Microboundles and Thom classes

Nicroboundles and Tho1 cl~sses

by

Per Eolm

by

P H ¹ 1 ) er o-'-m

In this note we introduce Them classes of microbundles.

We determine the Thorn class of the Whitney sum as the cup product of Thorn classes and state two applications; one to Gysin sequences of Whitney sums and one to the Atiyah-Bott-Shapiro duality

theorem for Thorn spaces (cf. Atiyah [~J

).

Thus our main result states that for microbundles ~t1 , (-l2_ over a compact (topological) manifold X, if T (X)<±.! 1'~1 ^{(B ~~}2 is stably fjbre homotopy

trivial, then

(.e

₁ and

1J2;

have S-dual Thorn spaces. Actt.:ally ou:r·

result is more Qeneral since it treats the relative case, i.e.

with relative Thorn spaces. This makes us able to handle manifolds with boundaries (by passing to the double) among other things.

Thus proposition

(3.2)

in Atiyah C~J has an extended version which just appears as another special case of our duality theorem,

The approach given her3 to the S-duality theorem shows very clearly that S-dJality of Thorn spaces is simply Alexander·- Spanier duality of compact pairs in the bas§ manifold lifted by

Thorn isomorphisms. Our approach does not make use pf imbectdings 1:han Atlyah's of manifolds and seems more ~elevant to the~result~ We also think it is conceptual! y easier than .4ti yah's method although there c..re some technical difficulties due to the fact that we work in a more general settingo

Throughout this paper all base spaces cf bundles and micro- bundles are assumed paracompact unless otherwise stated. Manifolds are manifolds without boundary, For net a tior;s and concepts see [Lif] ~

Generalizations and details will appear elsewhere.

1 \ 1 This work was done in Berkeley, California, in spring 1965

while the author was supported from NAVF (Norway) and NSS ( ccntrac:-

1. Throughout this paper R denotes a fixed principal ideal domain. By a local system on a space X we understand a local system of R-modules on X, i.e. a (contravariant) functor frcm the fundamental groupoid of X to the category of R~modules.

If ~ : X

2;,

E 1?7 X is an tRq~microbundle with total space E, write E⁰ = E-sX. Then there are local systems r.[J = t()( ~),

c{!

⁼ (0*(~} on X depending only on the equivalence class of

f-l

such that for x ~X, fOx= Hq(Eix, E⁰ !x), (D~ = Hq(E\x, E⁰\x) an.

for a path class

[w1

in X from x₀ to x1 , ^{(7, :}

Co .,. ^(D

Ll4\ X o X1

and G)* : '£)* -> (()

*

l<J.)] xo x1 are the isomorphismsinduced from the

(homotopic) map-·germs of fibers Ejx₀ ==±;? E\x 1 determined by

[cv1·

(Coefficients in homology or cohomology are taten in R if not indicated.) ~ and~* are constant over each trivializing subset of X but in general not constant over

x.

A Thorn class for the microbundle ~ is a cohomology class

q 0

* *

^{q 0 "}

U E. H (E,E ;p (Q) such that the restriction to H (Elx, E

I

^x; 0~~

=

Hom(Hq(E\x, E⁰(x), Hq(E\x, E⁰lx)) maps U to the generate~

corresponding to the identity automorphism. It is not difficult t

show the following

( 1) T}J§_2femo f:..JJY.. microbundle admits a uni~ Thorn class.

Clearly Thorn classes correspond functorial with resgect to pull-backs.

1 p 1 -

Let ~(1 : X ~:? E1 ^-:r X,

!J

2 : X bundles over X and let 1--1 : X sl E

E

7 X

TL1 ^; E ^-") E 1 and \l'2 : E ^-._? E2 be

under equivalences and ars

52 ~ p2

--7 .t.2 ^{- 7} X be two micro., be their Whitney sum. Let the canonical projections and <11

. .

_{E1 -: E, 42}

. .

_E2 -::> E their corresponding sect i. ^{j .1} s • The projections _{TL-1 '}

Tr

₂ define maps of pairs

which induce isomorphisms in homology. Given a pairing of local systems on X ~ : ~ ®

_t;

^->

^{J ,}

there is a natural pairing , ·* ( ^r: Eo

*

^~

)

^(y\ H.,* ( E Eo :;.co )

t1 ~ ^' 1 ; p 1 01 ·~ 2 ' 2 ; p 2 J2 -?

defined by commutativity of the diagram

- 2 -

_...._ H* ( c Eo . J~ <? ) r J....' , ,p ~

T

..

