ñòð. 8 |

:1

bundle is trivial is even easier.

˜˜

The proof proceeds by induction on dimension. Suppose we know the result -Jj

in dimension m - 1 and let M have dimension m > 1. It suffices to show that the '.J.

two Spin structures on BE agree when restricted to embedded circles. We can span

HI (M; Z /2Z) by embedded circles, Sf, i = 1,Â·Â·Â·, r, where all the circles except

,I

the first lift to disjoint circles in the double cover. The first double covers itself

j

if the line bundle is non-trivial and lifts to disjoint circles otherwise. The group

HI (BE; Z/2Z) is spanned by the collection of connected components of the covers

˜

from the circles in M. 11l

o˜

Let M o be the boundary of the tubular neighborhood of such a circle and let ˜

!VIo be a connected component of the corresponding double cover. It suffices to '1˜

Kirby & Taylor: Pin structures on low-dimensional manifolds 199

prove that the two Spin structures on BE agree when restricted to Mo. We can

restrict the line bundle to M o and consider the resulting ,total space Eo. First note

that Eo has trivial normal bundle in E and that it suffices to show that the two

Spin structures on 8E agree when restricted to BEo.

Consider first the Spin structure induced by the double cover map. This map

is an immersion, so Lemma 2.7 shows that inducing the structure on oE and then

restricting to oEo is the same as first restricting the structure to M o and then

inducing via the double cover map BEo ˜ Mo.

Next consider the Spin structure induced by restricting the Pin+ structure to

the boundary. We can restrict the Pin+ structure on E to Eo and then restrict to

oEo or else restrict to the boundary and then to BEo. These are not obviously the

same: if we let VI be the normal vector to Eo in E, restricted to BEo, and let V2 be

the normal bundle to oE in E, again restricted to BEo. We have a Spin structure

on TEJoE o ' and in the two cases we identify this bundle with TaE o EB VI EB V2 in

one case and with T8E o EB V2 EB VI in the other. By Lemma 1.10, these two ways of

getting the Spin structure via boundaries agree up to a reverse of Spin structure.

But we are using the orientation of M to keep track of all the other orientations,

so the structures turn out to agree.

Our inductive hypothesis applies over M o and we conclude that the two Spin

structures on BEo agree.â€¢

The other result relates double covers and the '11 2 â€¢ Let M be a manifold and let

E' be the total space of the bundle det TM EB det TM over M. There is a natural one

to one function w2:PinÂ±(M) ˜ 'Pin=f(E'). Let E C E' be the total space of the

first copy of det TM: note aE -+ M is a 2 sheeted cover. The embedding 8E c E'

has a normal bundle which we see as two copies of the trivial bundle, which happens

'Pin =F(E') ˜ PinÂ±(oE).

to be det ToE- This gives a natural function '11 2

Theorem 2.9. The PinÂ± structure defined above on aE is the same as the one

induced by the double cover map.

Proof: We begin by proving that certain diagrams commute. To fix notation, let

M o x Rl C M. Let Eo denote the total space of det TMo EBdet TMo and observe that

we can embed Eo x Rl in E. We can arrange the embedding so that on 0 sections

it is our given embedding, and so that (BEo) x Rl is embedded in BE. We begin

with

'11 2

˜

PinÂ±(M) P=f(E')

1 1L

L1 2

'11 2

PinÂ±(Mo) ˜ 'P=f(E˜)

where L 1 is just S-1 followed by the restriction map induced by the embedding of

Mo x R 1 in M and L 2 is defined similarly but using the embedding of Eo x R 1 in

E. This diagraIn ("(.. -.. __.... oft˜ 'h... T ....'rV'..-˜ 1 1 () 'i\To ""Q˜ fho˜ ˜oClfor;,.f +h;˜ Q+ ...l1˜tl1 ...˜ tn

Kirby & Taylor: Pin structures on low-dimensional manifolds

200

8E and then further to (BEo) x Rl. Since stabilization commutes with restriction

we see

PinÂ±(M) ˜ PinÂ±(8E)

1 1Â£4

La

PinÂ±(Mo) ˜ PinÂ±(8Eo)

commutes, where Â£3 is defined by restricting from M to Mo x R I followed by the

inverse stabilization map and L 4 is defined by restricting from BE to (8Eo) x R 1

followed by the inverse stabilization map.

The proof now proceeds much like the last one. First we check the result for

1 . Applying the last diagram to the 2-disk with boundary 51 shows the result for

8

the structure which bOWlds. Apply the Pin+ diagram to the Mobius band to see

the result for the Lie Pin+ structure. The result now holds for any Pin+ structure

on 8 1 . Hence it holds for Spin structures and hence for Pin- structures.

For M of dimension at least 2 we induct on the dimension. But just like the

proof of the preceding result, this follows from the commutativity of our second

diagram.â€¢

Â§3. Pin- structures on surfaces, quadratic forms and Brown's arf invariant.

In this section we want to recall an algebraic way of describing Pin - structures

due to Brown [Br].

Definition 3.1. A function q: HI (F; Z/2Z) -+ Z/4Z is called a quadratic enhance-

ment of the intersection form provided it satisfies q(x +y) = q(x) +q(y) +2Â· x.y for

all x, y E HI (F; Z/2Z) (here 2Â· denotes the inclusion Z/2Z C Z/4Z and. denotes

intersection number.

The main technical result of this section is

Theorem 3.2. There is a canonical 1-1 correspondence between Pin- structures

on a surface F and quadratic enhancements of the intersection form.

Discussion. One sometimes says that there is a 1-1 correspondence between Pin-

structures on F and HI (F; Z/2Z), but this is non-canonical. Canonically, there is

an action of HI (F; Z/2Z) on the set of Pin- structures which is simply transitive.

Once a base point has been selected, the action gives a 1-1 correspondence between

HI (F; Z/2Z) and the set of Pin- structures.

Note also that HI (F; Z/2Z) acts on the set of quadratic enhancements, by

q x , goes to q,., defined by

q-y(y) = q(y) + 2 . ,(y)

(3.3)

and note that this,action is simply transitive. The 1-1 correspondence in Theorem

3.2 is equivariant with respect to these actions. Indeed, the proof of Theorem 3.2 will '

Kirby & Taylor: Pin structures on low-dimensional manifolds 201

be to fix a Pin- structure on F and use it to write down a quadratic enhancement.

This gives a transformation from the set of Pin- structures to the set of quadratic

enhancements. We will check that it is equivariant for the HI (F; Z/2Z) action and

this will prove the theorem.

Before describing the enhancement, we prove a lemma that produces enhance-

ments from functions on embeddings. Specifically

Lemma 3.4. Let q be a. function which assigns an element in Z/4Z to each em-

bedded disjoint union of circles in a surface F subject to the following conditions:

(a) q is additive on disjoint union; if L 1 and L 2 are two embedded collections of

circles such that L 1 II L 2 is also an embedding then ti(L 1 lL L 2 ) = q(L 1 ) +

q(L 2 )

(b) if L 1 and L 2 are embedded collections of circles which cross transversely at

r points, then we can get a third embedded collection, L 3 , by replacing each

crossing: we require q(L3 ) = q(L}) +q(L2 ) +2 . r

(c) if L is a single embedded circle which bounds a disk in F, then q(L) = O.

Then q(L) depends only on the underlying homology class of L, and the induced

function q: HI (F; Z/2Z) Z/4Z is a quadratic enhancement.

--+

Proof: The first step is to show how given L, we may replace it with a single

embedded circle K such that the L and K represent the same homology class

in HI (F; Z/2Z) and have the same q. If L has more than one component, it

is possible to draw an arc between two different components. A small regular

neighborhood of this arc is a disk, and let 1<1 be its boundary circle. By (c),

q(K}) = o. The circle ]{l has two pairs of intersection points with L. Apply (b):

the new embedding consists of a new collection L 1 which has one fewer components

that L, and two small circles ](2 and K 3 , each of which bounds a disk. Condition

(b) says that (j(L I II ](2 Jl K 3 ) = q(L) + q(1(1) = (j(L). From (a) and (c) we see

that q(L 1 lL K 2 lL K 3 ) = q(L 1 ), so q(L) = q(L 1 ), and L and L 1 represent the

same homology class. Continue until there is only one component left.

Next we prove isotopy invariance of qin several steps. First, suppose A C F

is an embedded annulus with boundary K o II K 1 and core C. We want to show

q(Ko) = q(K}) = q(C). Draw an arc from K o to C and let K 3 be a circle bounding

a regular neighborhood of this arc. Apply condition (b): the result is two circles,

each of which bounds a disk. From conditions (a) and (c) we see q(C) = q(I<o). A

similar proof establishes the rest. We can also show that q(C) must be even. Let

C1 be a copy of C pushed off itself in the annular structure. Then q(C) = q(C1 )

since they are both q(Ko ). Let L = C II C1 â€¢ Then q(L) = 2q(C) by (a). On the

other hand, just as above, we can use (b) to transform L into a picture with two

circles bounding disks, so by (a) and (c) we see q(L) = 0 and the result follows.

lfence any curve in F with trivial normal bundle has even q. Finally, suppose that

C'l is embedded in A and represents the same element in mod 2 homology as C.

Kirby & Taylor: Pin structures on low-dimensional manifolds

202

We can find a third curve C2 which also represents the same element in mod 2

homology and which intersects both C1 and C transversely. Consider say C2 and

C. Apply (b): r is even as are both q(C) and q(C2 ). Hence q(C) = q(C2 ). Similarly

q(C1 ) = q(C2 ) and we have our result.

