On the union of two generalized manifolds

A NNALI DELLA

S ^CUOLA N ^ORMALE S UPERIORE DI P ^ISA Classe di Scienze

P AUL A. W HITE

On the union of two generalized manifolds

Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 3

^e

série, tome 4, n

^o

3-4 (1950), p. 231-243

<http://www.numdam.org/item?id=ASNSP_1950_3_4_3-4_231_0>

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Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

http://www.numdam.org/

by

^{PAUL A.}

WHITE (Los Angeles)

^{: .}

/

..

The

author

has

^{made a}

study

ⁱⁿ

his paper (3) of additive

^set

proper-

°

ties. An i-dimensional property p’

^{of a}

space was

^called

additive if

it sa

tisfied .

^the

following theorem.

^{’ :.. ;... ’}

,

Theorem

^A.

^If

^and

the

compact space M1 U 1Jl2, and if

^{has -then}

M1 U M2 p4.,

and the

s~t theoretic and « inter

.section », ’ respectively;

^thus

«2013)- &#x3E;&#x3E;

^{can be reserved}

for the group operation). ^,

In

this ^paper

^{it is} shown that the

properties ^{« to}

^be

a generalized n, manifolq with boundary »

^{and ’« to be}

an orientable generalized n-manifold with boundary » (see

dennitions 5 and

7) satisfy

a

modified

form of the

ad

ditive property.

^{. ’ ~ ~ ~ - ..}

We

shall

^{assume that all} sets to be

considered

are

subsets

of a fixed ^"

compact Hausdorff space S. Since no metric is assumed,’we shall use Cech cycles (chains)

^with

coeSciehts

i u an

arbitrary field G j instead of Vietoris cycles (with

^{mod 2}

coefficients)

as in

(3).

^A

knowledge theory

^.

will

be

assumed since most ^of the general

definitions

and results

needed

are discussed ⁱⁿ _chapter

^{8 of}

₍₂₎

^and

the specialized ^ideas

ⁱⁿ

sections 4 and 6ofE. G..B.egle’s

paper

(1).

^The

boundary operator ^will be denoted

by . « Q » . , t.- ,

-

Definition

^I. The closed

set

isi-lc

(local-ly

^tlre

point p E A

^{if for}

each open

^{set P of S}

containing p,

^there

exists an open set such

that

p E QC

P and such that every

i-dimensional cycle ^on Q n

^{A 0}

^on

^P

n

^{A. .’}

(A Cech

^{is « on » a set} if the nucleus of

each cell

of

each

coordinate

cycle

^of zn intersects the

set).

°

...

.

The following definition, which

^is

sometimes more convenient to use,

^is

equivalent

^to definition, 1. ^{’ .} ;

.

BeNnition

I’..

The

closed i-lc.

if corresponding to

each

open

^set

P of S containig

p

and covering U of /S’ _{by sets,}

there

exists an

,open set Q such,

^E

covering V

^of

6Bby

^.

open sets

such

^{that V is a}

refinement

^of

and such that

.

if zi

(V)

is any

cycle on Q n A,, then

on sim-

plicial projection from W into U)

- -

Definition

1". The .closed set A C

~S

is

i-’lc ^if

it is i-le at each

point

p E

^,

Definition 1’’’.

The

closed set A c ^S

is

loll if it ist i-le for all i

.

Definition 2. The closed

^set

Ae

S i s co?cnect ed if

each

i

cycle

on .A. is

~-o-~ 9

on. A.

°

,

.

Theorem 1.

^{i-lc i8}

for

^{e(tch i.}

C-orallary

^{1’. Tlie -}

-

Theorem

^2.

The simple-i-connectedness property

^{is additive each i.}

These theorems

were

proved

ⁱⁿ

(3)

^{with Vietoris}

cycles; they ^are

^still

true in

our present more general situation,

^{but the}

proofs are oniitted

since

they diffei- only in tlre mechanical

^{details and not}

in

^the

essen- tial idea

from the

first ^proofs.

^Similar mechanical details

will

^{occur in all the}

folIo.

wing proofs, and depend

^on

^certain

^lemmas

that .appe:i:, in ^R.

^{L. ,Wilder’s}

«

Colloquium »(4).’..,1 ^will

^state

-them here

^for

reference. ; ø,’

^’

’&#x3E;

’

,

Lemma

1.

If, L

^{subset and U is ac}

^of

⁸

by sets, covering V,

^a,

refinement of U,

such that

if the

nucleus of a cell

of

^V

meets

^{both L}

and L,

^{it meets}

F ( I;),

^{the boun-}

-dary of

^{l,. ’}

’

Lemma

2.

1./"

^{zi is a}

^K

^{then the}

collection (a

^~s

an

(i

^{-- 1 on}

K,

^{which we denote}

. .. Lemma

3. If

^{zi is It K 0 on}

Jl,then

^there

^exists

zi+l - on ill such that 0 zi on K.