_"

t

^I

The map u is the ordinary cup product, and L* is the

homomorphism induced from tQe composite L :'1t~p~

:f

₁

®

TL~P~

g

_{2 =}

p* $1

0

p*i2 ~ p*(

.f

₁

® 1'

_{2 )}

l?_;f'

p*L'5·· This pairing is inv;riant under equivalence of microbundles, but in general dependent on the germs of pi and si, i=1,2. Theimageofanelementof u(g1v will (by a slight abuse of notation) be denoted '1l ~ ( u) w "1t ~ ⁽ v).

It follows that there is always a pairing induced from the KUnneth

In*-

* *

formula pairing

v

₁ ~)(02 -> (D • The following is true ( 2) TQ£:.2.!'.3 m. If t--t ,

[-!

_{1 ,} t-f₂

u,u1 ,u2

and }--f =

)4.1

(3) ~2'

are microbundJ.es wi!_h Tho'""m:.;...;c;;..;;l;...:a.ssc:>s

' *- ...

*

then U = rr₁ (u_{1 )} u Tt ₂(u).

Similarly, for micrubundles ~ , ~ whose composite { = ~ o ~ is defined, there is a natural cup product pairing

I-I *-(E E 0 ··pK..9) 0l· .. H·*r,E_,,E.⁰ ,'o *(t) _.._

~' \-I. ' f.l-.j ^J y - v ·y • v ~ '-" '·* ( _H E _--~' E

_Yt

⁰ _;pY\.

* )

_~

*r .- /

*

associated to each pairing p _)-!. .!2 rx> o1 -> p G1 f of local systems

J Q '>J It- ·~v\

(jJ, 6[)., on the base of

0- , 4

on the base of \) ) • The two cup product pairings are actually equivalent; the one is turneu into the other by the isornorohism

v(

~- ~~ tt} S,~ V , ll 0:: \J~

1

^{0 p)(}

l1

_{2 ,}

' ! ,- f \.41 : ~

c f • [if

J ,

^{cor o 1.1 a r y 5 • If}

r

^{~ X ...§>- E ~ X , and ..J : E .r' '> E '} 2_&_> E , let '~rr: (E ¹ ,E ^{1 - Tl:-1} sX) ~?-.. (E,E0 ) be the map given by [;::- . Then we have

( 3) Corollary" Jf ).( , {-\, .. :J ^a~e microbundles with Thorn cJ.asses

----~---~-.

---

u1^~

,u

₁ⁱ

,uv

^{_§..D.Q '>{}

=

~to

v ,

^{then U} _{f =} '"·n.*rr _{I U\.J}

n.

By a spectral sequence argument (or by elementa~y piecing - together techniques using a Mayer-~~oris exact sequence) we can show the existance of a relative Thorn isomorphism:

( 4) Theorem: 1.et

t"- :

X ~ E p:> X be an tR q-microbundle, i-\ a subsei_Qf X and j .§._l_gca.l_§_ystem OQ.

x.

Tl)eil there are

naturaJ. isornorg_.b_j.sms of qrad.§d modules of degree_2. -q _and q I.§..§J?.ecti v~l v

¢

^{H(E,E0 \;J} E\A;p*-j') H(X,A;Clf(Z:I~)

¢*

t-r* (

x,

A; [0

® -8 )

t;::t H* ( E 'E ⁰

w

E

I

^{A; p} *~ ) defined by

~,(z)

=

p~(Ur"-n z),

Note that o2p and ~* are invariant under equivalence of microbundles.

From this theorem and the homology sequence of

(E ,o ,r: I...J E\' Eo' A, J and the five lemma follows that ~ E⁰ jEIA}

1.. is

an excicive couple with respect to any local system p*~ pulled up from X. If ~~ happens to be a bundle, there is an exact Gysin homology sequence

·~·

-:.'>H (E_n ⁰ ,E⁰ IA;p_, ⁰\)l}-:> H _n

(X,A;j')~

^{H ( X A ·}

<if (::\

³⁾ -'> • • ·

n-q ' ' 10'

obtained from the homology sequence of ( E, E⁰

w

E \A,E

I

A), and a similar exact Gysin cohomo~ogy sequence. Since by [ ] any

microbundle has a fundamental system of bundle neighbourhoods of the zero-section, any microbundle has a functorial exact Gysin homology sequence as above, except the R-module Hn(E⁰,E⁰IA;p⁰~~

should be replaced \Ni th lim Hn (V0 , V0 \ A; p ⁰*-3), where V runs through the neighbourhoods of f'- sX in E.