Next suppose that M c F is a Mobius band with core Co. We can push

Co to get another copy, C1 intersecting Co transversely in one point. We can

push off another copy C2 which intersects Co and C1 transversely in a single point

and all three points are distinct. Applying (b) to pairs of these circles, we get

q(Ci ) + q(Cj) = 2 for 0 ˜ i,j ˜ 2, i f= j. Adding all three equations we see

2(q(CO)+q(C1 )+Q(C2 )) = 2, so at least one q(Ci) must be odd. But then returning

to the individual equations we see that q(Co) = q( C1 ) = q( C2 ), so we see that q( C)

must be odd whenever the normal bundle to C is non-trivial. Let C1 be any

embedded circle in M which represents the core in mod 2 homology. It is possible

to find a third embedded circle, C2 which also represents the core and intersects

Co and C1 transversely. Since q( Ci) must be odd, it is not hard to use (b) to show

that q(Co) = q(C1 ).

To show isotopy invariance proceed as follows. Let !( be a circle with a neigh- .

I

borhood W. Any isotopy of K will remain for a small interval inside W and the

J

image K t will continue to represent the core in mod 2 homology. By the above dis-

cussion qwill be constant on !{t, the circle at time t. Hence, the subset of t E [0, 1] ˜I'

for which q(Kt ) = q(K) is an open set. Likewise the set of t E [0,1] for which .˜

q(Kt ) =1= q(K) is an open set, so we have isotopy invariance for a single circle. By

I

part (a), the result for general isotopies follows as above.

'I

Next we prove homology invariance. Suppose L 1 and L 2 represent the same

element in homology. By isotopy invariance, we may assume that they intersect :1

transversely. Let La be the result of applying condition (b). q(L a) = q(L 1 ) + o'J

:1

q(L 2 ) + 2 Â· r, and L 3 is null-homologous. If we can prove q(L 3 ) = 0 then we are

done. As we saw above, it is no loss of generality to assume that L3 is connected, 'IZ˜

and since it is null-homologous, it has trivial normal bundle, so q(L 3 ) is even. Also, ;;˜

since L 3 is null-homolo˜ous, there exists a 2-manifold with bo˜ndar! a single circle, Â·;'o.o˜ o.o

=

say W, and an embeddIng W c F so that oW L 3 â€¢ If W 18 a dIsk we are done :0,:".

by (c), so we work by induction on the Euler characteristic of W. If W is not a .˜˜

disk then we can write W = W j U V where oV = 00 V lL 01 V = Sl J.L Sl, V is o:t

!I

either a t˜i˜e punctured torus or a p˜ctured Mobi˜ band, ˜d W l has large˜ Eu.ler

characterIstIc than W. We are done If we can show q(8oV) = q(Ol V). We begIn WIth 0j'˜

the toral case. Using (b) and (c) as usual, we can see that q(BoV) = q(Sa) +q(Sb) oOil

where So, and Sb are two meridian circles, one on either side of the hole. Likewise <˜

q(8l V) = q(Sa) +q(Sb) so we are done with this case. In the Mobius band case we ˜

can again use (b) and (c) and see that q(80 V) +q(8 1 V) = o. Since they are both or,

'.˜

even, again they are equal.

This shows that Ii induces a function q: HI (F; Z/2Z) -+ Z/4Z, and (b) trans- ;j

J

lates immediately into the relation q(x + y) = q(x) + q(y) + 2Â· x-yo â€¢

1

Kirby & Taylor: Pin structures on low.. dinlcl1sionalnulnifolds 203

Now we describe our function. Let ,\ be a line bundle ()v(˜r F with WI ( A) =

w}(F) and let E(A) denote its total space. From Lemma 1.7, 1:\ Spin structure on

E(A) gives a Pin- structure on F. Let K be an embedded cir(˜le in F, and let T

denote the tangent bundle of E(A) restricted to K. A Spin structure on E(A) yields

a trivialization of T. It is also true that T = TSl Ef) VKcF â‚¬I' VFCE(>.), where v denotes

normal bundle. Note all three of these bundles are line bundles. Pick a point p E !<

and orient each of the line bundles at p so that the orientation on T agrees with

that coming from the Spin structure. Since TSl is trivial, the orientation picks out

a trivialization, and hence VKcF Ef) VFcE acquires a preferred even framing. (Note

that framings of a 2-plane bundle correspond to Z, while those of a 3-plane bundle

correspond to Z/2Z. Hence the framing of the 3-plane bundle picks out a set of

framing of the 2-plane bundle, a set we call even.)

Definition 3.5. Choose an odd framing on VKcF E9 VFcE and using it, count the

number (mod 4) of right half twists that VKcF makes in a complete traverse of

K. This is q(K). Given a disjoint union of circles, Lemma 3.4 (a) gives the value

of qin terms of the individual components.

. We first need to check that qreally only depends on the embedded curve and

not on the choice of p or the local orientations made at p or on the choice of odd

framing. It is easy to see that the actual choice of framing within its homotopy

class is irrelevant because we get the same count in either frame. H we choose a

new odd framing the new count of right half twists will change by a multiple of 4,

so the specific choice of odd framing is irrelevant. If we move p to a new point,

we can move around K in the direction of the orientation and transport the local

orientations as we go. If we make these choices at our new point, nothing changes

so the choice of point is irrelevant. Since we must keep the same orientation on T,

we are only free to change orientations in pairs. H we keep the same orientation

on K, the odd framing on the normal bundJe remains the same and so we get the

same count. Finally, suppose we switch the orientation on !{. We can keep the

same framing on the normal bundle provided we switch the order of the two frame

vectors. If we do this and traverse K in the old positive direction we get the same

count as before, except with a minus sign. Fortunately, we are now required to

traverse K in the other direction which introduces another minus sign, so the net

result is the same count as before. Hence qonly depends on the embedded curve.

Since q satisfies Lemma 3.4 (a) by definition, we next show that it satisfies

conditions (b) and (c) also. We begin with (c). In this case, all three line bundles

are trivial, hence framed after our choice of p and the local orientations. However,

this stable framing of the circle is the Lie group one, so it is not the stable framing

of the circle which extends over the disk, Theorem 2.1. Since the framing from the

Spin structure does extend over the disk, the framing constructed above is an odd

framing, and q is clearly 0 for these choices. To show (b), consider a small disk

neighborhood of a crossing. It is not hard to check that in the framing corning from

204 Kirby & Taylor: Pin structures on low-dimensional manifolds

that of the disk, we can remove the crossing without changing the count. However,

this is the even framing and we are supposed to do the counting using the odd

framing. This introduces a full twist, and so we get a contribution of 2 for each

crossing. This is (b).

Thanks to Lemma 3.4 we have described a function from the set of Pin-

structures on F to the set of quadratic enhancements on the intersection fonn on

HI (F; Z/2Z). Suppose now we change the Pin- structure by , E HI (F; Z/2Z).

The effect of this change is to reverse even and odd framings on K for which '.i:

,([() = -1 and to leave things alone for 1< for which ,([{) = 1. The effect on the

}

resulting q is to add 2 to q(x) if ,(x) = -1 and add nothing to it if ,(x) = 1. But .:;'

I

this is just q-yo

This completes the proof of Theorem 3.2.

Next we describe an invariant due to Brown, [Brl, associated to any quadratic

I

enhancement q. Given q, form the Gauss sum

i

L e 21riq (x)/4.

Aq =

;i

zEHt (F;Z/2Z)

˜J

˜˜

This complex number has absolute value JIH1 (F; Z/2Z) I and there exists an ele- .˜

;;˜

ment (1(q) E Z/8Z such that A q = JIB! (F; Z/2Z) I e21ri {J(q)/8.

il

Hence we can think of f3 as a function from Pin- structures on surfaces to

.˜

Z/8Z. It also follows from Brown's work, that (1 is an invariant of Pin- bordism:

two surfaces with Pin- structures that are Pin- bordant have the same (3.

'1

I

Lemma 3.6. The homomorphism

13: nfin- I

Z/8Z

--+

t

I

nr in

- !-.Z/8Z - Z/2Z is the mod 2 Euler

is an isomorphism. The composite

characteristic and hence determines the unoriented bordism class of the surface.

Proof: Brown proves that f3 induces an isomorphism between Witt equivalence . ˜

classes of quadratic forms and Z/8Z. One homomorphism from the Witt group is .:˜˜

:.! ˜. :

the dimension mod 2 of the underlying vector space. Since this is just the mod 2 Â·â€¢.'. :.;l

...

',Â·

.....

˜

Euler characteristic of our surface, the second result follows.

Hence, if f3(F) = 0, the manifold is an unoriented boundary, say of W â€¢ There '::;˜

3

aw;

is an obstruction in H 2 (W, Z/2Z) to extending the Pin- structure on F across .);. ;

.

W. If this obstruction is 0 we are done, so assume otherwise. There is a dual circle, .˜;

K C W - F and the Pin- structure on F extends across W - !{. The boundary

of a neighborhood of [( is either a torus or a I{lein bottle, so if f3(F) = 0, F is

Pin- bordant to a torus or a Klein bottle with f3 still O. Moreover, since the Pin-

Kirby & Taylor: Pin struclurcs on )uw..dimcnsionul manifolds 205

structure is not supposed to extend across the neighborhood of K, one of the non-

zero classes in HI has a non-zero q. For the Klein bottle, two of the non-zero classes

have odd square and the other has even square. It is the class with even square

that must have a non-trivial q on it to prevent the Pin- structure from extending

across the disk bundle. But the Klein bottle with this sort of enhancement has

non-zero (3, so the boundary of K must be a torus. For the torus, q must vanish

on the remaining classes in H l in order to have f3 = 0 and it is easy to find a Pin-

boundary for it.â€¢

= Â±1 E Z/8Z.

Exercise. Show that Rp2 with its two Pin- structures has {j

The relation between Pin - structures and quadratic enhancements is perva-

sive in low-dimensional topology. In [Ro], [F-K] and [G-M] enhancements were

produced on characteristic surfaces in order to generalize Rochlin's theorem. In

Â§6, we will show how to find an enhancement without the use of membranes. This

gives some generalizations of the previous work. In the next section we will study

surfaces embedded in "spun" 3-manifolds. An interesting theory that we do not

pursue is Brown's idea of studying immersions of a surface in R 3 . Since R3 has

a unique Spin structure, an immersion pulls back a Spin structure onto the total

space of a line bundle over the surface with oriented total space.

