0 ' 0

such that v(y

1) =f' (a')v(y

2^{)p' (0}

1 ) • Since 11.: A{a} -A'{a'}

0 0 ¹

p: D{d

0 } - D'{d~} are assumed tobe surjective, we have a ¹ = ;\ ( a) , o ' = p ( o) for some a E A { a

0 } , o E D { d

0 } • So

Since

v(y1) =f';\(a)v(y

2)p'p(o)

=

^\J f ( a) V ( Y

2 ) Vp (

o )

=

v (f (a)y

2p(o)) •

v · C{c} -_{. 0} C'{c'} ₀ is injective, i t follows that

~

~

and hence b

1 =6(y

1) =6(y

2) =b

2 by 2.2 (iv). o

For the following corollary we suppose given the situation of diagram (2.8).

- 16 -

2. 12 Corollary. Suppose :\ n p : 1T

OA n 1T

OD ..,. 1T

OA '-n 1T OD' is injective and for all objects (a

0 ,d

0) of B

:\: A{a

O} - A' {\a

O} , p D{d

O} ... D' {pd

O} are surjective and

Then

v :C{c} - C'{vc}

0 0 is injective, where C = f ( a ) .

0 O

µ · 1 T B - 1 T B '

• 0 0 is injective.

Proof. Let

~ ~

~ ~

b ,b E 1T B ,

0 0 and suppose ~ ~

µ (b )

=

^{µ (b) •}

0 Then

<P (b

0 )

=

^<P (b) by commutati vi ty of ( 2. 1 O) and injecti vi ty of

:\ 1"I P • Suppose that applies to show that

b

=

(a ,d ) .

0 0 0

~ ~

b

=

b _{0 •}

o

Then Theorem 2.11

2.13 Remark. It seems that an extra structure of some kind of operation is needed to give 5-lemma type conditions for µ: TT

0B - TI

0B' tobe surjective. This question is discussed in the next section.

- 17 -

3. An operation

The boundary mapping

a :

B{px} - ^TI F of the sequence

X O pX

(1.1) can be described as the restriction of an Operation of B{px} on ^TI F to the base point of ^TI F and so d6es not

o px o px

depend on the choice of ^X in its component of F px However, the following example shows that the boundary

!:::. : C { c } - ^TI B of Theorem 2. 2 does in general depend on the

0 0

choice of b

=

(a , d ) in i ts component in B .

0 0 0

3.1 Example. In Theorem 2.2„ Zet C be a group and let f be thi inclusion of a subgroup A of C Suppose that D is

simply connected„ and that !:::. ~s independent of the choice of b

0

=

(a 0 ,d

0 ) in its component. Then A is normal in C • Proof. Let a E A and let y E C . Let a, y l i f t to elements

ß ED(do,d;), o ED(do,d) . Let also y l i f t to 0¹ ED(d;,d') . By the assumption on ^{!:::. ,} there is an element

(Cn)ER((a

0 ,d), (a

0 ,d')) . Since D is simply connected no = o' ß • Hence p{n)y = ya . But p{n) =

s

^{E A,} and we have shown yay ^-1 E A • D

Thus ^!:::. cannot in general be described as the restriction of an operation of C{c}

0 on ^{1T B}

0 However, we can give an operation of a subgroup of C{c

0} on the subset ~(a

0) of

TIOB which lies over the component of a

O

In detail, let (2.1) be a pullback of groupoids where p is a fibration; let b

=

^(a ,d )

0 0 0

define C ₀

=

f ^{(a ) •} _{0 I} and let

- 18 -

be an object of B i

where p : TI B - TI A .

0 0

3. 2 Remarks. (i) Let j : F - B denote the inclusion of the fibre F of p over a

0 Since p is a fibration and (2.1) is a pullback, p is a fibration. Thus by exactness of the sequence 1T F

0

-

j ^{1T 0 B} --1 ~

'l' ( a )

=

p ( a )

=

Im j •

0 0

we have

Thus 'l'(a

0) is the subset of n

0B of those elements b which are of the form b = ( a , d) ~ , where

0 ( a

O , d) E Ob ( B) • (ii) If A is connected, then 'l'(a )=TI B .