I

,

-Lemma- 4. If’

^{zi is a}

cycle

^{))tod K on 11’1}

^that

^{r7 zi 0 on}

M,

^then

exists a

cycle yi

^{that zi -}

y’

^{1nod If.}

’. Lemma

5.,

zi is it

cycle

^{mod, K that zi N 0 inod} then there exists’ a K M such that xa N

y~

^1(..

: .

’

.. We. shall need

slightly stronger forms

^{of lemma 1 which. we now state}

and prove. ’

^{. , .}

’

.

.

, Lemmsi

1’.

If -L

^{rs subset}

of

^a’

^closed

^{subset M and. 9t is}

IS

by

open

-Rits, then

exists (f, q)

of’ ^b~ by

open

sets,

, a

refinemeitt of

^such

that if ^the

^{ac cell}

of

^{q) both Land}

then

it meet8,,

(L)

⁼

boundary ^of

^{’L with}

respect

^to

Jli.

.. Proof.

Let U1

^{be ,the} subcollection of U

consisting

^of

^sets

^that

meet ill, then M)

^for

all.U.

^{E is a}

covering

^{of M}

by

^sets

open

relative to M.

By

^{lemma 1 there exists a}

coverin~

^{of M}

^with sets open relative such, that

is . a refinement ^of and such that if ^a cell of

V’1

^meets botl L and

ill;-, L, ^it

^will

^{meet the} boundary

of L rel.a.

;tivè

^to

Corresponding

^{to each}

V’1

^E

V’1 there exists an :open

^B

set .of

such

^that:

V’ n

^{M ,-}

vi,

also a set such : that.

let ^{then 1 ;hus} is a

eoveriiig

^of

M by open sets of S

^{that is a,}

refinement of

.&#x26; ⁱ I ’. I .

,

?P 1’ i .. ^{Let the}

^sets

that meet S - M and let, _

⁴

for

all

^then

q)2

^{is tt}

coveri Jig of ,8

^{._ l~}

^{by open sets}

^of

^{S .that}

is _.a

refinement of U2.

^{Let is a}

covering of S -0y,

I - 1..- - Ll... :.._-_..._...,.~ -&#x26;1...4- lW ..

open ^{sets that}

^{is a} refinement of U and clearly has ^the properby ^U cell 1neets both L and ^it

meets FM (1~)..’

’

.. :

.

1"., A is

^a~

closed, ^and of’ ^{B, is}

^(I

c.overing S, ^there

^(1,.

such

the nucleus o

^a

^cell of U meets ^{both A}

^{then -}

^{it,, 8,}

^ory

Ile

specifccall y, ^{- 11 U B,}

^{it meets}

^{Fc(d) c A n}

^.

.

.

.

. - .. ...- - - - - -""’1111. n v

Proof.

^{I,et, If a. cell .}

:,. ,,

is on ^’ A and . conclusion is

already satisfied. ^{If a cell}

^{is on}

^A

- - - .-.

and B

^{then it is Oil}

^by

^{lemma J (it the}

same choice of q)

^is

made)...

Definition 3. The closed

^{set is at}

p6 A, if

^{for every}

^open

^{set 11 of 8}

containing

p

there ^{exists an} open ^set

9 ^of Bauch

^P

^and suph any

^A

(,~ - (~),~ A

^This

definition is equivalent

^{to the}

^folio

wing one: , , ~ , , ,

,

Definition 3’. The closed

set A

8 /d pEA ^if corresponding to.

each

open

^set

P containing p

^and

^covering

^{U of}

^{open sets,}

there

exists ^am open

set Q such ^{that p}

^E

^{QC P}

^{and a}

covering U ^{of 8.} by, , open sets such

^{that and}

^{such that if}

^{is any}

^{cycle on A mod}

(s -

^{A then zi}

^{~?~): N 0 (8}

^-

~ ~

,

Definition

3".

The closed ^set

^A

is i-c6lo if

it is

i-colc

^at

^each ’A.

’

Definition 3"". The closed

set ^{A is}

said to, be Ion if

^{it is i}

^colc,

^for

all

t(O~~M).’ ’" , ./’ ’ ~ "..’ ,-;." ~.~~-/.’~’..:.: ^..

, 3. I’he

,f’v~~

^{, .}

,,

Proof.

Let ^and be

compact subsets of the compact

witere

M1 and M2 ^are i.-col’c ^and M12 ⁽ⁱ

der

p E

MI ^U M2. ^{If p E M129}

^{we cya}

^{suppose the open}

^set

^1’

^of

^defini-

tiou 3’ is

chosen such that

^{P =}

0. The

^i-colc

property ^for M! at _p no w

implies

^{that jM}

^{is i-colc}

^{yt p.}

Similarly if p ^E lil2

^for

Mz at

im,plies

^{that M is}

^i-colc at,p. Finally consider a ’point p E M12, an ^set

^P

such _{that p}

E

P

and

^a

by open

^sets.