Another point we wish to investigate is the behavior of f3 under change of Pin-

structure. Hence fix a quadratic form q: V ˜ Z/4Z: i.e. V is a Z/2Z-vector space;

q(rx) = r 2 q(x) for all x E V and r E Z; and q(x + y) - q(x) - q(y) is always even

and gives rise to a non-singular bilinear pairing A: V X V ˜ Z/2Z.

= q(x) +2Â· A(a, x).

Given a E V, define qa by qa(X)

= f3(q) + 2Â· q(a).

Lemma 3.7. With notation as above, f3(qa)

Proof: There is a rank 1 form (1) consisting of a Z/2Z vector space with one

generator, x, for which q(x) = 1. There is a similar form (-1). It is easy to check

the formula by hand for these two cases. Or, having checked it for (1) and a = x

and a = 0, argue as follows. Given any form q, there is another form -q defined on

the same vector space by ( -q)(x) = -q( x). It is easy to check that p( -q) = -f3(q).

If the formula holds for q and a, it is easily checked for -q and a after we note

(-q)a = -(qa).

Given two forms ql on Vi and q2 on V2, we can form the orthogonal sum ql .L q2

on Vi â‚¬a V2 by the formula (ql .L q2)(Vl,V2) = ql(Vl) + q2(V2). Brown checks that

f3(qt .L q2) = f3(ql) + f3(q2). If ai E Vi, note (ql .L Q2)(at,a2) = (ql)al .L (q2)a2' so if

the formula holds for the two pieces, it holds for the orthogonal sum. Moreover, if

it holds for the sum and one of the pieces, it holds for the other piece.

Finally, note that if a = 0, the formula is true.

Now use Brown, [Br, Theorem 2.2 (viii)] to see that it suffices to prove the

formula for a form isomorphic to m( 1) + n( -1) and any a and this follows from the

206 Kirby & Taylor: Pin structures on low-dimensional manifolds

above discussion.â€¢

Next we present a "geometric" calculation of the Spin and Pin+ bordism

groups in dimension 2.

Proposition 3.8. Any Spin structure induces a unique Pin- structure, so f3 is

defined just as above for surfaces with a Spin structure. We have f3 defines an

n: pin

--+ Z/2Z. Any surface with odd Euler characteristic with any

isomorphism

Pin- structure is a generator for nfin- and the 2-torus with the Lie group Spin

structure is a generator for ni,in.

Proof: The proof is almost identical to that of Lemma 3.6. The surface F bounds

an oriented 3-manifold W and by considering the obstruction to extending the Spin

structure we see that F is Spin bordant to a torus with the same Spin structure as

in the proof of Lemma 3.6. Just note that the boundary constructed there is actually

a Spin boundary. It is a fact from Brown that (3 restricted to even forms only takes

on the values 0 and 4. The results about the generators are straightforward.â€¢

The Pin+ case is more interesting. We have already seen that the only way a

surface can have a Pin+ structure is for w˜ to be O. Hence the [nw;] map must

also be 0, so the [nWI] map is defined on all of 02 In + â€¢

p'

Proposition 3.9. The homomorphism [nWl]: nfin+ --+ n˜pin ˜ Z/2Z is an iso-

morphism. A generator is given by the Klein bottle in half of its four Pin+ struc-

tures.

Proof: A surface, F, has a Pin+ structure iff w2(F) = 0 iff F is an unoriented

boundary, say F = aw. The obstruction to the Pin+ structure on F extending

to W is given by a relative 2-dimensional cohomology class, so its dual is a 1-

dimensional absolute homology class. We can assume that it is a single circle, and

so F is Pin+ bordant to either a torus or a Klein bottle, and the Pin+ structure

has the property that it does not extend over the corresponding 2-disk bundle over

S1.

J

Since SI with either Pin+ structure is a Pin+ boundary it is not hard to see :˜

that the torns with any Pin+ structure is a Pin+ boundary. There are two Pin+

structures on the Klein bottle which do not extend over the disk bundle. If one cuts

the Klein bottle open along the dual to WI and glues in two copies of the Mobius

band, one sees a Pin+ bordism between these two Pin+ structures. Hence nfin+

has at most two elements. On the other hand it is not hard to see that the Klein :

bottle with the Pin+ structures which do not extend over the disk bundle hit the i

non-zero element in n˜pin under [nWl]' â€¢

For future convenience let us discuss another way to "see" structures on the

torus and the Klein bottle. We begin with the torus, T2.

Example 3.10. We can write T2 as the union of two open sets Uj = S1 X (-1,1)

so that U1 nU2 is two disjoint copies of 51 X (-1,1), say U1 nU2 = Vi2 lL V 12 0

Kirby & Taylor: Pin structures on 10w-dimcl1siol1ullllunifolds 207

We can frame'SI x (-1, 1) using the product structure and the framings of the two

I-dimensional manifolds, 8 1 and (-1,1). If we form an 80(2) bundle over T2 with

transition function 912 defined by 912(U1 n U2) = 1 then we get the tangent bundle.

H we think of 1 as the identity of 8pin(2) then the same transition functions give

a Spin structure on T2. This Spin structure is the Lie group one: clearly the copy

of 8 1 in the 8 1 x (-1, 1)'s receives the Lie group structure, and it is not difficult

to start with a framing of (-1, 1) and transport it around the torus to get the Lie

group structure on this circle. IT we take as Spin(2) transition functions h I2 defined

by h12 (Vi2) = 1 and h I2 (V I2 ) = -1 E Spin(2), then we get a Spin structure whose

enhancement is 0 on the obvious SI and 2 on the circle formed by gluing the two

intervals.

Example 3.11. We can write the Klein bottle, [(2, as the union of two open

sets Ui = 8 1 X (-1,1) so that U1 n U2 is two disjoint copies of SI x (-1,1), say

U1 n U2 = W12 lL W 12 â€¢ We can frame 8 1 x (-1,1) using the product structure

and the framings of the two I-dimensional manifolds, SI and (-1,1). If we form

an 0(2) bundle over K2 with transition function 912 defined by 912(Wl ) = 1 and

- (-1 0)

O l E 0(2) then we get the tangent bundle (we are writing the

=

912(W I2 )

tangent vector first). If we define hI2 (Wl ) = 1 and hI2 (W I2 ) = el E Pin(2), we

SI

get a Pin structure on the tangent bundle. The copy of 8 1 in the SI X (-1, l)'s

receives the Lie group structure, so if we are describing a Pin- structure, then we

get the bordism generator.

We conclude this section with two amusing results that we will need later.

Theorem 3.12. Let F be a surface with a Spin structure. Let q: HI (F; Z/2Z) --+

Z/2Z denote the induced quadratic enhancement. Let x E HI (F; Z/2Z). Corre-

sponding to x there is a double cover of F, F which has an induced Spin structure.

There is also a dual homology class a and

[F] = q(a) Z/2Z .

E

Proof: We can write F as T 2 #F1 where T 2 is a 2 torus and a is contained in

T2. Then F = Tl#Fl #Fl , where T{ is a double cover of T2 given by x E

HI (T 2 ; Z/2Z). Note (x, a) = 1 not -l,so a lifts to 2 disjoint parallel circles.

Moreover, HI (T{; Z/2Z) is generated by one component of the cover of a, say a,

and another circle, say bwhich double covers a circle, say b in T 2 â€¢

Note [F] = [Tf] + 2[F1 ], so [F] = [Ti]. The enhancement q: HI (T[; Z/2Z)

=

satisfies q(a) q(a) and q(b) = -1. Hence the Spin bordism class of T; in Z/2Z

is given by q(a).â€¢

Kirby & Taylor: Pin structures on low-dimensional manifolds

208

The second result is the following. Given any surface, F, we can take the

orientation cover, F, and orient F so that the orientation does not extend across

any component of the total space of the associated line bundle. Given a PinÂ±

structure on F, we can induce a Spin structure on F.

Lemma 3.13. The orientation double cover map induces homomorphisms

nPinÂ± nSpin

˜ 01Â£2

11Â£2

which are independent of the orientation on the double cover. The Pin - map is

trivial, and the Pin+ map is an isomorpbism.

Proof: H we switch to orientation on P, we get the reverse of the Spin structure

ni ˜ Z/2Z this shows that the answer is independent

pin

we originally had. Since

of orientation. By applying the construction to a bordism between two sunaces we

see that the maps are well-defined on the bordism groups. Since addition is disjoint

union, the maps are clearly homomorphisms.

In the Pin- case, Rp2 is a generator of the bordism group. The oriented cover

is S2 which has a unique Spin structure and is a Spin boundary. This shows the

Pin- map is trivial.

In the Pin+ case, a generator is given by the Klein bottle. Consider the

transition functions that we gave for this Pin+ structure in Example 3.11. This give

us a set of transition functions for the torus which double covers the Klein bottle.

We get 4 open sets, but it is not difficult to amalgamate three of the cylinders into

one. The new transition function, h 12 , takes the value 1 on one component of the

overlap and the value e˜ on the other. Since el E Pin+(2), e˜ = 1 so we get the Lie

group structure on T 2 by Example 3.10.â€¢

Remark. H we started with a non-bounding Pin- structure on the I{:Iein bottle,

then the above proof would show that the double cover has Spin transition functions

given by 1 on one component of the overlap and -Ion the other, and, as we saw,

this Spin structure bounds (as Lemma 3.13 requires).

Â§4. Spin structures on 3-mallifolds.

Let M 3 be a closed 3-manifold with a given Spin structure. We begin by

generalizing some of the basic ideas in the calculus of framed links in S3.

Given any embedded circle k: S1 -+ M 3 , the normal bundle is trivial, and

therefore has a countable nwnber of framings. If the homology class represented by

k is torsion, we can give a somewhat more geometric description of these framings.