0 0

Let N(a

0) denote the normalizer of fA{a

0} in

C{c0 } , i.e.

N ( a )

= {

y E C { c } j y • fA { a } • y -1

=

f A { a } } .

0 0 0 0

We say f (i.e. if

is normal at N(a ) =C{c })

0 0

a0 if fA{a

0}

and we say f

is normal in C{c

0 }

is normal if i t is normal at all a

0 E Ob (A) . 3.3

such

for

Theorem. There is an operation of N(a) ₀ that !::.(y) =y•bo ~ for eaah y E N (a ) _{• 0}

.

Let q> be the aanoniaal map ^1T B -nAnTI

0 0 0 D

.

b,b

¹ E 'l' (a ) we have q>(b) =q>(b') ~ i f there is

0

~ ~

on 'l' ( a )

0

Then a yEN(a

0 ) such that y•b=b¹ The aonverse is true i f f is normal at a₀ •

- 19 -

1 ! , j 1 '

P':ti'o6:tL 'Let y E N ( a ) , b = ( a , d) ~ E ^{1±' (} a ) .

0 0 0 Then we define

y•b

tobe (a

0 ,d

1^)~ where

a

₁ is the end point of a l i f t

e

ED ( d, a₁) of y . For other choices ( a ~

0 , d ¹⁾ E b ·and y , there is a morphism

0' E D ( d' , a

1)

^lifting

(a.,o) : (a

0 ,d) - (a

0 ,d') Then p ( ö ' ) = y • f ( a.) • y - ¹

in B • Let

ö' =

⁰

'öe-

¹ : d

1 .... d

1

which belongs to fA{a

0 } because yEN(a

0) . Thus p(o') =f(a') for some a' EA{a

0 } and

(a' ,ö') is a morphism (a

0 ,d

1) - (a

0 ,d

1)

^{in B. Hence}

y•b ~ is well defined.

Note also that y•b =6(y)

0 for

The axioms for an operation are easily verified, as is the fact that </J(y •

b) = <fJ(b)

when y • b is defined.

Suppose conversely that b,b'E'l:'(a)

0

and that b = (a

0 ,d) ~

,b'

= (a

0d')

~ ~

~ ~

satisfy cp (b)

=

cp (b' ) ,

so d

=

d' Let

e :

d - d' in D . De f ine y

=

p (

e )

in

c {

c

O}

~ ~

(by assumption) ana· y • b = b' by defini tion of the operation. •

The operation of 3.3 is natural. In the situation of diagram (2.8) with given base points b

0

=

(a

0 ,d

0 ) etc. we have in analogy with 2.9:

~

~

3.4 Proposition. I f bE'l:'(a

0 ) then µ(b) E'!'(a~).

- ~

I f y EN(a

0 ) and v(y) EN(a;) then µ(y•b) =v{y)•µ(b) • o

This naturality of the Operations enables us to prove a 4-lemma type result, namely that part of the 5-lemma dealing with surjectivity. We suppose given the situation of the previous proposition, and consider the diagram

- 20 -

(3.5)

where

C{c}

0

V

l

C' {c'}

0

6. ¹

'IT B

0

p

<P'

'IT _o

B'---

T is defined by the maps

• 'IT An 'IT D

0 0 ^'IT C

l

^{An p} 0

T'

lv

• TI A'n 'IT D' ^TI C'

0 0 0

3.6 Theorem. Let f be normal at a , f'

0 be normal at a' _{0 •} Suppose V·TIC-TTC'

• 0 0 is injective, p: ^TI D - 'IT D' are surjective. Then

0 0

µ:'±'(a) - '±'(a')

0 0

is surjective.

Proof. Let b' =(a~1d ¹ )~E'±'(a~), and consider cp'(b') =(a',d') ~

0 Since

we can find d E TT

O^D with

p : TI D - ^TI D'

0 0

~ ~

p(d) =d' •

vf(a)

=C

¹ ='Vp(d) •

0 0

is surjective, We have

and

Since V is injective i t follows that i.e.

