ñòð. 9 |

Let us begin by considering the effect of changing the orientation on This switches

the orientation on 77 and so our bundle map remains the same over BE - Nand

_ ˜ ). This has the effect of putting

over N it is multiplied by the matrix ( -˜

full twists into the framing around any circle that intersects F n V geometrically

s

t times where s == t (mod 2). Hence the class in HI (F; Z/2Z) that measures the

change in Pin- structure is just the class dual to F n V. If F n V has several

eover

components and we switch the orientation of only one of them then the class

in HI (F; Z/2Z) that measures the change in Pin- structure is just the class dual

to that component of F n V.

eand on M - V. This time the

Now suppose that we switch the orientation on

aE by multiplication by the matrix ( - ˜ _ ˜) .

two bundle maps differ over all of

The effect of this is to change the Pin- structure on F via w}(F). This follows

from Lemma 1.6.

From the two results above the reader can work out the effect of the other

possible changes of orientations. Finally, the diligent reader should work through

the relative version.

This ends our description of the Pin- Structure Correspondence.â€¢

As an application of the Pin- Structure Correspondence and Reduction 6.10

we present

Kirby & Taylor: Pin structures on low-dimensional manifolds 227

Theorem 6.11. There exists a homomorphism R: n˜ ˜ n˜˜2- (BO (2Â»). Given an

object, x E Q˜, let F denote the submanifold dual to W2 + wf. This manifold has a

map F ˜ BO(2) classifying the normal bundle. Use the above construction to put

a Pin- structure on F: R(x) is the bordism class of this Pin structure on F.

Variants of this map enter into the discussions below.

Corollary 6.12. If MFKr denotes the r-th bordism group of Freedman-Kirby,

then there exists a long exact sequence

nSpin i MF}r? R nSpin(B ) a nSpin

80(2) ---+˜'r-l --+ â€¢â€¢â€¢

â€¢ â€¢ â€¢ --+ ˜Â£r Â·\.r---+˜'r-2

-+

where R takes the Spin bordism class of the classifying map for the normal bundle

to F in M, and a takes the Spin structure we put on the total space of the associated

circle bundle. The V we always take is the empty set.

Remark 6.13. There are definitely non-trivial extensions in this sequence.

Remark 6.14. The Freedman-Kirby bordism theory is equivalent to the bordism

theory Spine, the theory of oriented manifolds with a specific reduction of W2 to

an integral cohomology class. This bordism theory has been computed, e.g. Stong

[Stong], and is determine˜d by Stiefel-Whitney numbers, Pontrjagin numbers, and

rational numbers formed from products of Pontrjagin numbers and powers of the

chosen integralization of W2.

Remark 6.15. There are versions of this sequence for the bordism theory studied

by Guillou-Marin and for our bordism theory. In both of these cases we replace

n Spin by the Pin- bordism groups nPin -. We also replace n;?:.;n(BSO (2Â») by the

bordism groups of O(2)-bundles over Pin- manifolds with some extra structure.

The bordism groups of O(2)-bundles over Pin-manifolds can be identified with

the homotopy groups of the Thom spectrum formed from BPin- X BO(2) using the

universal bundle over B pin - and the trivial bundle over BO(2). The associated

bordism groups are denoted n˜˜2- (B O (2Â»). In the Guillon-Marin case we define

BGM as the fibre of the map BPin- X B O (2) --+ K(Z/2Z, 1) where the map is the

sum of WI of the universal bundle over B pin - and WI of the universal bundle over

BO(2). In our case we let BE be the fibre of the map B pin - X B O (2) ˜ K(Z/2Z, 2)

where the map is the product of two I-dimensional cohomology classes: namely 0

WI of the universal bundle over B pin - and WI of the universal bundle over B O (2).

Over either BGM or BE we can pull back the universal bundle over BPin- plus the

trivial bundle over BO(2) and form the associated Thom spectrum. The homotopy

groups of these spectra fit into the analogous exact sequences for the bordism theory

studied by Guillou-Marin and by us.

Remark 6.16. All the bordism groups defined in Theorem 6.11, Corollary 6.12

and its two other versions are naturally modules over the Spin bordism ring, and

all the maps defined above are maps of n˜pin-modules.