Recall that there is a non-singular linking form

˜

l: torH l (M; Z) (8) torHI (M; Z) Q/Z .

Let x E H l (M; Z) be the class represented by k, and assume that x is torsion.

Kirby & Taylor: Pin structures on low-dimensiol1111 Inanifolds 209

Lemma 4.1. The framings on the normal bundle to k are in one-to-one corre-

spondence with rational numbers q such that the class ofq in Q/Z is f,(x, x).

Proof: We describe the correspondence. A framing on the normal bundle of k is

equivalent to a choice of longitude in the torus which bounds a tubular neighborhood

of k. Suppose r E Z is chosen so that r Â· x = 0 in HI (M; Z). Take r copies of the

longitude in the boundary torus and let F be an oriented surface which bounds'

these r circles. Count the intersection of F and k with signs as usual. H one gets

p E Z, then assign the rational number; to this framing. It is a standard argument

that ˜ is well-defined once the framing is fixed. It is also easy to see that ˜ mod

Z is f(x,x), and that if we choose a new framing which turns through t full right

twists with respect to our original framing, then the new rational number that we

get is ; +t.â€¢

A Spin structure on M gives a Spin structure on the normal bundle to k

as follows. Restriction gives a Spin structure on the tangent bundle to S1 plus

the normal bundle. Choose the Spin structure on the normal bundle so that this

Spin structure plus the one on SI which makes Sl into a Spin boundary gives the

restricted Spin structure.

Definition 4.2. We call the above framings even.

Hx above is torsion and M is spun, then the Spin structure picks out half

as

of the rational numbers for which the longitude gives a framing compatible with

the Spin structure on the normal bundle. Given one of these rational numbers, say

q, the remaining ones are of the form q + 2t for t an integer. Hence we can define a

new element in Q/Z, namely ˜. This gives a map

˜

,:torH1 (M; Z) Q/Z

which is a quadratic enhancement of the linking form:i.e.

,(x+y) ,(x)+,(y)+f(x,y)

=

= r 2 â€¢ ,(x) for any integer r

,(rx) .

Suppose now that x is zero in HI (M;Z/2Z), but not necessarily-torsion in

HI (M; Z). Then any Spin structure on M induces the same Spin structure in a

neighborhood of k, and hence the notion of even framing is independent of Spin

structure for these classes.

Theorem 4.3. A knot k which is mod 2 trivial as above, bounds a surface which

does not intersect k. This surface selects a longitude for the normal bundle to k,

and this longitude represents an even framing.

Proof: Let E be a tubular neighborhood for k with boundary T 2 â€¢ (This T 2 is

often called the peripheral torus.) We can select a basis for HI (T2; Z/2Z) as

Kirby & Taylor: Pin structures on low-dimensional manifolds

210

1

follows. One element, the meridian, is the unique non-trivial element in the ker-

˜

nel of the map HI (T2; Z/2Z) --+ HI (E; Z/2Z). One calculates that the sequence .,

HI (T2; Z/2Z) ˜ HI (M - k; Z/2Z) ˜ HI (M; Z/2Z) is exact, and that the im- ::1

˜

age of HI (T2; Z/2Z) in HI (M - k; Z/2Z) is I-dimensional and generated by the

I

meridian. Hence there is a unique non-trivial element, the mod 2 longitude, in

,

the kernel of HI (T2; Z/2Z) -+ HI (M - k; Z/2Z). An even longitude for k is an

element t E HI (T2; Z) which reduces in mod 2 homology to the mod 2 longitude.

˜

Fix an even longitude, t. It follows that there is an embedded surface, F2 C M

I

such that of = k. This surface can be chosen to intersect T2 transversely in the

even longitude. The southeast corner of Corollary 1.15 assigns a Pin- structure to

F. Restricted to k, the normal bundle to Fin M is trivial, so the surface frames the

,

normal bundle to k in M. Hence the Spin structure on M restricted to k is seen as

the Spin structure on the circle coming from the restriction of the Pin- structure

on F plus the Spin structure on the normal bundle coming from the framing. We

.˜

saw in the proof of Theorem 2.1 that, regardless of the Pin- structure on F, the

I

boundary circle receives the non-Lie structure. This is the definition of the even

framing.â€¢

Remarks 4.4. ':J

(i) In S3 with its unique Spin structure, the framing on k designated by an evenJ

1

number in the framed link calculus is an even framing in the above sense.

(ii) IT the class x has odd order, then Â£(x, x) = ; with r odd. There are then two \˜

sorts of representatives in Q for f( x, x): the p is even for half the representatives :j

Â·1

and odd for the other half. The framings that the Spin structure will call even :˜

:1

are the ones with even numerator.

(iii) If we change the Spin structure on M by a class 0 E Hl(Mj Z/2Z) the even;

:˜:=::˜˜n c˜c˜˜cle change iff 0 evaluates non-trivially on the fundamental,

(iv) If we attach a handle to a knot in a 3-manifold, M 3 , we get a 4-manifold W :˜

with H 2 (W, Mj Z) = Z. If our knot in M 3 is torsion, we get a unique (up to :J

si?tnh){clas˜ x E H 2Q ; Q)Lwhich h it s 0h rel˜tive clatss.. If lfwe ˜tthtachala hanfdle i.˜

ur

(W

ramIng q E f rom emma 4.1, t en x Intersec S Itse WI a v ue 0 q. J

WI

Hence the signature of W is sign (q), where sign (q) = 1 if q > 0; -1 if q < 0 ˜

: d;o:˜:O˜.15 the surface F we used in the proof of Theorem 4.3 inherits!

a Pin- structure from one on M. This suggests trying to define a knot invariant '.˜; .˜

in this situation. Indeed, for knots in S3, this is one way to define Robertello's Arf J

invariant, [R]. The situation in general is more complicated and needs results from :1

Â§6, so we carry out the discussion in Â§8.

An invariant of a 3-manifold with a Spin structure is the It-invariant. We

nf

pin

discuss in Theorem 5.1 the classical result that = O. It follows that any

Kirby & Taylor: Pin structures on low-dimensional manifolds 211

3-manifold, M 3 , is the boundary of a Spin 4-manifold, W.

Definition 4.5. The signature of W, reduced mod 16, is the J.t-invariant of the

manifold M with its Spin structure. It follows from Rochlin's theoreln that J.L(M)

is well-defined once the Spin structure on M is fixed.

Remark. Some authors stick to Z/2Z homology spheres so that there is a unique

Spin structure and hence a p, invariant that depends only on the manifold.

We now turn to a geometric interpretation of some wprk of Turaev [Tu]. Inter-

section defines a symmetric trilinear product

r: H 2 (M; Z/2Z) X H 2 (M; Z/2Z) X H 2 (M; Z/2Z) ˜ Z/2Z

We introduce a symmetric bilinear form

which is defined as follows. Let F x and F y be embedded sunaces representing two

classes x and y in H 2 (M; Z/2Z). To define A(X, y) put the two surlaces in general

position. The intersection will be a collection of embedded circles. The normal

bundle of each circle in M has a sub-line bundle, ex, given by the inward normal

to the surface F x â€¢ Define A(x, y) to be the number of circles with non-trivial ex.

Here is an equivalent definition of A. Any codimension 1 submanifold of a

manifold is mod 2 dual to a I-dimensional cohomology class in the manifold. If

this cohomology class is pulled-back to the submanifold, it becomes WI of the

normal bundle to the embedding. Hence, if x* and y* are the Poincare duals to

x and y, A(X, y) = x* U x* U y*[M], where [M] is the fundamental class of the 3-

manifold. This follows because x*Uy*n[M] is the homology class represented by the

ex

intersection circles, and to count the number with non-trivial we just evaluate

WI of the normal bundle on these circles. But WI = x* so we are done. We can

also prove symmetry using this definition. Since M is orientable, 0 = WI (M)x*y* =

SqI(x*y*) =(X*)2y* +x*(y*)2.

Yet another definition of A is

= r(x,x,y)

A(X,y) .

Hence A is symmetric and bilinear.

Given a Spin structure on M, we can enhance A to a function

˜

f: H 2 (M; Z/2Z) H2 (M; Z/2Z) Z/4Z .

X

To begin, we define f on embedded sunaces F x and F y in M as above, but now use

the Spin structure to put even framings on the intersection circles and then count

Kirby & Taylor: Pin structures on low-dimensional manifolds

212

ex.

the number of half twists in each (Since the collection of circles is embedded,

there is no correction term needed to account for intersections.) Note if we defined

e ex,

y in the obvious manner and counted half twists in it instead of in we would

get the same number, so f is symmetric.

Here is another description of f(Fx,Fy ). In M 3 , Fy is dual to a cohomology

class, a E HI (M; Z/2Z), and we could take Q and restrict it to Fx , getting ax E

HI (Fx ; Z/2Z). The Poincare dual of ax in F x is just the class represented by our

collection of circles, which we will denote by y. Associated to our Pin- structure

on F z, there is a quadratic enhancement tPx. Note

(4.6)

In particular, note f(Fx , F y ) only depends on the homology class of Fy , and hence

by symmetry also only on the homology class of F x â€¢

Once we see the pairing is well-defined, it is easy to see that f(x, 0) = f(O, x) =

ofor all x E H2 (M; Z/2Z). We have lost bilinearity and gained

= f(x, y) + f(x, z) +2T(X, y, z)

f(x, y + z) .

(4.7)

Proof: With notation as above, we apply formula 4.6. We need to show tPx('y+˜x) =

t/lx(Y) + t/lx(.z) +27(X, y, z), which is just the quadratic enhancement property of'l/Jx

and the identification of y.z in F:r; with r(x,y,z).â€¢

If we change the Spin structure on M by a E HI (M; Z/2Z), then we change f

as follows. Let fa denote the new pairing and let a E H2 (M; Z/2Z) be the Poincare

dual to Then

0:.