In the cohomology Gysin sequence of ~ there is a map -~ * : H* ( X,J\; x ) H* (X, A; of degree q which determ.:pes

j·L ^{A 1E} -.q

* ¢

^{..D :¥'}

an element -'.Aft= \f~

(

^{1) E} H (X;(D ) (for A.

= .

and .::)

=

(£) )

_called the cl}aract~ri s_t~~ class of

\-t .

^Generally

~(z)

=

il('--ln z and '1'v..(v)

=

vu

fit-t.

As an application of (2:

we get the following

re~ations

between the Gysin sequences of a Whitney sum to those of the components. (For simplicity of

notations we consider the bundle case only.) ( 5) '[}:le or em. Le_!_ ~ i, i

=

1 , 2, be

X and

1J =

~~ EfJ

i;

₂ their \fvhi tn~y su~

q. l

1R -bundles over a space

- 4 -

s. p.

X 2:>-E . ~~ X,

l

The canonical inje~tions

gysin~_qy_§nces of ~ i

i

=

1 '2

--> E

, (J::o Eo\A o*J) ....

,..

_{Hn (X ,1-q ~) H ( X , A ; r[/@}

J )

-~> • " •

. . .

^-> H n '--'"' . l l ;p.-l ^-> n-q. l

~ 'iQ-¥-

at-d.

^l _-.It ^~· _~-

*"'

'

..

^{-> I'} _~n (Eo Eol ". ox:f1 ' .

K.'

^{p . ' -7 l H} _n ^{(I /\.' A·8) ' --.;> H} _n-q ^{(X,A;{[/~ j} ) -:> •• ⁰

where the maps \.fix are giveQ..__Qy

Simila:r:-l.Y..L in the di~rams of the Gvsin cohomology seg_t.:enGe'l the correspondinq mags

f

^~

*

^ar.§_qi \~_g_n by

The verification is easy except for determining the

y;*

^{and ~~~}

which is done directly by means of theorem 2. Theorem 5 has also been found by Chern and Lashof (unpublished) in the case of

orientable orthogonal bundles over a compact manifold.

2. This section considers some relations between !"Rq· ':::·.mdles

q q

and D -bundles. In the following

P

denotes the closed Gnit balJ.

centered at the origin in (R.q. A !Dq-.QundlE;. is a fibre bundle with fibre Vq and a zero-section. Thus the structure group of a vq-bundle is the group of all homeomorphisms of

vq

^leaving

the origin invariant. Denote this by G(Uq,O). Any Dq-bundle ~

0-1 ~ ^I

determines an S ^-! -bundle ~ , • its b'"2_1..2:.[lda...I..Y.J2_t.~ICUEt.

Converse! y, ~ is determined by ~1 up to equivalence in the Sq -·1

following way: Any homeomorphism of can be radially extended to a homeomorphism of ~q leaving the origin invariant. If

G( sq·-1 ) is the group of homeomorphisms of Sq- 1 , this gives a canonical imbedding i : G(Sq- 1 ) • G(Dq, ). By the invariance of domain theorem restricting maps from

~q

^to

~q-

¹ determines a

retraction r G(tDq,O) ^-'> G(Sq- 1 ). The Alexander radializati process ( ~ shows immediately that it is homotopic

rel G(Sq- 1 ) to the identity on G(Uq,o), i.e. G(Sq- 1 ) is a strong deformation retract of

.

G (IDq, 0) • It follows that any ftc;·- bundle ~ is isomorphic to M~, the mapping cylinder of it~

boundary bundel, by an isomorphism which is the identity on ~

In particular, if ~

1

^,~

2

^are ~q-bundles over the same base ^t then any isomorphism ~

1

^~

q

2 extends to an isomorphism

~1 ^{~ ~} 2 • This is also true in the category of PL-bundles, where the argument is more delicate, cf. Hirsch [3]

For any integer q ~ 0 let IRq c. Sq be an embedding pf Rq into its one-point compactification, fixed throughout this paper. Let Oo denote the complementary point of rRq itl Sq.