( a

₀ ^{, d) E TI} ₀ ^{A n 'IT} ₀ ^{D •}

Since is surjective, there is a b in

~ ~

that cp (b)

=

(a

0 ,d) • µ(b) E'±'(a') .

0

~ ~

We see bEp-1 (a)='±'(a)

0 0

TI B

0 such whence

N ow cp ' µ ( b)

=

cp ' ( b ' ) Hence by 3.3 there is a y' in C' {c'}

0 such that y' • µ (b)

=

^{b' •} Since ... C'{c'}

0

is surjecti ve, there is a y in C { c

0 } such that v (y) = y' Then by 3. 4

~ ~

µ(y•b)=y'•µ(b)=b'.

- 21 -

This proves, µ l!'(a) - l!'(a'Y' is surjective.

•

0 0 f

!

For the following corollary let as before (2.8) be a commutative diagram in which the inner and outer squares are pullbacks and p and p' are fibrations.

3~7 Corollary. Suppose V:1TC-1TC ¹

0 0 is injective„

A: 1T A - 1T A'

0 0 and p:1TD-1TD¹

0 0 are surjective. Assume„

f,f' a:t'e no:t'mai and v:C{f(a )} -c'{vf(a )}

0 0 is surjective.

Then JJ : 1T B - 1T B' -is surj e ctive.

0 0

Proof. The method is to start with an element of and then choose appropriate

that 3.6 can be applied. o

a' E Ob (A ¹ ) , a E Ob (A)

0 0

- 22 -

1T B'

0

so

4. Applications to homotopy theory

If X,E are pointed topological spaces, then let TTEX denote the pointed track groupoid whose objects are the pointed maps X - E and whose morphisms are the homotopy classes of pointed homotopies rel end maps. Then the set of components may be identified with [X,E] ,

the set of pointed homotopy classes of pointed maps, and the vertex group TTEX{·} , where is the constant map,

may be identified with [EX,E] , where EX is the reduced suspension of X (cf.[3]).

4.1 Proposition. Suppose we are given a pullback of pointed spaces

XnE

f

_E

pl lp

X B

f

in which p is an h-fibration in the category of pointed topological spaces. Then for any pointed space Z the canonical map ljJ :TT((XrrE)Z) - TT(XZ)nTT(EZ) induces a bijection

The proof of 4.1 is a pointed version of [15], 2.3, and is left to the reader.

In the following corollary TT

0Y denotes the pointed set of pa th componen ts of the poin ted space Y and, f or n ~ 1 ,

'ITnY denotes, the n-th homotopy group of Y .

- 23 -

4.2 Corollary. In the situation'of 4.1 there is an infinite sequenae

TT2E TT

1 E TT

0E

· ~

••• ;n2B

^{- T T6 1 (XnE)}

1/\

^TT ₁ ^B

_Ln

⁰

_(XnE)~/. Y\

^{TT B •}

TT

~I

2X TT

1 X TT X

0

The sequenae is exaat in the sense of 2.2. Furthermore there

the subset of elements of ^TI (XnE)

0 whiah projeat to zero

Proof. The seven sets/groups to the right come from a direct application of 2.2 together with the information

given before 4.1 in the case z is the 0-sphere s^{0 •} The infinite continuation is given inductively by putting

0

Zn+1 =rzn in a manner similar to [4], 6. Finally the operation is a straightforward interpretation of Theorem 3.4.

o

Clearly there are results in the vein of Corollary 4.2 corresponding to Corollaries 2.4 to 2.7. By replacing

f : X - B by i ts associated mapping track fibration the

result corresponding to 2.4 allows one to deduce the results of [9], 1.

In the following example, we give an application to Eilenberg-MacLane spaces generalizing [12], 17. Exercise 10.