= f(x, y) + 2r(x, y, a) ,

fa(x, y)

or

= f(x, y + a) -

fa(x, y) f(x, a) Â·

Using 4.6 we see that the first formula is

Proof: We prove the first formula.

equivalent to tPcr(Y) = 'l/J(y) + 2T(X, y, a), which follows easily from formula 3.3.â€¢

Finally, we have a function

{3: H2 (M; Z/2Z)

(4.8) Z/8Z Â·

-+

We define f3 by taking an embedded surface representing x, using the Spin structure

on M to get a Pin- structure on F x , taking the underlying Pin- bordism class,

and using our explicit identification of this group with Z/8Z.

We need to see why this is independent of the choice of embedded surface.

Given two such surfaces, there is a bordism in M x [0, 1] between them. Let W C

M x [0,1] be a 3-manifold with the two boundary components representing the

same element in H 2 (M; Z/2Z). Since M x [0,1] is spun, we get a Pin- structure

Kirby & Taylor: Pin structures on low-dimensional manifolds 213

on W which is our given Pin- structure at the two ends. Since Brown's Z/8Z is a

pin- bordism invariant, we are done. It further follows that {3(0) = o.

Reduced mod 2 f3( x) is just the mod 2 Euler class of an embedded surface

representing x, and hence /3 is additive mod 2. We have

f3(x + y) = /3(x) + {3(y) + 2f(x,y) Â·

(4.9)

which we will prove in a minute. It follows that f(x,x) = -f3(x) reduced mod 4.

Note that, mod 4, f3(x + y) = f3(x) + f3(y) + 2r(x, x, y).

How does f3 change when we change the Spin structure by a E H t (M; Z/2Z)?

The principle is easy. Given a sudace, F, restrict O! to F and consider it to be a

change in Pin- structure on F. Compute the Brown invariant for this new Pin-

structure, and this is the value of the new f3 on F. It follows from Lemma 3.7 that

= f3(x) + 2f(x, a)

f3a(x)

(4.10)

with notation as above.

Given the theorem below, we now prove formula 4.9. From this theorem we

get: U - U a = 2f3(a) and U - U Ot = 2f3(at). Also U a - U 01 = 2{30/(al - a). Hence

f3a(al - a) = f3(a}) - f3(a). Set at = x + a and use formula 4.10.â€¢

The main result concerning /3 is

Theorem 4.11. Let M be a spun 3-manifold with resulting function f3 and J.t-

invariant U in Z/16Z. Let O! E HI (M; Z/2Z) be used to change the Spin structure,

and let UO/ be the new J.t-invariant. Then

= 2f3(a) (mod 16)

UO/

U-

where a E H2 (M; Z/2Z) is the Poincare dual to O!.

Proof: The proof is just the Guillou-Marin formula, [G-M, Theoreme, p. 98], or

our discussion of it in Â§6, 6.4. On M x [0,1] put the original Spin structure on

M X 0 and put the altered one on M x 1. We can cap this off to a closed 4-manifold

by adding Spin manifolds that the two copies of M bound to either end. The

resulting 4-manifold has index U a - u. Let F be a surface in M representing a.

Then F X 1/2 is a dual to W2 for the 4-manifold. Since F is in a product, F.F = 0

and the enhancement used in the Guillou-Marin formula is the same as the one we

put on F to calculate f3. By formula 6.4, U - U a = 2f3(a).â€¢

As a corollary we get a result of Turaev, [Tu]

Corollary 4.12. The quadratic enhancement of the linking form gives the J.t-

invariant mod 8 via the Milgram Gauss sum formula.

Proof: This was proved in [Tal for rational homology spheres. Pick a basis for the

torsion free part of HI and do surgery on this basis. The resulting bordism, W, has

Kirby & Taylor: Pin structures on low-dimensional manifolds

214

signature 0; both boundary components have isomorphic torsion subgroups of HI;

and the top boundary component has no torsion free part. Put a Spin structure ˜n

the bordism, which puts a Spin structure at both ends. The two enhancements on

the linking forms are equal, and they stay equal if we change both Spin structures

by an element in HI (W; Z/2Z). Any Spin structure on M can be obtained from

our initial one by acting on it by an element of the form x + y, where x comes

from HI (W; Z/2Z) and y comes from HI (M; Z). But acting by this second sort

of element does not change the mod 8 IJ-invariant or the quadratic enhancement of

the linking form. â€¢

Â§5. Geometric calculations of n˜˜nÂ± â€¢

We begin this section with a calculation for the 3-dimensional Spin, Pin- and

Pin+ bordism groups.

ni ˜ 0; nfin- ˜ 0 and [nWI]: nfin+ ˜ n˜pin ˜ Z/2Z is an

pin

Theorem 5.1.

isomorphism.

Proof: The Spin bordism result is classical: [ABPl], [Ka] or [Ki].

Given a non-orientable PinÂ± manifold M 3 , we will try to find a PinÂ± bordism

to an orientable manifold which then PinÂ± bounds by the Spin case. The dual to

wl(M) is an orientable surface F by Proposition 2.3. The first step is to reduce to

the case when F has trivial normal bundle. If not, consider F intersected trans-

versely with itself. It can be arranged that this is a single circle C, which is dual int

F to wI(M) pulled back to F. The normal bundle to C in M is VFCMlcEBvFcMlc j

which is also VCCF E9 VCCF which is trivialized. Hence the PinÂ± structure on M .,'.l.:.l

induces a Pin=F structure on C. Suppose C with this structure bounds y2; let E j

denote the total space of ( EB ( over Y, where ( is the determinant line bundle for .˜

Y. Note that inside BE there is a copy of (BY) x B 2 , and E has a PinÂ± structure}

extending the one on (aY) X B2. We can form M x [0,1] UE by gluing (ay2) X B 2 .I ;.˜ Â·

to C x B2 X 1 where C x B2 is the trivialized disk bundle to C above. Clearly the

PinÂ± structure extends across the bordism, and the "top" is a new PinÂ± manifold

"..

â€¢.˜

M1 with a new dual surface F1 with trivial normal bundle.

In the Pin- case, C has a Pin+ structure which bounds (nfin+ = 0, Theorem ˜

2.1) so we have achieved the (M1,FI ) case. In the Pin+ case an argument is needed j

;

to see that we never get C representing the non-zero element in ni in - = Z/2Z, iJ

˜

Le. C does not get the Lie group Spin structure.

j

To show this, let V be a dual to WI and let E be a tubular neighborhood of :˜

V. By the discussion just before Lemma 2.7, since E as a Pin+ structure, there is ˜

an inherited Spin structure on V (in fact there are two which differ by the action ..˜

of x E HI (V; Z/2Z), where x denotes the restriction of WI to V). Note x also :˜

describes the double cover BE ˜ V. The boundary, BE, also inherits a Pin+ )

structure and we saw, Lemma 2.7, that, if we orient aE and V so that the covering ,

Kirby & Taylor: Pin structures on low-dimensional manifolds 215

map is degree 1, the Spin structure on BE is the same as the one induced by the

covering map. The Spin structure on BE bounds the Spin manifold which is the

closure of M - E, so if C is the dual to x and q is the quadratic enhancement on

HI (V; Z/2Z), q(C) = 0 by Theorem 3.12. Recall that the normal bundle to V in

M, when restricted to C is trivial. Hence the framing on C as a circle in V is the

same as the Pin- structure on C as V intersect V in a Pin+ manifold. Hence C

has the non-Lie group Spin structure and hence represents 0 in n˜pin.

Hence we may now assume that F has trivial normal bundle in M. Therefore

F inherits a PinÂ± structure from the one on M, and hence, after choosing an

orientation, F has a Spin structure. If the Spin structure on F is a bou˜dary then

it is easy as above to construct a PinÂ± bordism to an oriented manifold. In the

Pin+ case we are entitled to assume that the surface bounds because that is what

the invariant [nW1] is measuring. In the Pin- case, the Klein bottle X S1 with

the Lie group framing is an example for which the F has the non-bounding Spin

structure. But if we add this manifold to our original M, for the new manifold, F .

will bound and we are done.

We have now proved that [nWl] is injective in the Pin+ case and that nfin-

is generated by !{ x Sl, where K is the Klein bottle and the Pin - structure comes

from some structure on the surface and the Lie group Spin structure on 51. In some

Pin- structures, !{ bounds and hence so does K X Sl. In the others, K is Pin-

bordant to two copies of Rp2, so !{ X SI is bordant to two copies of Rp2 x Sl.

Hence, if we can prove that [nW1] is onto and that Rp2 X Sl bounds, we are done.

If we take the generator of nfin+ and cross it with 8 1 with the Lie group Spin

structure, we get a 3-manifold with [nW1] being the 2-torus with Lie group Spin

structure so by Proposition 3.8, [nW1] is onto.

Consider Rp2 in Rp4: it is the dual to w˜ +W2 so there is a Pin- structure on

Rp4 - Rp2 which restricts to the Lie group structure on the normal circle to RP2.

An easy calculation of Stiefel-Whitney classes shows that the normal bundle v of

Rp2 in Rp4 is orientable but W2(V) f: o. So we take the pairwise connected sum

(RP4, RP2)#(CP2, CP1) and then the normal bundle of Rp2 = Rp2#Cpl in

Rp4#Cp2 has W1 = W2 = o. For a bundle over Rp2 this means that the bundle

is trivial, so its normal circle bundle is Rp2 X 51. The two Pin- struct˜res on

Rp4#Cp2 - Rp2 bound two Pin- structures on Rp2 X 51 which have the Lie

group structure on Sl. Since this is all the Pin- structures that there are with the

Lie group Spin structure on the S1, we are done.â€¢

Next we turn to the 4-dimensional case. The result is

nfin- =

Theorem 5.2. The group n˜pin ˜ Z generated by the Kummer surface;

0; and the group nfin+ ˜ Z/16Z generated by RP4.

Spin result may be found in [Ki, p. 64, Corollary].

Proof: The first lemma

OUf

determines the image of n;pin in the PinÂ± bordism groups.