^tjetQ

^be

chosen accor-

,

ding

^to

definition

3 for the

property

^of

(i

^oll

M12

^mod

^{(8 - P} n

^{N-0 mod}

(8

^-

M12.

By

^the

property

^for

and 1112 accordyng

^to

dehn.itjol. ^3’J

^we

choose

open

^sets

R’i ^and B2

^and

coveri ngs qJt

and U2 such

ttia p E 9,.

,

,

i

.

t

,,’ ’ .’ ^... ,-

, "

. . , ;,; " ^.’ ,I ...!,

and such

that (Vi)

^{is an}

^i-cycle

^on

Mj ^{mod (S ,} Q)U

.. &#x3E;

^such

^that

^if

zi

^{is an}

i-cycle on Mj mod S.

:-

- ’ ^-

, ".---- ^,: :,:&#x3E;...»...i ,’, ^, ... .".J’

"

’

’

" ’

.

234

mod

"

on M Let R be an

open set

, ., ^. , ^{.. -} . ..

such

^{that and} let V be a refinement of

Yj ( j = 1,2). ^By

lemma

1"

^{we can}

suppose

^is

^chosen

^{with the}

property

^{that i’f a}

cell is on both

lJl

t

and M2,

^{it is on}

B’y applying

^{lemma 1"}

’again

we can

require

^{that has the} additional

property

^{that if one of}

its

cells

intersects

^both

(is

^-

1)) n M~

^and

(8

²⁰¹³

P) n u2, it

intersects

(8

^.-

P) n M,2 . Finally

^{we can}

^{suppose V}

^is a refinement of CJl

with the property

^{that the}

projection into

^{9f of}

any cycle

^oii M mod (S-P) U M ^{is the}

coordinate

of a

Cech cycle

^{on ‘M mod}

^(The

^{existence of}

^such

^a

^refinement

for

any covering U

^is

established in

Wilder’s

« Colloquium »). ^{Now consi-} der any i-cycle

^zt

^(q))

^on

^{i+I .mod (s} U2J12). By our choice

of

’2~,

zi

^{is the}

coordinate in

’2t of a Cech

cycle

^Zi

(,~0))

^{on M mod}

We can write for each 2

-

I

Where z s

the _part

of

zi (Qv)

^on t

(i.

^e.

forme

._1~ ^{I I I J.’ I I - . . - .} ..

I

from -- - ...._^on

311

and incladed

(90)

with the ^same coefficients that

they

^rave

in

zi and z

hence 2

^A;

Bv .

hypothesis a

s ;

cycle

^on

Let z ^{vhere ~ s} the

part

of z

In (,

1

and 2

^{hence A}

is ou (

...

Finally

^let

Since

I ^{, -}

on 1t14

and

^{9 OH}

a I

it follows that

, I 1. ^’ - , 11

is’ on

Furthermore

^thus

s on

both 0 ^and

^{( and}

is,

^there

~ I ^"

fore. I on (,, I , y.., ~ I~ , ^We have,

thus.

shown ^that

0)

^{is a}

cycle

^on

~

~ , . ~ I -

M.~ mod. (8 ^- j)) f1 for

each

but ^we

have

not

yet showii

^that

s a relative Cech cvcle. To this end let ~ C)O, ^II

then

since øi is a Cech

cycle

^{on iJll mod}

(S

^-

P) n

^{there exists a chain}

on M

^{and a}

(~~~1) ^{on (S}

^-

^{1’) n}

^M

^{such that (1)}

simplicial projection

^from

Q02

^into

Let I Mhere ^I s the

part of

I ..., . I

(

Similarly

^{let~ c}

here

^{a s}

the.part of a

on

(~

^{and a}

B# iaking

boundaries in

(1’)

^we

^obtain

I ^{- -}

_

3y expanding

^and

algebraic ma.nipu-

r ^.

lation,

^this

becomes 1’

which we will denote

by ;

^,,

^Since

^,the

left

1’:;; . ,. ^{Q. _} ,

,

hand

side is

on

(S -

P) U M

and the right hand side

is

on

(

~~-- ~.~- ^{-- --} %I- - ~ ~ ,, _ _ ~

-

we conclude

that eacli side,

^i.e.

Is on (

^:f

we expand

and rearrange (1),

^we

^obtain

where’the left hand

side)s

on

M

1 and the

right hand

^{side is on} hence both sides are on

That a .Cech

cycle

^mod

(8- P) n

^on

M12

^now

^{follows. since .}

1 1,1.% 01 B 1’; ; i ~nv ~ I : ~nn~ ~ ~ ~nnt ~, ^. 1 n ; rnm ~ n ~ : .~nw . ’B

which uy

mod ^I Oil ~I By the ^cnoice ot

y, we

^conclude

that 71 - ^{B1 -} WH ’1119 ()Il every covering ^--.