- 24 -

4 3 . E xamp e. 1 Let X _f_ B ^....J2_ E b e a d. 1.,,agram o f E'? 1.,, 1,,en e1>g-b MaeLane spaees of types (G,n), (L,n), (H,n), respeetively

(n :2:: 1) • Assume that p is a fibration and k : G x H - L, (g,h) ._.. f*(g) •p*(h) ^.2. is surjeetive. Then the pullbaak of f

of is

and p is a

f p*

*

G

-

^L

-

surjeetive.

K(G nH,n) , where G n H is the pullbaek H. In partieular, this holds i f f* or p*

Proof. The result follows from 2.4 and 4.1 which gives an exact sequence

rri+1 (B) - rri (X n E) - - rr. (X) n ^1T. (E) - 1 •

l l

The assumption on k is needed to show ^7T 1 (X n E)

=

O . o n-

General remarks. There are also free versions of the above results, as well as dual results, both pointed and free.

All four cases can be obtained by working more generally in a suitable abstract category with homotopy system as indicated in [15]. That is, the arguments go over formally to such general categories and so can be applied there.

Some of the results in the dual, pointed case can be found in [24] under stronger assumptions than given here.

An example of the formal argument in the category of topological pairs is given in [10].

- 25 -

5. Non-abelian cohomology and van Kampen's theorem

Let X be a space with base point x

0 and let G be a group, not necessarily abelian. Olum in [23] defined a cohomology set and he showed that if X = U u V; W = U n V; x E W; U , V, W are pa th- conne cted , and

0

suitable local conditions hold (for example U,V are open), then the square induced by inclusions

( 5 • 1 )

H 1 (X,x ;G)

0

H (V,

, l

x

0 ; G)

H (U,x ;G) 1

0

H (W,

, l

x

0 ; G)

is a pullback. Olum also proved that there is a natural bijection

(5.2) H (Y,y ;G) ¹

0

for each path-connected space

Horn ( ^1r

1 ( Y, y

O) , G) Y and y E Y .

0 The two results easily imply the van Kampen theorem, for path- connected U, V, U n V .

Another useful consequence of (5.2) is the 1-dimensional Hurewicz theorem. Suppose that G is abelian. Then

H (X,x1

0 ;G) is the usual 1-dimensional cohomology with coefficients in G. Also there is a natural surjection

( 5. 3) _H 1 _{(X,x ;G)}

0 Hom(H

1 (X) ,G)

which is bijective if X is path-connected. From (5.2) and (5.3) we deduce easily that if X is path-connected, then H1 (X) is ^1r

1 (X,x

0 ) made abelian.

- 26 -

The pullback square (5.1) was generalized in [1] to the non-connected case, by establishing a sequence of the form given in Theorem 2.2 above. This gave for example a

determination of TT

1 (X,x

0 ) when U,V are open and 1-connected a")d W

=

U n V has n + 1 path-components.

In particular, i t gave a new proof that TT

1^(s1 ,1);;, 2Z. It seemed unlikely, though, that such a Mayer-Vietoris sequence would determine TT

1 (X,x

0 ) completely in the general case.

Let X

0 be a subset of X. In [2] the fundamental groupoid TTXX

0*) of the pair (X,X

0 ) was defined as the set of homotopy classes rel I of maps (I,I) - (X,X

0 ) ,

with groupoid operation induced by composition of paths.

The van Kampen theorem was then generalized to a pushout theorem for nXX

0 the conditions of connectivity of U,V,W being replaced by the condition that X

0 meets each

path-component of U,V and W. Remarkably, this result does determine TT

1 (U

uv

,x

0 ) completely even when U n V

is not path-connected. It is not clear how such a result could be obtained by standard cohomological methods, which usually

relate invariants in different dimensions by an exact sequence.

Also, the pushout theorem looks like a relative result (on the fundamental groupoid of a pair (X,X

0) ) , but does

not fit into standard notions of relative results.

The purpose of this section is· to show that by combining the non-abelian cohomological methods of [23] with the groupoid methods of [2], [5] we are able to resolve the above anomalies.