216 Kirby & Taylor: Pin structures on low-dimensional manifolds

Lemma 5.3. The Kummer surface bounds a Pin-manifold hence so does any 4-

dimensional Spin manifold. Twice the Kummer surface bounds a Pin+ manifold,

but the Kummer surface itself does not. Hence a 4-dimensional Spin manifold

Pin+ bounds iff its signature is divisible by 32.

Proof: The Enriques surface, E, [Hal, is a complex surface with 1rI(E) ˜ Z/2Z

with w2(E) =F 0. Habegger shows that H2 (M; Z) ˜ ZlO E9 Z/2Z and w2(M) is

the image of the non-zero torsion class in H2 (M; Z), see paragraph 2 after the

Proposition on p. 23 of [HaJ. If y E HI (E; Z/2Z) is a generator, then from the

universal coefficient theorem, y2 = W2(W). If L is the total space of the line bundle

over E with WI = y, then it is easy to calculate that L is Pin- (but not Pin+),

and oE is the I{ummer surface. This proves the Kummer surface bounds a Pin-

n: pin

˜ Z generated by the Kummer surlace, this proves any Spin

manifold. Since

4-manifold bounds as a Pin- manifold.

Let M 4 is a Spin manifold and let W 5 be a Pin- manifold with oW = M

as Pin- manifolds. Consider the obstruction to putting a Pin+ structure on W

extending the one on M 4 â€¢ The obstruction is W2(W) = w˜(W), so the dual class

is represented by a 3-manifold formed as the intersection to a dual to Wl pushed

off itself. As usual, this 3-manifold has a natural Pin+ structure and it is easy to

see that we get a well-defined element in nfin+ ˜ Z/2Z. IT this element is 0, then

we can glue on the trivializing bordism and extend its normal bundle to get a new

Pin- manifold WI which still bounds M and has no obstruction to extending the

Spin structure on the boundary to a Pin+ structure on the interior. Hence, if our

element in nfin+ is 0, M bounds. From this it is easy to see that twice the Kummer

surface bounds. Hence any 4-dimensional Spin manifold with index divisible by 32

bounds a Pin+ manifold.

Suppose that W is a Pin+ manifold with oW = M orientable. Let V C W

be a dual to Wl contained in the interior of W. Let E be a tubular neighborhood

of V with boundary oE. As' usual, BE is orientable and the covering translation is

orientation preserving. Since V is orientable with a normal line bundle, if we fix an

orientation, Spin structures on V correspond to Pin+ structures on E. Since W is

a Pin+ manifold, E has an induced Pin+ structure and Y acquires an induced Spin

structure. The bordism between M and 8E is an oriented Pin+ bordism, so M

and oE have the same signature. But oE is the double cover of V so has signature

twice the signature of V. Since V is Spin, the signature of V is divisible by 16, so

the signature of M is divisible by 32. This shows that the Kummer surface does

not bound a Pin+ manifold and indeed that any 4-dimensional Spin manifold of

index congruent to 16 mod 32 does not bound a Pin+ manifold.â€¢

Since nfpin ˜ Z generated by the Kummer surface this lemma calculates the

n: in nfinÂ± and our next goal is to try to produce a PinÂ± bordism

pin

image of

from any PinÂ± manifold to an orientable one.

To this end let M be a 4-manifold with y 3 a dual to WI. Consider the dual

Kirby & Taylor: Pin structures on low-dimensional manifolds 217

to WI intersected with itself. It is a surface F C M and the normal bundle is

two copies of the same line bundle. Indeed, the transversality condition gives an

isomorphism between the two bundles. This line bundle is also abstractly isomorphic

to the detenninant line bundle for F. A PinÂ± structure on F gives rise to a PinT

structure on the total space of the normal bundle of F in M by Lemma 1.7. Hence

we can use the PinÂ± structure on M to put a PinT structure on F and it is not

hard to check that we get a homomorphism nfinÂ± -+ nfin T â€¢ If F bounds in this

structure, one can easily see a PinÂ± bordism to an new 4-manifold M I in which

the dual to WI has trivial normal bundle. This puts a PinÂ± structure on VI. By

orienting Vi we get a Spin manifold and if VI bounds in this Spin structure, M 1

PinÂ± bounds an orientable manifold.

Consider the Pin- case. Any element in the kernel of the map [nwl]: nfin- --+

nfin+ is Pin- bordant to a Pin-manifold whose dual to WI, say V, has trivial

normal bundle. Orienting this normal bundle gives a Pi n - structure on V, and since

ni = 0, we can further Pin- bord our element to an orientable representative.

pin

It then follows from Lemma 5.3 that the map [nw˜] is injective.

To show that this map is trivial, which proves nfin- = 0, proceed as follows.

Let V c M be a dual to WI (M) and let F2 denote the transverse intersection of

V with itself. Since the normal bundle to F in M is 2 copies of the determinant

line bundle for F, F acquires a Pin+ structure from the Pin- structure on M.

Let E C V be a tubular neighborhood for F in V. Theorem 2.9 applies to this

situation to show that the Pin+ structure on DE induced by the double cover map

BE ---+ F is the same as the Pin+ structure induced on DE C M from the fact that

its normal bundle is exhibited as the sum of 2 copies of its determinant line bundle.

Since the normal bundle to V in M is trivial on V - F, V - F has a Spin structure

which restrict˜ to the given one on vE. By Lemma 3.13, the oriented cover map

ni

nfin+ -+ pzn is an isomorphism, so F is a Pin+ boundary, which finishes the

Pin- case.

So consider the Pin+ case. This time our homomorphism goes from nfin+

to nfin- ˜ Z/8Z and the example of Rp4 shows that it is onto. Just as in the

Pin- case, any element in the kernel of this homomorphism is Pin+ bordant to

an orientable manifold. This together with Lemma 5.3 shows that 0 ---+ Z/2Z ---+

n4p.an+ ---+ Z /8Z ---+ 0 is exact.

To settle the extension requires more work. Given a Pin+ structure on a

4-manifold M, we can choose a dual to WI, say V eM, and an orientation on

M - V which does not extend across any component of V. We can consider the

bordism group of such structures, say G4 â€¢ There is an epimorphism G4 ---+ n,fin+

defined by just forgetting the dual to WI and the orientation. There is another

homomorphism G 4 ---+ Q/32Z defined as follows. Let E be a tubular neighborhood

of V with boundary BE. The covering translation on DE is orientation preserving,

so V is also oriented. The normal bundle to BE in M is a trivial line bundle,

Kirby & Taylor: Pin structures on low..dimensional manifolds

218

oriented by inward normal last, where inward is with respect to the associated disk

bundle. Hen˜e 8E acquires a Spin structure, and hence a p, invariant in Z/16Z.

The manifold BE is a 3-manifold with an orientation preserving free involution

on it, hence there is an associated Atiyah-Singer a invariant, a(8E) E Q. Define

t/J(M, V) = O'(M - int V) +o:(oE) - 2p,(V) E Q/32Z. It is not hard to check that

1/J depends only on the class of (M, V) in G4 and defines a homomorphism. We

can make choices so that t/J(RP4,Rp3) = +2. Hence 'Â¢'(8(Rp4,Rp3)) = 16 with

these choices. The Pin+ hordism of 8 copies of Rp4 to an oriented manifold is

seen to extend to a. bordism preserving the dual to WI and orientation data. This

oriented, hence Spin manifold has index congruent to 16 mod 32, and so we have

constructed a Pin+ bordism (with some extra structure which we ignore) from 8

copies of Rp4 to a Spin manifold which is Pin+ bordant to the Kummer surface.

This shows 0 4 In + ˜ Z/16Z.â€¢

p'

Â§6. 4-dimensional characteristic bordism.

The purpose of this section is to study the relations between 4-manifolds and

embedded surfaces dual to W2 + w˜ â€¢

Definition 6.1. A pair (M, F) with the embedding of F in M proper and the

boundary of M intersecting F precisely in the boundary of F is called a charac-

teristic pair if F is dual to W2 + wl. A characteristic pair is called characterized

provided we have fixed a Pin-structure on M - F which does not extend across

any component of F. The characterizations of a characteristic pair are in one to

one correspondence with HI (M; Z/2Z).

We begin by discussing the oriented case.

Lemma 6.2. Let M be an oriented manifold with a codimension 2 submanifold F

which is dual to W2. There exists a function

Char(M,F) Pin-(F) .

--+

The group HI (M;Z/2Z) acts on Char(M,F), the group HI (F;Z/2Z) acts on

Pin-(F) and the map is equivariant with respect to the map induced on HI (; Z/2Z)

by the inclusion F eM.

Remark. Later in this section we will define this function in a more general situ-

ation.

Proof: There is an obvious restriction map from characteristic structures on (M, F)

to those on (E,F), where E is the total space of the nonna! bundle to F in M,

denoted v. Hence it suffices to do the case M = E. In this case we expect our

function to be a bijection. After restricting to the case M = E it is no further

restriction to assume that F is connected since we may work one component at a

time.

Kirby & Taylor: Pin structures on low-dimensional manifolds 219

We begin with the case that F has the homotopy type of a circle. In this case

v has a section, so choose one and write v = AE9 e1 . Orient e1 and use it to embed F

in 8E. The normal bundle to 8E in E is oriented; E is oriented; so 8E is oriented.

The normal bundle to the embedding of F in 8E is A so the orientation on E plus

the orientation of e1 pick out a preferred isomorphism between A and det TF. From

Corollary 1.15, there is a Pin- structure on F induced from the one on 8E.

We want to see that this Pin- structure is independent of the section we chose.

It is not difficult to work out the effect of reorienting the section: there is none.

Suppose the bundle is trivial. We divide into two cases depending on the

dimension of E. In the I-dimensional case, we may proceed as follows. The manifold

F is a circle and since the bundle has oriented total space, it must be trivial. Hence

8E = T 2 and HI (T 2 ; Z/2Z) has one preferred generator, the image of the fibre,

otherwise known as a meridian, denoted m. Let x denote another generator. Since

the Spin structure is not to extend over the disk, the enhancement associated to

the Spin structure on T 2 , say q, satisfies q(m) = 2. The Spin structure on the

embedded base is determined by q of the image, which is either x or x +m. Check

q(x) = q(x + m).