Any homeomorphism of

D

q leaving the origin fixed extends to a homeomorphism of Sq also leaving 09 fixed. Therefore the embedding [Rq r; Sq induces an embedding G(rR.q,o) c G(Sq,O,ea).

It follows that any Rq-bundle ~ has a functorially associated Sq-bundle ~"'~ (over the same base) with two sections s , s ,

V'-' 0 I)V

and that

J?

is naturally imbedded in

"'3b.0Q

with zero-section corresponding to s₀ and total space E corresponding to E"_- im s.,._,., ( cf. remark at the end in ) •

~· v - q 0

For any

_p

^{-bundle ~ let ~1} denote the interior ('Rq- bundle of ~ , ~ ~ ~ We state and indicate a sho:rt proof of a result of Hirsch and Mazur (PL-category) in the general

topological case.

1. Iheo~em. ,1et

1o

be an [Rq-bundle over

X .

^The.Q_

^-&

⁶

^t·

is isomor.Qhic to the interior bundle of thfl, 1Dq+ 1 -btmdle M~

00

^•

0 0 Lei

Vh

^,~

₂

be !Dq-bundle:? over X. Iben any isomorpt~i-~.!1:._

t~

₁

~

_{Vj 2} extends to an isomorphism

~1 1

^(jj

b

¹

~ ~1 2

^G)

S

^{1 •}

Here E ¹ =

£.

1 (X) means the trivial 1R 1 -bundle over

X

and

6

^{1 -}

S

^(X) means the trivial \D1 -bundle over ";{ . According to Hirsch [ ~ _]

w.

Browder has shown the existence of a submani- fold of Eucledian space h2ving a normal ~q-bundle which is not the interior of a 1Dq-bundle. The Hirsch-llilazur theorem shows th2 :-

suchp~hology cannot occur for split bundles. We indicate a proc~

of theorem 6: The mapping cylinder bundle Mfo~ has a section

s

0 defined by

S'

₀ (;t)

=

[s₀ (x) ,~J· Form the ¹¹dentedii mapping

.. 6 -

w

cylinder bundle M~ ... ~ from M~ by collapsing the radius to the

..,..., 00 fV

0.0 -point in each fiber. Then S0 defines a section of Mf"~ ,

N 1 ^·c~

With ~

17

as zero--section Mf-;:~/}0 is a roq+ -·bundle whose interior is easily seen to be isomorphic to ~ ^(!)

.S

¹ On the other hand

r-J

the boundary bundle of M~ is clearly isomorphic to the boundarv ca

bundle of M~~ and this isomorphism extends to an isomorphism

""

M}, ~ MY.;~o.:? by the remarks in the first part of this sectionc

[X> q

Finally, if

1

is any tD -bun₀dle one establishes immediately natural isomorphisms between ~oc and the boundary bundle of

~

81

&

^{1 ,} which again extends to isomorphisms

M~ ~

^VI®

6

^{1 • This}

. ~ I

gives the last part of theorem 6. ' We make a remark about Thorn spaces. If

t·:

q q s

an IR -bundle with associated S -bundle

f1o:

^X--~

(A,B) is a pair of subsets of

X

the Thorn space defined to be t;Je pointed space

X s E p v -.:;? • -,? 1\

p~ -

E ^---,.7 X,

T~(A,B)

/

is and is

By the representation theorem in Holm

[iJ

this definition carries over to microbundles, cf. last part of [

4 J .

^Sur=-pose

~ contains or is the interior of a [)q-bundle ~~ • Then, again

. 0

by the microbundle representation the boundary in each fiber of ~

theorem, ~~ ~ Collapsin'J gives rise to an Sq-bu~dle

W • ^I

1

/v\

o.nd a ca2onical bundle map 1,1 ->

Vl 11

which is an natural isomorphis~of

imbedding on ~ Moreover, there is a

.;; ' ^'

bundles ~~ ~ Vt!/0 which is the identity on

~~ I

that we have homeo;norphisms

i1 .,

It fallows

I

When B

=

^)D I this gives the classical definition of the Thorn space of a vector bundle.

We next show that T.;: (X ,B) ref l.ects the homology

J:y

properties of the bundel pair (~,~~B) ~at least when B is

nicely imbedded in

x.