-1 . , 111, l I , "., .1. , ,. I - -1 - 1.

vnere ) ^{is a chain on}

...’N 11, 1 ir; ir

and S

1M It CllIblIl OIL I

,-,., ...

1V ow Jet

r1 I., .

then

n : ^’"

cycle

^on.

Mj

^moo (

Ci W101v

for

which is

on ~ ( In particular ¹

,-., B. ,

is a

cycle on Mj

^mod

, --... - a -

therefore ^o

_ .".,

mod k

by

the ^{choice of}

"""’, ...

1(, ana V. Since

AO : aB.

= -

we conclude

that

mod (

^an

lfinally

^{since z’ is a}

cycle

z ,~,,, ^~ nr _ ":,, . 1,

I

-111 - 1..-

on mod )n ^I

11 aaw’i

.~4- -.

on

M,

^which

by.

definition 3’

.

- ’ ’ ’ . = ...

.

Corollary 3.

^I’li-e

p,rope;.ty lell

^i-v _,

’

, The

following

^{is a}

slightly generalized

^form

of the definition

of a local Betti

uumber (see [1]).

^{It is}

equivalent

^{to the}

ordinary

definition

when

B = ~0. ^{’ .}

Definition

4. lf H A are

closed subsets of S, then

^{we shall}

say

that

the. ^local

.Betti

of’,A

^at

^{p niod B}

^{is tlte}

.’finite positive integer

^k

(denoted

^{if k is the}

smallest positive integer

^{with the}

property that corresponding

^to any open

set

^L’

such there

^exist8 an

open such that

^and

^{such -} that if k+

^{t are}

cycles ^{on A}

^-

I’)

^then

there

^exists

integers m, , i 2 &#x3E; : ..ra &#x3E; k+1

...£B.4- n such that 4. AU I AU _.11 I - I _u _.M n. - .

not all U such that t mod 1

¿.&#x26; . °

. , .

,

. ’

This is

equivalent

^{to the}

following definition.

Definition

^{4’. If B} audj4

are ctoséd

subsets

of S, ^then

if k . is the smallest

positive integer

^{with the}

property that corresponding

to any open set P with

pE

^{1’ and}

^covering

^{U of S}

^by open sets, there

exists ^an

open set Q

^and

covering V

^{of S}

by open sets

^{such that}

:p,e

such that

if ~i (Q», zi (Q» ; .. , 2J§+1 (Q»

^are

^{cycles On .A Inod}

(8

^there exist

integers m1 m2 . , ,

^{not all}

LL.:.l u- I . . ’’’’B rt. : " ".,., .. "Y"’to... .

that n mod &#x3E;ri A~.: ^.

236

In some of

the following theorems the following aasumptions and

^no

tations will be assumed. ^{.. ,} .. ^{; ..}

,

As usual P-~, ~ ,~,,; ~,nd~

^{__ __ 1tL _} ,

-~ M-, be compact

^{subsets of the}

compact space S. Also

^{. let I}i ’1’" i ^and

ind assume vbere

122013’’-*-y*’*i;;~!t’’:B 17, ’+OUI£.a,-’ v

-M B-,-I/

--

-~MB2013-2~ 7--Ig -- - -

-

boundaries v nf A -relative to resp. Note ^.

I -k" / ^I

that ,that .1’

11 .- ^.

’Thi- follows sinCe

..

I

uy. yam,:

implies S

...- - .- i,

therefore F

^{Fol’ reference we}

., -.- , -~’-;- -- -:- - - -; - 1,(; 1~’’

vi11 1 Ai

..

Theorem 4. A i/’ (p , Mj i) &#x3E;

^{:~ ,}

p .E ( j =1,2) ^{(p, 411}

^{p E the)t}

~ ., .. ^..: .... &#x3E;

/

’

: , ..- Proof.

If p % m 12

¹

&#x3E; a implies (1~ ~ :~ ~ ^~’) &#x3E;

^0.

Tv p E M -..;.. lot P be an arbitrary

olden

^:- set of Since

_ V . -

1)

there

exists

^acl

open

^set

Q

^such

that p

--,.-, ,.c 7 ~.-"1’: 1 .1 . = 11 , , ..

I

I I - -

,that. for any two Cech cvcles zn-1

m-od f(

~.’.~. ~1L&#x26;o"’MA.’ ava ~awa.~J. :v ...~ ^- -~~_.- -tl ^{---.- ’-1} 1 7 ._~

. ._

Lbere exist both zero

such

inod ((

Llso since R

22 ^-

iere "exists

- -

.. ,... I r ^{i , ,}

an

open

^set

P C p

such that _{f’or any}

open

^{set that p E v C}

^l’,

there exist

cycles zj

^on

Mj

^mod

^[(8

^--

^{P) (j :Mj] U}

^{such that}

m z ~ 0 mod [S-R) U M1]

(where integer) implies m =

⁰

(j=1,2).