Thus we consider relative non-abeZian cohomoZogy with coefficients in a groupoid to establish simplicial analogues of the main

results on Mayer-Vietoris sequences and van Kampen's theorem

*) The notation is borrowed from [3] and differs from [2].

- 27 -

in [23], [1], [2], and [3]. The section closes with an indication of how the topological and simplicial theories are related.

In the following let K be a simplicial set with face maps

a. :

^{K - K} ₁ ^{, i}

=

0, ••• ,n ,

1.. n n- and degeneracies

s . : K

1 - K , j = 0, ... , n-1 ; let G be a groupoid.

J n- n

5.4 Definition. A 1-cocycle z of K with coefficients

~n G consists of two functions

satisfying (i) z

1(k) EG(z

0 (a

1k),z

0 (a

0k)) for kEK

1 , (ii) z

1 (a

1k) = z 1 (a

0k) z

1 (a

2k) for k E K 2 .

0

2

a

k

0

1

Note that the composition on the right hand side of (ii) is defined and that (ii) implies

(iii) z 1 (s;a) = 1 z (a)

0

for aEK

0

5.5 Definition. Let z,w be 1-cocycles of K with coefficients in G . A homotopy c : z ^c:! w is a function c : K - G satisfying ..

0

( i) c ( a) E G ( z

O ( a )_ , w

O ( a) ) f or a E K

O ,

(ii) c(a

0k)z

1(k)=w

1(k)c(a

1k) for kEK 1 .

- 28 -

c ( i3 k)

~o

By z¹ (K;G) we denote the groupoid which has as objects the 1-cocycles of K with coefficients in G, and as morphisms the homotopies of 1-cocycles, with the obvious composition.

5.6 Definition. The 1-dimensionai cohomology of K with coefficients in G is defined as

A map f : L - K of simplicial sets induces a morphism of groupoids f* : Z 1

(K; G) .... Z 1

(L; G) in an obvious way. Thus z1 (K;G) becomes a functor of K.

5. 7 Proposition. Let f : L .... K be a map o f simp licia i

sets such that K

0 is injective. Then f*:

z

1 (K;G) ....

z

¹ (L;G)

is a fibration of groupoids.

Proof. Let z be an object of Z (K; G) , and let d : f ( z) :::: ¹

*

w be a homotopy in

z

1 (L;G) • Then a homotop:y c of z satisfying

*

f (c)

=

d is given by

c (k) =

Next we define relative cohomology. We cannot define a relative theory l?Y ·cansidering cocycles vanishing on simplicial subsets since for groupoids as coefficients there is no

standard meaning tobe given to 'vanishing'. Instead, we have relative cocycles for each choice of cocycle on a simplicial subset.

More generally, consider a map i : K' - K of two simplicial sets. Let ~' be a 1-cocycle of K' with coefficients in the groupoid G. Define

z

¹ (K,~';G) to be the fibre of _i

·*

_{: Z} ¹ _{(K;G) - Z} ¹ _{(K' ; G) over}

The groupoid

z

¹ (K,~';G) has a set of components

and for each of its objects ~ (so that ^~ is a 1-cocycle relative to ~' has a vertex group at ~. The set

of components is written H1

(K,~';G) and called the

1T 0

1-dimensional relative cohomology set. The vertex group at

~ is written H~(K,~';G) and is called the 0-dimensional relative cohomology group at ~ .

We need a relative version of 5.7.

5.8 Proposition. Let

L' f' K' j

l

_{L K}

^l

ⁱ

f

be a commutative diagram of simplicial maps such that the corresponding diagram in dimension 0 is a diagram of

1 1

inclusions. Suppose

'1 ' ; ¹ 1'

11 i ! 1 1

L'=L nK'.

0 0 0

Then, i f <P' is a 1-cocycle in

z

¹(K';G) and 1/J' =f'*(<P) the induced morphism

f*: Z 1

(K,<j)' ;G) - Z 1

(L,1/J' ;G) is a fibration of groupoids.