^A ~ad~al ne~~p~our~ of a closed set D in X is an open set U ) B such that U is homeomorphic to the mapping cylinder of a map

. .

f :

U

-~> B

(U

⁼ boundary of U) sending

.

U onto B, by a homeomorphism which is the ide~tity on U u B. Call

this case B

r-imbedded in

f the _qefininq rf@_Q of the radial neighbourhood. In

· ~ d · 11 · b"" ·d'-'d · b dd d ·

x·

I- ^f\

lS ,;[; l_a..::;fl_ .;t.m '-2..:.-~-- or r-1m e -~...§._ ln ~ t r t

X and U c. X is~rc:'.d:lal neighbourhood of A,

~ - . i

is

then again X-U is r-imbedded in X and H(U) ~ H(A), H(X-U) ~ H(X-A). We have the following

(2) Lemma. 1et ~: X ~~E E>X be an Rq-bundle and. let

B c. X be a closed set, r-imbedded with radial ns;ighbourhood U and defininq maQ f. Then EIB, E~\B are r-imbedded in E,E~

resQectively, with radial neighbourhood..2_ E \U, ELlti\U and d,?fi_[L\,Q_q maQS f',f~ such that p o f ' = f ^€> p, p₀₀ o fO:,= f o p_{0, /}

q s p

Given an IR -bundle ~-: X ->-E ->X and a space Y, a

I

^{-1 -1}

fibrewise QrOQer ma.2 f : Y _,... E is a map such that f f p x is a proper map for any fiber p -1 ^X. A f ibrewise QrOQer fibre r..:.:~-

tOl2..'i I-I : Y x I ---?- E is a fibre homotopy which is a fibrewise proper map. Then we have

( 3) Le!T'ma. Let ~, ~ be two bundles ana f : -}_; ^--7 -~' a bundle maQ• Then f extends uniquely to a bundlg map

f_{00 :} ~c.::; ^{- 7}

-B

::r'lhich maQS im sCl>D to im s h.:, if and only if f is fibrewise proQer. Similarly~ if H : f ~ g is a fibre

homotoQy between bundle maps f, g : ~ ^{- 7 { , ' ,} then H extends uniguely to a fibre homo.i.Q.Q.y HIW ~ f_{00 N} g<X? mao12ing (im s¢<..) x I to irn s ~ if _ind only if H is fibrewist: QrOQer.

Using lemma 2 and ordinary homotopy-type reasoning we get

( 4) Corollary. Let ~ ^{: X} ~> ^E ~X be an 1Rq-bqndle and B c X an r-imbedded closed set. Then there is an isomor~hism

"-' . 0

H(Tf(X,B)) ~ H(E,E u EIB)

~

natural with res.Pect to f ib .. :?rwise QI'qJ2_er bundle maQ§

( ~' ~JB) ^-J- (~I '

-G I

^{B) •}

,., S p S I 0 I

Let 1b> : ^{X -7 E ->X and ~ 1 : X 1 -7 E'} -'--;:> X tvvo non- linear bundles (or microbundles) and let (X,B),(X',B') be pair~

of subsets of X,X' resp. Then CX,B) x (X',B') is a pair of subsets of X x X' and we may consider the Thorn space

T ~ x ~ ( ( X , B ) x ( X ^{1 ,} B ^{1 )} ~ One e stab 1 ish e s a natura 1 con t in u o us bijection (of pointed spaces)

- 8 -

T ~X ~; ⁽ (X 'B ) X (X ^I 'B ^{I ) ) -:>} T ~ (X 'B ) ;, T ~I (X ^I 'B ^{I )}

which always induces isomorphisms in homology. This map is a relative homeomorphism, and if one of (X,B), (X ¹,B ^{1 )} is a compact pair or both of them §...D.Q their product a:Le compactly generated, t~en the map is a homeomorphism. In this case the two pointed spaces will be identified"

For any space X and any !Rq-bundle -~ over X there is a natural isomorphism ~~ f:n(X) ~~ xl.(pt); thus if B c. X, then (t$~(X))lB ~ ~\B x Sn(pt). It follows that

T ~ G) f-.n ( X , B ) = T ~ (X , B ) /\. T e,0 ( p t ) • ( We write T ~ ( p t ) for

T n (pt,¢)). But T n ~ Sn, therefore we have

£: (pt) £:.-(pt)

T (X,B)

=

Sni.,e (X,B).