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

Â§7. Geometric calculations of characteristic bordism.

In this section we will calculate the characteristic bordism introduced in the

last section up through dimension 4.

The first remark is that any manifold M of dimension less than or equal to 4

has a characteristic structure. Hence !-bordism is onto unoriented bordism through

dimension 4. We show next that

Theorem 7.1. The forgetful map

ni ˜ 0.

f2b ˜ n˜ ˜ Z/2Z and

is an isomorphism for r = 0,1, and 2. Hence

Proof: Since the forgetful map is onto, it is merely necessary to show that the

!-bordism groups are abstractly isomorphic to Z/2Z or o. We begin in dimension

0. The only connected manifold is the point and it has a unique characteristic

structure: F and V are empty. Hence n˜ is a quotient of Z. It is easy to find a -:(

characteristic structure on [0, 1] which has 2 times the oriented point as its boundary:

{lh

F is empty and V = {1/2}. Hence ˜ Z/2Z given by the number of points mod 2.

In dimensions at least 1, it is easy to add I-handles to show any object is

bordant to a connected one. Hence in dimension 1, the only objects we need to

consider are characteristic structures on 51. Here F is still empty, and V is an even

˜˜

number of points. The circle bounds B 2 , the 2-disk, and it is easy to extend V

˜

to a collection of arcs in B 2 and to extend the orientation on 8 1 - V. The Pin-

!i

structure on the circle either bounds a 2-disk, in which case extend it over B2, or

it does not, in which case take F to be a point in B 2 which misses the arcs and A

ni˜ 0.

extend the Pin- structure over B 2 - pt. Hence It

j

In dimension 2 we can assume that M is connected and that it bounds as

:j

an unoriented manifold. The goal is to prove that it bounds as a characteristic

>,'˜.l,Â·

structure. Note V is a disjoint union of circles, and F is a finite set of points with

F nV being empty. Since every surface has a Pin- structure, F is an even number .:˜

˜

of points. Let W be a collection of embedded arcs in M x [0,1] which miss M x 1

wi, ˜˜

and have boundary F. Since W is a dual to W2 + there is a Pin- structure

j

on M X [0,1]- W which extends across no component of W. This induces such a

structure on M x O. Since HI (M; Z/2Z) acts on such structures, it is easy to adjust :4

tJ

".i

to get a Pin- structure on M x [0,1] - W which extends across no component of W

}˜

and which is our original Pin- structure on M x O. Given V C M x 0 we can extend

.˜

to an embedding V x [0, 1) in M x [0,1]. The orientation on M - V extends to one

on M x [0,1] - V x [0,1]. Clearly this orientation extends across no component of .:1

.˜

V x [0,1], so this submanifold is dual to Wt. Hence we may assume our surface has

empty F with no loss of generality: i.e. M has a fixed Pin- structure.

Let Ek denote the total space of the non-trivial 2-disk bundle over the circle.

The boundary of Ek is 1(2, the Klein bottle and HI (K; Z/2Z) is spanned by a fibre

Kirby & Taylor: Pin structures on low-dimensional manifolds 229

circle,C/, and a choice of circle which maps non-trivially to the base, Ct. Consider

the Pin-structure on K2 whose quadratic enhancement satisfies q(C/) = 2 and

q(Ct ) = 1. This structure does not extend across EK so let F be the core circle in

EK. Let V be a fibre 2-disk. Orient the normal bundle to V n F in F any way one

likes. It is easy to check that this gives a characteristic structure on Ek extending

,. the one on K2 which does not bound as a Pin- manifold. By adding copies of this

structure on K2 to M, we can assume that M is a Pin- boundary, so let W 3 be a

Pin- boundary for M.

Inside W we find a dual to WI, say X2, which extends V in M. There is some

orientation on W - X which extends across no component of X and this structure

restricts to such a structure on M - V. Since M is connected, there are only two

such structures and both can be obtained from such a structure on W - X. Hence

our original characteristic structure is a characteristic boundary assuming nothing

more than that it was an unoriented boundary.â€¢

The results in dimensions 3 and 4 are more complicated. We begin with the

3-dimensional result.