Let- P !set’!!of

(2), let Q

be the open ^{set from}

( 1) corresponding

to

P,

^{let R be}

any arbitrary (but fixed)

open set such. _{that p}

pr

aii:d let be

the corresponding cycles

^from

(2).. By

^lemma

1",

^con-

only

have coordinates ^{on a}

confinal family

^{of cove-}

property that if a, cell is on [(S -P) fl U F-’,.alyd on 4 n

is, on { 1

--- - - I I I

ltnd

--_. --. -,.. ,- t,-- ^,

-

i ii - J

ia

ⁿⁿ M.

and

is on 1 ^{Av, H-} i

. ""--1- ...."’--

--27 &#x3E; ^{-- ’. - .}

$ aL

Cech cycle

^on

- 7 B - ~_ ^{I ’-- I I} For each cove-

J

Iiz

11

vhere. a s the

part.

----

of fi

i ^I

Mid 1

i ^’

Since I

and y

______

~ ’ ’ --i ^{1 11 ~} 1 ^{1 1}

~on

it must ,be on ~I~~~. ^Fur-

,

thermore

.... ’-- 00.7.... ~rv ~ w.. ....- 1~. --- ^.

where; the first term

.,... ""... ’" ^..

is on

und

the second is on

~Vl ~ ~ ; therefore each ^{is on} Iii.,

particular

12

cycle

^on

.&#x26;.

To show ⁱ s a Cech

cycle, ^consider.a

L’ ....&#x26;... - . - - ^- ,- ^- . ^,

Since a

Cechcvcle. there

existsacbain t

’"’,’!’J’ v o

on

.- i

’

such that i ^ls in ^,

the ,~ where s-

andj

son’ This gives

It follows ^as

before

that ^{-- a}

chaiii

^on

hence

’)

^{Oil . This shows}

that

is

^a

Cech cycle ^on M"12 ^moc

There

exist _integers

_1ni

uot

^both

zero such thu

mol 2

aud let

us

suppose that iii, ⁺

⁰ for conveuience.

2 , -

.

- - . , ; -

. L ^B ..." ... - ,., ^. ,

This

implies

^the

existence

^for

each

^{of a}

(U)

^on

^and

^a

chai

:)

^on

12 such that

Let

then

)

^which

By

^{our choice}

^{of R} =t=0y

^we

^{know tha}

^mod

" .I , e, there exists a

covering V such

moc ^. Now consider

any coverin

,.. -_.~.AIo.."""..

.

W -

J,

^{then for any}

- ... - ..,

integer

^w

~ 4

^{we havi}

)~. It follows tha’

^{LJ mod}

"

^for

otherwise there exist

chains L) ~

^{of dll 1 ) 01} such tha

Let ) where

’ ) is ^{Oil LVil and}

p

^is

)

^where

))

^{is ol and ,I}

).

^It

follows

^tha

/) ^{wnere side}

... ’W!’8

°

is

on

Mt ^aud

^the

^right

^{side is ou}

^il-12;

^{thus ootn sides are oi}

If we

denote

^this

^chain by (’1’), ^then

^{we haT}

) ; thu

^{U 0!i}

I

Since zn1

I ^{is a}

^{relative Cech}

_

^cycle

_,

,/ ) OJ1 _i ^~u

I , ; therefor

^{0 mod}

¹

which implies

~

0, a contradiction ^{since and ’Ill ~}

^0. concludes the

proof tha

^{0 oil}

^M

^{mo r for} . _.

^any

^Integer, .

0. ^We have, ^thus,

^{found an}

open’set Q ^Dp

^{and a} covering V

of S such

that for _any and

covering 90 &#x3E; ’-1’, ^there

^{exists a}

^cycle

^{ztl on}

M

mod JP such ^thi 0 oil M mod

for all

integers &#x3E;n # 0 ’

^°

^This

^{is the - statement or the negative 01}

- -- ..." -- ....,..

’

(p , - M , ^/1’)

⁼

⁰ according

^to

4’; hen

Theo;eni 5.

assumptions A ,

¹

^for

^a

- ,.

1 an ^U for a,

the

1 1

. Proof., According to definition ^4" that for any open ^o.v P

p and covering 2f there exists

’ai’t open set B ^such that and

-

covering U&#x3E;U such that

ⁱ

I I

’

V).ain

^{,), are cycies on M moo}

6. Annali della Scuola ^Norm. Sup. - ^Pisa.

238

;

^then

there

^exists

integers

m, and not bottu 0 such that

0 oil M mo . I , tlaeu

1

implies

^{. I}

,

^{let P be}

arbitrary

- · ...’ , , - _ , , = .J. 1# / ^{- - _} - - - .-- - - tl

and

choose

Q

^{such that} p ^E

according to

definition 4 of

(p ; M12,

^..