~EB

8.--n

⁰

Next, de£ ine two non-line2r bundles ~, ~~ to be J-

ekvivalent or to have the same stable fi.::l):r:-e homoi££Y~S- if fer some integers m,n there exists fibrewise proper bundle maps g ^! ~ $ tffi --;., . ~I t8:) f_.n' h: ~~ 6

£!

^{~) -~ (B ~} and fibreWiSe

proper fibre homotopies G: 1 ,., rJ hog and H: 1 r._l

~ij-) ~m f;,¹ C-0

sn

g o h. We then have the following result similar to (2.6) in Atiyah [1.]

( 5) Lemma • If ~;; , %-~

X, then for any B c x, same S-!_yp~.

are j -equivaleQ..LQ.9n-linear bu_nd{.es; ... S.Y_~;

T-!,(X,B) and T-f..,' (X,B) are of the

/ /

It is not hard to see (using the Hirsch-Mazur theorem and lemma 3) +~hat the above definition of j-equivalence is equivalent to the classical on~ in the case of vector bundles. We skip the details.

3. Let X be q-manifold and let _{~{ 1} X s1

E p1

a -:> _-":? X'

s2 p

~2

^{: X -:.~ E ~>-X} be microbundles over X of dimension q1 'q2

respectively. Assume the Whitney sum I. tfJ

tl

1 ^{(£; ~}2 trivial, whA.re

6 pri

~ : X ~>X x X -~ X is the ~angent micr0bundle of X. Consider the composite microbundle ~ ~ T _o(p1 ~2^{). We have}

s p

\-t:

^X-> _E1 ^:x _E2 ^{~x with s}

=

(s 1 x s2 ) o& and

p

=

pr ₁ o ( p₁ x p_{2 ) •} After Milnor 1--t is equivalent t.o

1" ® .1*(1~1 \

p

2 )

=

L

EtJ

~1 ^{@ ~}

2

and thus trivial (cf. (5) in [ :¹¹ Therefore there exists a neighbourhood W of sX in E1

x

_E2 ~r

a homeomorphism

which fits into a commutative diagram

JC

w

There is a map

s /

X

X

(W,W-sX) t;:::· X><

(1DO,IDO-o)P~J(o0,ol..!_o)

^-> (tDQ/tDQ,oQ/fl)O_o)

~

(sO,sO .. o)

with extensions

where

• Q) ID •

( E 1 ^K E 2 , E 1 ^'>< E 2 -s X ) -> ( S Q , S 0 -0 ) ( ( E 1 ^X. E 2 ) ^{i'JO ,} i m ( s 1 ^{X s} 2 \,,) -> ( S Q , Oo )

De is the antipodal point of 0 Passing to quotients give a map

(= the collapsed boundary

whore

*

is the basepoint of the Thorn space T ~y

=

T . t1 ^X ~~~2

Tu

u

(X x

x,'if;).

If X is compact, ₁₁

r ·1 " ( 2 tJ1 .>(

r

"2

'1\i /\

r_{1 1 •} In this case we there£ ore get a map

1 ¹ r2

(I~ 1 ^{A I~} 2 ^,

^{*) -<> (SQ,Oo)}

· this is an S-duality map in the sense of Spanier-Whitehead"

More generally, let (A,B) be a compact r-pair in the

(possibly non-compact) manifold X with a radial neighbourhood

(u,v),

say, and defining map (f,g) (i.e. U and V are

- 10 -

radial neighbourhoods of the compact subsets A and B in X with defining maps f and g respectively, and V n A is a radial neighbourhood of

(boundary of V ~ A in

B in A with defining map A)). Then there is a map ( TU ( A , B ) !'. T l!n ( X-V , X-U) , x ) -7- ( S Q , 00)

r

^{·1 ' L}

To show that this is an S-duality map we must show that slanting with the spherical cohomology class in

"'Q( H TtJ A,B ( ) A TH ,x-v,X-U) ^{I )} gives ,;:. duality isomorphism-:,

H

^(T1.t1

(X-V,X-U)

)~

^'H0-n(T11 (A,B)). According to corollary 4 in

n ~-2 r·1

section 2 there are natural isomorphisms in homology and

cohomology of the pair (T\A. (A,B) ^A T~1 (X-V,X-U), *) and the pair (E

1

^{1A,E~IA u E11B)}

^x.