In the higher dimensional case, there is an 8 1 embedded in F and the normal

bundle to this embedding is trivial. Over the 8 1 in F there is an e˜bedded T 2 in

8E and the bundle projection, p, identifies the normal bundle to T2 in 8E with the

normal bundle to 8 1 in F. Fix a Spin structure on one of these normal bundles

and use p to put a Spin structure on the other. The Spin structure on 8E restricts

to one on T2 and it is not hard to check that the Pin- structure we want to put on

F using the section is determined by using the section over 8 1 and checking what

happens in T2. We saw this was independent of section so we are done with the

trivial case.

Now we turn to the non-trivial case, still assuming that F is the total space of

a bundle over 8 1 â€¢ The minimal dimension for such an F is 2 since the bundle, v, is

non-trivial. In this case F is just a Mobius band. Since E is oriented, the bundle

we have over F is isomorphic to det v $ el . Sitting over our copy of 8 1 in F is the

Klein bottle, K2, and the normal bundle to K2 in 8E is just the pull-back of v.

One can sort out orientations and check that there is an induced Pin - structure

on K2 so that the Pin- structure that we want to put on F is determined by the

enhancement of the section applied to SI as a longitude of j{2. This calculation

is just like the torus case. In the higher dimensional case, v is a non-trivial line

bundle plus a trivial bundle so we can reduce to the dimension 2 case just as above.

Now we turn to the case of a general F.

Since we have done the circle case, we may as well assume that the dimension

of F is at least 2. If the dimension of F is 2, then we can find a section of our bundle

over F - pt. The embedding of F - pt in 8E gives a Pin- structure on F - pt and

this extends uniquely to a Pin- structure on F. This argument even works if F

Kirby & Taylor: Pin structures on low-dimensional manifolds

220

has a boundary and we take as the function on the boundary the function we have

already defined. Now if we restrict this structure on F to a neighborhood of an

embedded circle, we get our previous structure. Since this structure is independent

of the section, the structure on all of F is also independent of the section since Pin-

structures can be detected by restricting to circles.

The higher dimensional case is a bit more complicated. We can define our

function by choosing a set of disjointly embedded circles and taking a tubular neigh-

borhood to get U, with H t (U; Z/2Z) ˜ HI (F; Z/2Z) an isomorphism. We then

use our initial results to put a Pin- structure on U and then extend it uniquely to

all of F. Now let V be a tubular neighborhood of a circle in F. We can restrict

the Pin- structure on F to V, or we can use our "choose a section, embed in aE

and induce" technique. There is an embedded surface, W 2 , in F which has the core

circle for V as one boundary component and some of the cores of U as the others.

Let X be a tubular neighborhood of W in F. The bundle restricted to X has a

section so we can induce a Pin - structure on X using the section. This shows that

the two Pin- structures defined above on V agree. It is not hard from this result

to see that the Pin- structure on F is independent of the' choice of U.â€¢

Remarks. Notice that the proof shows that the Pin- structure on a codimension

osubset of F, say X, only depends on the Pin- structure on the circle bundle lying

over X. It is not hard to check that our function commutes with taking boundary,

P, from the rth Guillou-Marin bordism group

we get a well-defined homomorphism,

to n˜˜˜-.

Theorem 6.3. Let M 4 be an oriented 4-manifold, and suppose we have a charac-

teristic structure on the pair (M, F). The following formula holds:

(6.4) 2 Â· (3(F) = F.F - sign(M) (mod 16)

where the Pin- structure on F is the one induced by the characteristic structure

on (M, F) via 6.2.

Proof: By the Guillon-Marin calculation, their bordism group in dimension 4 is

Z E9 Z, generated by (84 , RP2) and (CP2, 8 2). The formula is trivial to verify

for (CP2, 52). For (84 , RP2) we must verify that RP2.Rp2 = 2 implies that the

resulting q is 1 on the generator. Now Rp2 has two sorts of embeddings in 8 4 â€¢

There is a "right-handed" one, which has RP2.Rp2 = 2, and a "left-hand" one

which has RP2.Rp2 = -2. The "right-handed" one can be constructed by taking

a 'right-handed" Mobius strip in the equatorial 8 3 and capping it off with a ball

in the northern hemisphere. For our vector field, use the north-pointing normal.

The "even" framing on the bundle to Vk, the core of the Mobius band, is the one

given by the o-framing in S3. Hence we may count half twists in S3, where the

right-hand Mobius band half twists once.â€¢

Kirby & Taylor: Pin structures on low-dimensional manifolds 221

It would be nice to check that the Pin- structure we put on the character-

ized sudace agrees with those of Guillou-Marin and Freedman-Kirby. For the

Freedman-Kirby case we take an embedded curve k in F and cap it off by an ori-

entable surface, B, in M. We start B off in the same direction as our normal vector

field, so then the normal bundle to B in M, when restricted to the boundary circle,

will be the 2-plane bundle around k we are to consider. The Guillou-Marin case is

similar except that B need not be orientable. Since B is a punctured sudace, the

normal bundle to B in M splits off a trivial line bundle and so is a trivial bundle

plus the determinant line bundle for the tangent bundle. Having chosen one section,

the others are classified by HI (B; ZWl) , where ZWI denotes Z coefficients twisted

by WI of the normal bundle. When restricted to the boundary circle, this gives a

well-defined "even" framing of the normal bundle.

If B does not intersect F except along BB, Theorem 4.3 shows that the framing

on 8B is the even one in the sense of Definition 4.2. We can assume in general that

B intersects F transversally away from BB. The surface iJ = B - Jl D2 lies in

M - F and each circle from the transverse intersection has the non-bounding Spin

structure. Hence, in general, the framing on BB is even iff the mod 2 intersection

number of F and B is even. Moreover, the number of half right twists mod 4 is just

the obstruction to extending the section given by the normal to k in F over all of B.

This shows that our enhancement and those of Freedman-Kirby and Guillou-Marin

agree when both are defined.

The enhancement above is defined more generally since we do not need the

membranes to select the Pin- structure and hence do not need the condition that

HI (F; Z/2Z) ˜ HI (M; Z/2Z) should be o. One nice application of this is to

compute the Jl-invariant of circle bundles over surfaces when the associated disk

bundle is orientable.

Any O(2)-bundle, TJ, over a 2 complex, X, is determined by WI (1]) and the Euler

class, X(77) E H2 (X; ZW1), where ZWl denotes Z coefficients twisted by WI(77). In

our case, X is a surface which we will denote by F; the bundle 77 has the same WI

as the surface; and the Euler class is in H 2 (F; ZW1) ˜ z. Let S(1]) denote the circle

bundle. One way to fix the isomorphism is to orient the total space of 77 and then

F.F = X(1]). The signature of the disk bundle is also easy to compute. We denote

it by u(77) since we will see it depends only on 1]; indeed it can be computed from

Wl(77) and X(77). If WI(1]) = 0 then U(77) = sign X(",) (Â±1 or 0 depending on X(TJÂ»):

if WI (17) f; 0 then u(77) = O. By Lemma 6.2, Spin structures on S('1J) which do not

extend across the disk bundle are in 1-1 correspondence with Pin- structures on

F.

Theorem 6.5. With notation as above fix a Spin structure on S(TJ). Let b(F) = 0

if this structure extends across the disc bundle and let b(F) = f3(F) if it does not

and the Pin - structure on F is induced via the function in Lemma 6.2. We have

(6.6) (mod 16) .

Kirby & Taylor: Pin structures on low-dimensional manifolds

222

Proof: The result follows easily from 6.4.â€¢

We want to describe a homomorphism from various characteristic bordism

groups into the Pi'n - bordism group in two dimensions less. Roughly the ho-

momorphism is described as follows. We have a characteristic pair (M,F) and we

will see that, with certain hypotheses, F is a Pin- manifold. We then use the char-

acterization of the pair to pick out a Pin- structure on F. The homomorphism

then just sends (M, F) to the Pin- bordism class of F.

To describe our hypotheses, consider the following commutative square

F BO(2)

---)0

1 1

M TO(2)

---)0

Let U E H2 (TO(2); Z/2Z) denote the Thom class and recall that U pulls back

to W2 in H 2 (B O (2); Z/2Z). The 2-plane bundle classified by v is just the normal

bundle to the embedding i: F C M, and f*(U) E H 2 (M; Z/2Z) is the class dual to

F. Let a denote the class dual to F. Then we see that i*(a) = W2(VPCM), where

VPcM is the normal bundle to the embedding. Let us apply this last equation to

our characteristic situation. The class a is w2(M) +w˜(M) and we have the bundle

equation i*(TM) = TF EB VPCM. Now i*Wl(M) = WI (F) + WI(V) and i*W2(M) =

w2(F)+W2(V)+Wl(F)Â·Wl(V). Hence i*(W2(M)+wi(M)) = w2(F)+W2(V)+Wl(F).

wl(v)+w˜(F)+wi(v) and using our equation for W2(V) we see that w2(F)+w˜(F)=

Wl(V) Â· i*WI(M). Hence F is Pin- iff the right hand product vanishes or

Lemma 6.7. The surface F has a Pin- structure iff

To study Wl(V). i*WI(M) we may equally study Wl(V) n (i*Wl(M) n [F,8F]).

The term i*WI(M)n [F, 8F] can be described as the image of the fundamental class

of the manifold obtained by transversally intersecting F and a manifold V in M

dual to WI' Hence, the product Wl(V) â€¢ i*Wt(M) vanishes if the normal bundle to

F n V c V is orientable. This suggests studying the following situation.