F12) 1

^such

^that

^{for any two}

(n -1)

²⁰¹³

dimensional

Cech

cycles

^on

M12 mod

a .. -

there

exists’ a

non tri vial

linear cotnhma tion that

is N

^{0 on 1nod}

-...

.

By

definition 4’ of

---

.. -- ^-

? we can choose au

open

^{set R such that}

covering ’1)

^{that is a}

normal refinement

^{of’ ’21 with}

respect

^to

cycles mod

and such that if is a

cycle

^mod

),

^then _{, I} I ^{t mod} -

.... ~.

’Furthermore we shall assume

by lemma

1" that all

cycles

^are

defined only

on a ^confinal

family of coverings ^with

^the

property

^that any cell on M,

. and

M2

^{is also on}

J.tJ 12’

^{and that}

any

^{cell 011}

0

~

^and

on

)

^{is also 01}

First

consider a

cycle z n (,V)

^{on M}

mod [(2013

^{Since V is}

’

a normal refinement of

2f z "I (-T)

^is

the

coordinate

in

^{-2.f of a Coch}

cycle

I

^{on J/ movl I. For}

each

) wlierE

^{) is on and}

~

^{is . on Since}

I)

^is

I - - I

on can write

where

) is on

I - - I

and

is on

.. _ ,

. 1 ^{- ~ . .}

It follows

by

^{the usual}

argulnent

^that

| is on

; hence

. &#x26; _

is a

c,ycle

^oil

Mj

^mod Furthermore

therefore

I

2 and

. .

, I z ^{I , , I}

)

^{is ~~}

cycle

^Oil

M~ ~

mod .

By

ⁿⁿ

entirety analogous argument

^it

follows

thut

)and

^{are Oech}

cycles,

^mod

and

, respectively.

Let us suppose for the moment that mod

i.e. for each qO there

exist

^{chains oil}

.1112

^{and Oil}

,, such that

~ ^and then

where 1’1

^{is on and}

x12 is on

and

where 7’2

^{is Oil}

This

shows

^{that Ct}

is

^a

cycle

^on

Mj ^nod

... ’

1 tl particular

^{this is}

trne

^for

Cl (Ql) ; hence

^mod

) and

^{mod Since}

we conclude that ^mod

Finally since zn is

^{a Cech}

cycle ^{on M}

^mod

mod and ^mod

Now let ("1’) (#)

^be

any ^two cycles

^{mod As}

in the

argument

^{of the}

preceding

^two

paragraphs we

^{can write}

and

cycles and

^mod

By

^our

^choice of R , ,

^there

exist ^integers.

^’l))21

^not

^{both zero, such}

that on.

mod

^The

^argument

of the

preceding paragraph ^with

^{is now}

applicable.

This leads to the

conclusion

^that

mod

^{wich is the} conclusion of the

theorem.

^’

Theorem ^6. assumptions A , ., ^{if for all .}

^.1

and

for

^{a 11 then}

for all

Proof., This follows directly

^from

^{theorem 4 and}

^{5, ’}

" ’

.

Definition ^6.

^A

^closed

^{set’ M C S is calle8. a}

generalized

with boundary

^{relative to}

S

^{f ’ ...}

.

a,) .dim

^{M = ~z "}

’

,

.

b)

^{is l en - 1}

/

c)

^{M is . ’ .}

.

d,)

R~

( p ""M, F)=’l for , all p .E

^{M . ;}

e)

^Rn

~p ~ :- R,z (p ~

^{0 for}

all p E

^{~’ =}

the boundary

^{of M}

relative

to

s

Definition 6.

^A

compact space M

^{is called}

closed n-manifold

.

if it satisfies a , and

^{c of}

definition 5, together M)

¹

EM., ^{’ ’ ’ .}

’

, . Note

that

^{this is}

equivalent

to’ definition 5 where

--- S, ^{for then} causing

^{condition d to reduce to} d’ and condition ^e to be

meaningless.

This

is

^the definition

giveu by

^E.

G. Begle

^in’

[1] ^without

^the

orientability.

Teorem 7.

of having

^{dint n}

is additive.

’

Proof." -This

follows directly ^{from the}

^{« SUm}

theorem

^for

Diin

^{n »}

[see 2, . 30 which tell ut

^{that the union}

^’closed

sets each with dim ⁿ

also has diin

n (rega.rdless ^{of their}

^~

Theorem

8.

vnder

^.

aS8umptions A, if Mj is .a’ ge)iei-alized

_ , .. II - ... ’I J- -... - --

boundary relative

^to

⁽ⁿ

^{wita voun-}

dary

^{relative tc}

for,all

then M ^{is (t}

generalized ^{with boundary ’relative}

240

Proof.

Property a) follows

since di1n M ^{= n} is additive.

Properties b)

and c),follow

from

theorems 1’ and 3’.

Property d)

^for

Mi gives J

for

atl p

E

31; . Property ^d)

^for

^M,2 gives

^{J or all I}

since ·N

boundary

^{- of - relative to , To see tlmt the}

ter statement

holds

consider and

any neighborhood

Since

p ^E

F, there exists

ⁱ

^heuce.