^(E

2

J(X-V),E~f(x-V) ^u E₂\(X-U)). Notice also that (E

1

^{\A,E~]A u E1!B) x (E}

2

[(X-V),E~I(X-V) v E₂\(X-U)) is contained in the pair ( E _{1 x:.} E_{2 )} E₁ :X... E₂-sx) • Denote the inlus ion map by ^... l

( 1 ) Lemma. Under the natural iso_morQhi..§..l}l_ of 2.4 the _ SQQgiicaJo c 1 ass in

HQ (

T ~{I ^{( r\} ~ ^{B ) A T}

tJ

2 ( X-V, X·· U) ) cor r~o n d s :to

'uu

= '-t-*ufl'

th_EL.]hQDl_c__lass_gf

fJ

restricted_iQ (E¹.

1

^{IA,E~!A u E1}!B) x.(E 2

!

(X-V),E~[(X-V) u E2¹ (X-U)).

Moreover

'uw

^has

' · _{~1 ~} r \ E 1[h' ^{1 '} E⁰ 1'1-''U

I

~

f f . . ^{J.. \ '}

*

^{Kr,.){ \ .}

* * *

c o e l c len ~... s -t p v

=

-t-₁ p ₁

tJ

^X R where E₁1B) C (E₁ ,E~). Therefore there is a map (for each n)

'¥}.~:

_{Hn(E 2}¹

(X-V),E~!(X-V)

^u E2 \(X-U)) ^->

I-(~-n(E 1 ^\A,E~\A

^l:

^t.:

^1\B;

,•:R *r*\

't-1P1 •.D ^J

defined by '~~(z)

=

'~./z, corresponding to the map

fJ

^~~

w wr, n

d (T_ti (X-V,X-U)) -> Hl..(- (Tu (t.,B))

n r2

r1

defined by slanting with the spherical class in

HQ (

^T

u (

A , B ) 1\ T

u (

X-V , X-U ) ) • ( Not ice that

tJ *

hence

' · *

~1^p

~)/\*

1^w lS cons an

·

^{1 2t t} Slnc~ -~~ ^·

l1 ·

lS t · · rlvla 1 ·) •

~

tlna ^{· 11} y no tee ^{t · t h '} a~

the r-structure on (A,B) induces an isomorphism in homology

according to 2.2. Therefore there is a map

(for each n) corresponding to '\ '( The following result is the one we have been heading for: ~

(2) Iheorem. The rna.12_ "l'

=

^l' (A,B) is an isomorphism. More Qrecisely we have the commutative diaqrarn

qq .

Hn ( E2 \ (X .. B) ,

E~

\(X-B) V E2 \(X -A) ) ( - 1 )

--J ^{i d~-}

n ( E 1 {A, E

~\A

u _{E 1}

!

^B)

~~. ^¢2* ¢7 t ^·~

H _n-q (X-B,X-A;~2^*)

2

~here ~7, qp

2

*

are the relative Thorn isomorphisms and QfX

ih~

Alexander-Spani,..er duality mao for the manifold X.

Using 2.5 we get

( 3) Corollary. Let X be a man if old which is a CliV-c£llli2..le~

and let ~1^,~2 be two rnicrobundles over X such that

L ®

tJ

_{1 (±)} ^~₂ has trivial stable fibr~. Let (A,B) be a compact relative CVV-.£Qrn_f2lex in X such that P.. is a

~ ubc omplex of X. Then (;,,B) is an r-Qa ir in X with radi;?-1 neiqhbourhoods (U,V), say_,___ and Trt1 (A,B) and TIJ.2 (X-V,X-U) are S-duals.

References

1 A1exa.r:der, J.vJ.: QQ...t_he Ciefor ¹.''l_!lon::_9f an n~ce1L

Proc. :·.·~lt. Sci. U.'~).J,. 9 (1'="'23), 4o6-Y-U7.

2 Atiyab., 1i. F.: Th.Qlll_co:'~~e~.

Proc. Loncl • . ,:0t:1. Soc. 11 (1961), 291-310.

3 Hirsch, Ii.~'v. ~ On nonlinear qs11 bLmdles.

(Un)L:tblisbed)

l1- Holm, P.: NicT'obo1Jndlss And bun~ 1es.

Seminer Reports Matematisk Seminar 106~, Oslo.