Definition 6.8 . Let M be a manifold with a proper, codimension 2 submanifold

F (proper means that oM n F = of and that every compact set in M meets F

in a compact set). A characteristic structure on the pair (M,F) is a collection

consisting of

a) a proper submanifold V dual to wl(M) which intersects F transversely

Kirby & Taylor: Pin structures on low-dimensional manifolds 223

b) an orientation on M - V which does not extend across any component of V

c) a Pin - structure on M - F that does not extend across any component of F

(so F is dual to W2 + wi)

d) an orientation for the normal bundle of V n F in V.

Let Char-(M, F) be the set of characteristic structures on (M, F).

The next goal of this section is to prove a "reduction of structure" result, the

Pin - Structure Correspondence Theorem.

Theorem 6.9. Tbere exists a function

which is natural in the following sense. If we change the Pin- structure on

M - F which does not extend across any component of F by acting on it with

a E HI (M; Z/2Z), then we change W oftbe structure by acting on it with i*(a) E

HI (F; Z/2Z), where i: F C M is the inclusion. If X denotes a collection of com-

ponents of F n V, then the dual to X is a class in x E HI (F; Z/2Z). If we switch

the orientation to the normal bundle of F n V in F over X and not over the other

components, then we alter \lI by acting with x. If we change the orientation on

M - V which does not extend across any component of V, we do not change W

of tbe Pin- structure. Finally, if MI c M is a codimension 1 submanifold with

trivialized normal bundle such that F and V intersect M 1 transversely (including

the case M 1 = 8M), then the characteristic structure on M restricts to one on MI.

The Pin- structure we get on F I = M 1 n F is the restriction of the one we got on

F.

Remark. The observation that characteristic structures restrict to boundaries al-

lows us to define bordism groups: let n˜ denote the bordism group of characteristic

structures.

Reduction 6.10. Given a closed manifold M with a characteristic structure, let

E C M denote the total space of the normal bundle of F in M. The associated

circle bundle, 8E, is embedded in M with trivial normal bundle and without loss

of generality we may assume that V intersects fJE transversally. Hence E acquires

the above data by restriction.

This reduces the general case to the following local problem. We may deal

with one component at a time now and so we must describe how to put a Pin-

structure on a connected Pin- manifold F, given that we have a 2-disc bundle over

F with total space E; a Pin - structure on fJE which does not extend to all of E; a

codimension 1 submanifold V which is dual to WI (E) and intersects F transversally;

an orientation on E - V which does not extend across any component of V; and an

orientation for the normal bundle of Fn V in V. We must also check that the Pin-

Kirby & Taylor: Pin structures on low-dimensional manifolds

224

structure that we get on F is independent of our choice of tubular neighborhood.

Note for reassurance that Pin- structures on F are in one to one correspondence

with Pin- structures on 8E which do not extend to E.

Let us consider the following situation. We have a circle bundle p: BE ˜ F

e.

e.

over F with associated disc bundle We let E denote the total space of We have

a codimension 1 submanifold, V, of E which is dual to wl(E) and which intersects

F transversally. We are given an orientation on E - V which does not extend

across any component of V and we are given an orientation of the normal bundle

to F n V in V. We are going to describe a one to one correspondence between

Pin- structures on F and Pin- structures on DE which do not extend across E.

Furthermore, suppose that U C F is a submanifold with trivialized normal bundle.

Suppose that U intersects V transversally and let Eu denote the total space of the

erestricted to U. Then over U we have our data. Notice that any

disk bundle for

Pin - structure on F restricts to one on U, and any Pin-structure on BE restricts

to one on 8Eu. Let Pin-(F, U) denote the set of Pin- structures on F which

restrict to a fixed one on U. Define Pin-(8E,8Eu) similarly except we require

that the Pin-structures do not extend across the disk bundles. Below we will

define a 1-1 map '11: 'Pin-(8E, 0) ˜ 'Pin-(F, 0). If we fix a Pin- structure on U,

which comes from one on F, and use \It for U to pick out a Pin- structure on 8Eu,

then we also get a 1-1 map

There is an isomorphism, p*: HI (F, U; Z/2Z) ˜ HI (8E,8E u U 8 1 ; Z/2Z), in-

duced by the projection map, p: DE --+ F, where 8 1 denotes a fibre of the bundle

(if U :F 0 then 8Eu U 8 1 = 8Eu). The group HI (8E, 8Eu U 8 1 ; Z/2Z) acts in a

simply transitive fashion on Pin-CoE,8Eu) and the group HI (F, U; Z/2Z) acts in

a simply transitive fashion on Pin-(F, U). The map q, is equivariant with respect

to these actions and p* .

The relative version of the Pin - Structure Correspondence gives the uniqueness

result needed in Reduction 6.10 since any two choices are related by a picture with

our data over E x I with structure fixed over E x 0 and E x 1.

Note first that F has a Pin-structure by the calculations above.

Recall that there is a sub-bundle of ToE, namely the bundle along the fibres,

7]. This is a line bundle which is tangent to the fibre circle at each point in 8E. The

quotient bundle, p, is naturally isomorphic to TF, via the projection map, p. Our

first task is to use our given data to describe an isomorphism between 7] EB det(ToE)

and det(p) EB e1 â€¢ To fix notation, let N be a tubular neighborhood of V in 8E and

fix an isomorphism between p E9 7] and ToE.

On 8E - V we have an orientation of ToE. This describes an isomorphism

between det(ToE) and el . Furthermore, the orientation picks out an isomorphism

Kirby & Taylor: Pin structures on low-dimensional manifolds 225

between '1 and det(p) as follows. These two line bundles are isomorphic since they

have the same WI, and there are two distinct isomorphisms over each component of

aE - V. Pick a point in each component of BE - V, and orient '1 at those points.

The orientation of TaE picks out an orientation of p, and hence det(p), at each point.

We choose the isomorphism between '1 and det(p) which preserves the orientations

at each point. It is easy to check that if we reverse the orientation at a point for "l,

we reverse the orientation for det(p) and hence get the same isomorphism between

these two bundles. The isomorphism between "l â‚¬a det(T8E) and det(p) ED fl is just

the sum of the above two isomorphisms.

eis

We turn our attention to the situation over N. Over F n V, the normal

bundle to F n V in V, and hence it is oriented. Hence so is p*(e) in aE, and p*(e)

is isomorphic to ", ED fl. The -outward normal to DE in E orients the fl, and hence

." is oriented over p-l(F n V), and hence over N. This time det(p) and det(T8E)

are abstractly isomorphic, and we can choose an isomorphism by choosing a local

orientation. Since "l is oriented and 0 ˜ "l --. T8E ˜ P ˜ 0 is exact, there is a

n&tural correspondence between orientations of T8E at a point and orientations of

p at the same point, As before, if we switch the orientation on T8E, we still get the

same isomorphism between det(p) and det(T8E). As before, the orientation for ",

defines an isomorphism between '1 and fl, but this time we take the isomorphism

which reverses the orientations. We take the sum of these two isomorphisms as our

preferred isomorphism between ", ED det(TaE) and det(p) EB fl â€¢

Now over N - V, we have two isomorphisms between '1 Ef>det(TaE) and det(p)E9

fl. If we restrict attention to a neighborhood of aN both bundles are the sum of two

trivial bundles, and our two isomorphisms differ by composition with the matrix

(˜ -˜).

Parameterize a neighborhood of aN in N by aN x [0, 1r/2] and twist one bundle

Ism bif the matnx sin(t) -sin(t)) 't1:T can now g1 our two Isomor-Â·

,(cos(t)

1somorph'

, cos(t)' vve ue

phisms together to get an isomorphism between TJ EB det(TaE) and det(p) â‚¬a fl over

all of 8E.

Finally, we can describe our correspondence between Pin- structures. Suppose

that we have a Pin- structure on F. This is a Spin structure on TF E9 det(TF).

Since p is isomorphic via p to TF, we get a Spin structure on p â‚¬a det(p), and

hence on p E9 det(p) ED fl. Using our constructed isomorphism, this gives a Spin

structure on p EB "l EB det(TaE). Choose a splitting of the short exact sequence

o--. "l -. TaE ˜ p --. 0, and we get a Spin structure on TaE EI1 det(T8E).

H we choose a different splitting, we get an automorphism of TaE and hence an

automorphism of TIJE E9 det(TaE) which takes one Spin structure to the other. But

this automorphism is homotopic through bundle automorphisms to the identity,

and so the Spin structure does not change.

226 Kirby & Taylor: Pin structures on low-dimensional manifolds

Finally, let us consider the Pin- structure induced on a fibre SI. We will look

at this situation for a fibre over a point in F where we have an orientation of TaE.

Restricted to 8 1 , the bundle TaE splits as fJ plus the normal bundle of 8 1 in BE, so

1] is naturally identified as the tangent bundle of SI and the normal bundle of SI in

aE is trivialized using the bundle map p. The trivialization of the normal bundle

of SI in aE plus the Spin structure on TaE ED det(TaE) yields a trivialization of

77181, which then yields a trivialization of the tangent bundle of SI. Since SO(1)

is a point, any oriented I-plane bundle has a unique framing, which in the case of

the tangent bundle to the circle is the Lie group framing. The Pin- structure that

results from a framing of the tangent bundle of 51 is therefore the one that does

not extend across the disk, so our Pin- structure on aE does not extend across E.

RecaJl that Pin- structures on BE that do not extend across E are acted on

by HI (F; Z/2Z) in a simply-transitive manner by letting p*(x) E HI (aE; Z/2Z)

act as usual on Pin- structures on BE. If we change Pin- structures on F by

x E HI (F; Zj2Z), we change the Pin- structure that we get on BE by the p*(x)

in HI (aE; Z/2Z) so our procedure induces a one to one correspondence between

Pin- structures on F and Pin- structures on BE which do not extend across E.

Next, we consider the effects of changing our orientations. We wish to study

eeffect

how the choices of orientations on BE - V and on the resulting map between

Pin- structures on F and Pin- structures on BE which do not extend across E.

ñòð. 8 |