^iillud

p E

boundary

^of

311z ^relative

^{to 8 Jl}

^U M12 - Conversely

^if p ^E

boundary

^of

relative to ,

then p E M,.,

^since

Mt9.

is closed and _any

ueigh-

J..

borliood ^{be a}

neighbourhood

of q,

then

there exists an

thus

~, which

together with implies

^{I Since}

is

assiimd,

^condition

d), ^for

^all

^p’E

^{Jf follows}

from

theorem

6.

Condition e)

^{for and}

gives

^Rn

(p,

^{= 0 for all}

p E Fj

^{and R,,-,}

^{(p ,} Mt2)

^{= 0 for all}

^{p E} .F’12 .

^We

must

show that R,,

( p , ill)

= 0

for all this

follows directly

^from

Rn ( y ~ M~) = 0 ^for

^{all the}

result follows exactly

^{as in}

theorem

^{3 since}

^the

condition n-colc _{at p} and

R~ ( p ,111 ) = 0

^{are so}

nearly

the. same. We have now showin that M satixfies all the

properties

^{of an}

n-manifold with

boundary

^{relative to F.}

’

If we

require

^{that Al} be imbedded in ^a

compact

^{subset of Eaclidean} n-space. ^{then the}

assumptions A fo1l0w

^{from the} dimensionalities of and

m t2 7’

^i.e.

^{FM (M~)}

⁼

^{~’M = n?12..}

^Since

^{FM (Mj)}

^{C in}

^any

case, consi- der

p E M12

^such

^that p E ^Vim

^or

^FM (,112) ;

t.herefore there exist

neigh- borhood U.

^and

U2 ^ofp

^{such that}

^iY’n UjC Mj. Choose U &#x3E;p

^such

that Mn

^{7 bnt it is}

impossible

ⁱⁿ

Euclidean n-space

^for

the

- 1)-dimensional set Mi2

^{to contain a set}

wich is

^{open in}

J112 (see

theorem IV ³

of ~2~ ).

^,

Theorem

^9.

If

^and

n2

^are

relative to then qpace tha,t

lVlc n M 2’

^{is tlte common boun-}

t

a1td M2

^{and is a closed,}

⁽ⁿ

^{- M is a}

generalized

^closed

Proof. By hypothesis

⁼

^Ft2

^{= F= 0}

Conditions

a, 7 b

^{7 and}

^c

^for ,11 follow as in theorem 7.

Condition

^{d for}

Mj

and d’ for

M12 ^give

^{=1 for}

^{;ill p}

^E

M’ ,

^and

Rn-1 ( ~ ., Mc~)

^{= 1}

for all p ^E

lU12.

^Also

^{since ~’ =}

⁰ the condition R’1

(p

^{= 0 for}

all

p E Mt2

^is

equivalent

^to

.R" ( p ,

^{= 0}

^which

^is

^gi

^ven

by

^coudition

^e)

Now all the

hypothesis

^{of theorem 6 are satisfied}

^{and we conclude}

that

Rn (F, ~ y") =

^{for all ] which is condition}

^d.’)

^{for M.}

.

Deflnition 7.

^A

generalized

n-manifold M with

boundary

^{relative is}

called orientable

^{if the n}

dimensional

Betti number

of

^{M mod}

irreducibly

^’

(i. e: p" (M , F) ---1;

^{but if}

^{L c E , L c M,}

^{are clo.}

, ;&#x3E;;

’ ’

sed sets such that at least

one

of

the

^{last two} inclusions is

proper, then pn (L , E) = ^0)....

I)efinit-ion

^8..A

generalized ^closed n-manifold

^{M is} called orientable ^if

pn (M)

⁼¹

irreducibly (see [1J ^j.

’

.. =.

. ^B

10. ^{A, i f} -Mj

^{is acn} orientable

generalized

.

_ _ ’...’w

n-manifold relative

^to

is

1. - I - ". I-^an

lized

(it

^with relative to S-M U ^and

. .. - 48." , ,

for

^{all then M an} orientable

generalized

with

^relative

to

^{fj. ,}

Proof.

Everything hut ^the orielJ.tabilit,y

^{follows froin theorem}

^7.

^{To show}

M is orientable

we

see

that

⁼¹

( j 1- , 2) ^since Mj

^is

^orientable

aod

since MJ2

^is orientable the observatiiun ^{as in}

.