Theorem 7.2. The homomorphism R of Theorem 6.11, followed by forgetting the

map to BO(2) yields an isomorphism

Proof: We first show that Ris onto and then that it is injective.

Let Ek denote the disk bundle with boundary the Klein bottle as in the-last

proof. The Pin- structure received by F in this structure is seen to be the Lie

group Pin- structure. There is a similar story for the torus, T 2 â€¢ There is a 2-disk

bundle over a circle, E}, and a Pin- structure on the torus which does not extend

across the disk bundle so that the core circle receives the Lie group Pin - structure.

Indeed, E} is just a double cover of Ek. IT we take two copies of K2 with its Pin-

structure and one copy of T 2 with its Pin- structure, the resulting disjoint union

2

bounds in nf in -. Let W 3 denote such a bordism. Let M 3 = JlE˜. II E˜ lL W 3

with the boundaries identified. Let F be the disjoint union of the three core circles,

and note F is a dual to W2 + wf since the complement has a Pin-structure which

does not extend across any of the cores. Let V be a dual to WI and arrange it to

meet F transversely. Indeed, with a little care one can arrange it so that V n F

consists of 2 points, one on each core circle in a Ef(. This is our characteristic

ni

in - .

structure on M. Our homomorphism applied to M is onto the generator of

It remains to show monicity. Let M be a characterized 3-manifold. By adding

I-handles, we may assume that M is connected. First we want to fix it so that

V n F is empty. In general, V n F is dual to W2WI +wr and, for a 3-manifold, this

Kirby & Taylor: Pin structures on low-dimensional manifolds

230

vanishes. Hence V n F consists of an even number of points. We explain how to

remove a pair of such points.

Pick two points, po and PI, in V n F. Each point in F has an oriented normal

bundle. The normal bundle to each point in V is also trivial and V is oriented,

so the normal bundle to each point in V is oriented. Attach a I-handle, H =

(BI X [0, 1]) X B2 so as to preserve the orientations at po and Pl. Let W 4 denote the

resulting bordism. Inside W 4 , we have embedded bordisms, Vl and Fl beginning

at V and F in M. Notice that at the "top" of the bordism, the "top" of VI and

the "top" of F 1 intersect in 2 fewer points. Moreover, the orientation of the normal

bundle of V n F in F clearly extends to an orientation of the normal bundle of

Vi n F I in Fl.

Since F 1 is a codimension 2 submanifold of W, it is dual to some 2-dimensional

cohomology class. Since H* (W, M; Z/2Z) is 0 except when * = 1 (in which case

it is Z/2Z), this class is determined by its restriction to H2 (M; Z/2Z). Hence

wi,

F I is dual to W2 + so choose a Pin - structure on W - F I which extends

across no component of Fl. This restricts to a similar structure on M, and since

HI (W; Z/2Z) --. HI (M; Z/2Z) is onto, we can adjust the Pin- structure until it

extends the given one on M - F.

The above argument does not quite work for Vi, but it is easy in this case to see

that W - Vi has an orientation extending the one on M - V. Any such orientation

can not extend over any components of VI. Hence we have a characteristic bordism

as required.

We may now assume that V n F is empty. Since F is a union of circles and

V n F = 0, F has a trivial normal bundle in M. If our homomorphism vanishes

on our element, F is a Pin- boundary, which, in this dimension, means that it is

a Spin boundary: i.e. F bounds Q2, an orientable Pin-manifold. Glue Q2 x B 2

to M x [0, 1] along F X B 2 C M x 1 to get a bordism X 4 â€¢ Since Q is orientable,

V x [0, 1] is still dual to WI, and it is not hard to extend the Pin-structure on