•

The proof is given by the same forrnula as in the proof of 5.7.

Note that the assumption on L'

0 cannot be dropped.

If K = L = K' = { *}, L' = !Z> , and G is a group, then the

induced rnorphisrn of fibres is the group hornomorphisrn {1} - G which is not a fibration if G is non-trivial.

For the applications we are interested in an interpretation of cohomology in the case of Kan cornplexes.

Recall that if K is a Kan cornplex then the fundamental groupoid TTK is defined ([11],IV.5.2). Recall also that i f H,G are groupoids, then there is a groupoid (HG) whose objects are the rnorphisrns H - G and whose arrows are the homotopies of morphisms ([4],1). We write

[H,G] = TT

0 (HG) •

5.9 Proposition. I f K is a Kan complex, then there is a natural isomorphism of groupoids

r: z

¹ (K;G) - ((TTK)G) •

Proof. Because K is a Kan complex, each element a of TTK (x,y) is represented by a 1-simplex k E K

1 , and i f k, 1

- 31 -

are two representatives of a , then there is a 2-simplex k2 suchthat

a

₁k

2 =k,

a

₂k

2 ^=1, 3

0k 2=s

0y . So5.4implies that if z is a 1-cocycle, then the function f(z) TTK - G, cls k ^{i -} z

1(k) , is well defined. The definition of composition in TIK, and the cocycle condition, together imply that f(z) is a morphism of groupoids.

Let c : z ~ w be a homotopy in Z 1

(K;G) Then C

determines also a homotopy

r (

z) ~

r

(w) . Thus

r

is a function z¹ (K;G) - ((TIK)G) which is clearly a morphism.

An inverse 0 to

r

is defined on objects by 0(f): k...,. f(cls k)

for any morphism f : TTK - G ; and 0 is defined on homotopies as the inverse of

r

o

5.10 Corollary. I f K is a Kan compZex, there is a natural bijection

f : H ( K; G) -1 [ TIK, G] . D

In the relative case, let K and K' be Kan complexes, and let i : K' - K be a map. A cocycle <P ' in z¹(K';G) defines a morphism ^{<P 1} TIK ¹

-

^{G Let} ( (TIK) G) </i, be the fibre of ^l

·*

^: ( (TIK) G) ... ((TIK')G) _{over <P ' 1} and let

Clearly the isomorphisms Z 1

(K; G)

~

( (TTK) G) and

z¹ (K';G)

~

((TIK')G) induce an isomorphism z¹ (K,<P';G) - ((TTK)G)<P'

- 32 -

and hence a bijection

'1

^{1 ,i 1}

..

(5.11) H1 (K,q>';G) -[TTK,G]q>,

It is the latter set for which we wish to give an explicit description in some use_ful cases.

We assume that Then the groupoid

i : K' - K

0 0 is an inclusion mapping.

TTKK'

0 is defined as the full subgroupoid of TTK an K~ . Let Hom(rrKK~1G)q>, denote the set of morphisms from TTKK'

0 to G which give q, ¹ on composition wi th i : rrK ' - rrKK ^{1 •}

0

5.12 Prooosition. Suppose in addition to the above assumption that K'

0 meets eaah aomponent thePe is a bijection

Proof. Since

a: [TTK,G]q>,;; Hom(rrKK~,G)q>, .

K' meets each component of

0

K • Then

K , the inclusion

fact makes

i : rrKK ^{1 -} TTK

0 is a homotopy equivalence andin

Let r : rrK

rrKK' a deformation retract of rrK.

0

- rrKK' ₀ be a retraction.

Define a. by a. ( ;)

-

= ; i Then a. is well defined

because any homotopy c : i;:::: n in ( (rrK) G) ^<P is, by definition of this groupoid, constant on ^K' _{0 .}

Define an inverse S to a. by S (;')

= (;'

r) ~ • Then

a.ß

(i;') = ; ' ri

=;' ,

and ßa,(i;)

=

(;ir) ~

=

^~ since

ir :::: 1 rel rrKK~ . o

- 33 -