- -- ... I.. .. ’T_L- LL__L’

theorem 7 that

F12 = boundary

^of

M12

^{relative to . iiote that}

but ^{, To}

verify

^{this suppose, lor exam}

·

ple, 11112 C Ft

^which

^{implies n}

^{it 101-}

lows that

Fi2

^{= M}

¹² hence Pn-’ (Mt2,

^=--- 0. which, is con-

trary

^to

hypothesis. This

^shows

thut, Mj U F is a proper ^subset of ^{Fj; liencle}

p"

^{= )}

(j I , 2). Now ^{(M , F)}

^{= 1}

followsfrom the ana-

logue

^of

^theorem

⁶

ⁱⁱⁱ

^the

^large. Actually the ^sane proof ^could be used

where

^’all

open

^{sets chosen}

ⁱⁿ

^the

various definitions are taken as the

^{inte. ’}

rior of

At (which

^.is non-vacuous

since p" (Mj ,Fj) =1

^....

- - - ,... A ."

To, show the rest

of the orientability condition,

^consider

I where one of the first two

inclusions

^is

proper,

^{und let}

. Note aisc

F .r.

that ,

and

i ^therefore

This

also

shows

. We

Sha1l vtirst ’consider

^the

^case

^{where L is a proper subset}

_ _, _

anl

,then ij

^{is a}

proper

^{. subset} - .. - - ^ot . 1 -- - :.t....^{=_ 1. for}

otherwise

2 for at least

one j.

^Also

L1z

^{is a}

^proper

^subset

_ , _ .

of

JI12

^{sinc .2}

^b3

^.

^Using

^{the orien-}

tability. hypotheses ^and

^{we 4}

since

7~

^and

^{L 12}

^me proper subsets

of Vj ^and respectively

^{such that}

and That

p",(L, E):’:;_- 0

^{now follows}

from a proof

^that

is almost identical with that ^of theorem

3 (where

^the sets chosen ^{are all}

equal

^to

the

interior of

L)....

~"

Finally

^consider

^the

^case

wh

. c

o«’-w:1

hence

^{* Since one or tile inclusions}

.. - -- ... n ",,1 ~ 1~_~_:,.

must be

proper,

^{it follows that} one of tile ^tour inclusions

..

-

.

)

^{must be}

proper. Suppose

^{that eithe}

1 2013

is

proper,

^which

implies,

^{since 7}

242

or is

proper. Suppose

that

the ^cycles

^used

^only

have coordinates on the confinal

family

guaran- teed

by

^lemma

1"

^{with the}

property

^{that a cell on}

.L1

^{and on}

L2

^{is also}

on cel I on

~..9

^{and on is on .}

Since it follows that

is a proper subset

^of

All, (= ⁱⁿ

^this

case). This in

^turn

implies

^that

En M2’is a proper

^{subset of -1 for otherwise}

wich is a contradiction

since Fj2 is a

proper

subset

^of

It follows from the

orientability

^of

J12 relative

^to

F2

^that

p"

^I

Now let us choose _any

covering 9.f,

^{then there exists}

(both from

the

above

^defined coufinal

fainily)

sucl that if

G~2 (?~) ,

^is

ally cycle.

^on

L2

mod

En M21 then V- ^{(~.J) N}

^{0 on}

^L2

^mod Furthermore

suppose

that #

is

a normal refinement of ’1t with

respect

^to

cycles

^{on L mod .E .}

Now let ~

(’1’) be,

any

cycle

^{on .L}

mod E ,

^{then ; is the}

coordinate

^{on Qf}

of

^{a Cech}

cycle 2;"= x’~ (~~) ~ ^mod

^{E.. For each ’10}

let ; ¹

..,

where is the

part ^of

on

1,1

^and

is on -

By hypothesis

^{is on E and we}

can write 1 where

i g the

part

^{of ou and}

is It follows that

where ’ the left hand side is

on _L,

and the

right

hand side is

on

L2

^{i hence}

^by

the choice of ^our

coiifiiial family,

^both

are

on

Lz ^for

^each This shows that

z"(-10)

^{is a}

^cycle

^on

L, mod El ’

for vhere the first term is and the

&#x26; I - - ^-

second

is

^on

L12;

^hence

^both

^are

,DM1) ^U L12

⁼

E1. By

^an

^entirely

analogous. argument

it follows that

z§’ = (~~~J) ~ is

^{a Cech}

cycle

^on

lit

^mod

El.

^Since

Mi

^{is a}

generalized

^manifold

^with boundary

^and

b.,- hypothesis

either

.L~

^{I is a proper subset}

of

^for

_.Et is

^{a proper subset of}

.F~ ~~

^{we have}

pn E!)

^{= 0. In}

particular _{z~’ ~,}

^{0 on}

L,

^mod

.l%~ ,

and there exist chains

on Lt

^{and i oil}

E1

^{for each such}

^that

^I

By taking

boundaries on both

sides,

^{we see that}

Also let where ¹ is ,on

L12

^{and I}

is

ou then

It follows that I

where

the left hand side is ^on-

L

^{and the}

^right

^{hand side is on}

thus by

the choice of our confiml

family

^both

sicles,

^--

a chain on

Ei2.

^In

particular

⁴

then

wich is on F

thus

is a cycle

^on

L2

^’mod proper closed

subset

^of

L2 .