M - F to one on X - Q which extends across no component of Q. Since Q and

V x [0, 1] remain disjoint, the "top" of X is a new characteristic pair for which the

dual to W2 + wi is empty: i.e. the "top", say N3, has a Pin- structure. Since

nfin- = 0, N3 bounds a Pin- manifold, y4. Since M was connected, so is Nand

there is no obstruction to extending the dual to WI in N, say VI, to a dual to WI

in Y, say U, and extending the orientation on N - Vi to an orientation on Y - U

which extends across no component of U. The union of X 4 and y4 along N3 is a

characteristic bordism from M 3 to o.â€¢

The last goal of the section is to compute n˜. Since the group is non-zero,

we begin by describing the invariants which detect it. Given an element in n˜,

we get an associated surface F2 with a Pin - structure, and hence a quadratic

enhancement, q. We may also consider 1], the normal bundle to F in our original

4-manifold. We describe three homomorphisms. The first is {3: n˜ -+ Z/8Z which

Kirby & Taylor: Pin structures on low-dimensional manifolds 231

just takes the Brown invariant of the enhancement q. The second homomorphism

01 ˜ Z/4Z given by the element Q(Wl("'Â» E Z/4Z. The third homomorphism

is W:

w2:n1 ˜ Z/2Z given by (W2(1J), [F]) E Z/2Z. We leave it to the reader to check

is

that these three maps really are homomorphisms out of the bordisID group, n˜.

Theorem 7.3. The sum of the homomorphisms

f3 ED WEf) W2: n˜ ˜ Z/8Z EB Z/4Z EB Z/2Z

is an isomorphism.

Proof: First we prove the map is onto and then we prove it is 1-1. Recall from

Lemma 6.7 that a surface, M, with a Pin- structure and a 2-plane bundle, 7], can

be completed to a characteristic bordism element iff (WI (M) + WI (7]Â» U WI (7]) = O.

Notice that this equation is always satisfied since cupping with WI (M) and squaring

are the same. Hence we will only describe the surface with its Pin- structure and

the 2-plane bundle.

First note that Rp2 with the trivial 2-plane bundle generates the Z/8Z and

maps trivially to the Z/4Z and the Z/2Z.

The Hopf bundle over the 2-sphere maps trivially into the Z/8Z and the Z/4Z

since 52 is a Pin- boundary and \II vanishes whenever the 2-plane bundle has

trivial WI. However, 8 2 and the Hopf bundle maps non-trivially to the Z/2Z.

Let [(2 denote the Klein bottle, and fix a Pin- structure for which K2 is a

Pin- boundary. Let 1] be the 2-plane bundle coming from the line bundle with

WI being the class in HI (K2; Z/2Z) with non-zero square. Since K2 is a Pin-

boundary, (3(K2) = O. Since TJ comes from a line bundle, W2(TJ) = O. However,

Q(Wl(7]Â» is an element in Z/4Z of odd order and is hence a generator.

This shows that our map is onto. Before showing that our map is 1-1, we need

a lemma.

Lemma 7.4. There exists a 2-disk bundle B 2n over the punctured 8 1 x 52, 8 1 X

S2 - int B 3 , whose restriction to the boundary S2 has Euler class 2n, n E Z.

Proof: Start with the 2-disk bundle En over 8 2 with Euler number n and pull it

back over the product 8 2 xl. Now add a I-handle to 8 2 xl, forming 8 1 xS 2 -int B 3 ,

and extend the bundle B", over the I-handle so as to create a non-orientable bundle

B 2n â€¢ Then X(B 2n l s 2) = 2n.â€¢

Suppose M 4, V 3, F2, TJ2 is a representative of an element of n˜ and that

(3(F 2) = 0, 'I1(W] (1]Â» = 0, and W2(7J) = 0. We need to construct a !-bordism to 0.

Since we may assume that F, M and V are connected, there is a connected

I-manifold, an 8 1 , which is Poincare "dual to Wl(7]); then the normal vector to

8 1 in F makes an even number of full twists jp. the Pin- structure on F as 8 1

is traversed. It follows that we can form a !-bordism by adding to F a B 2 x B l

Kirby & Taylor: Pin structures on low-dimensional manifolds

232

where 51 x B l is attached to the dual 8 1 to wl(71) and its normal Bl bundle.

Clearly the Pin- structure on F extends across the bordism. Since the dual to 8 1

has self-intersection zero in F, 71 restricted to 51 is orientable, so 1] extends over

B2 x Bl â€¢

Since W2(71) = 0, it follows that X(7])[F] = 2n for some n E Z. By Lemma 7.4

there is a bundle B_ 2n over a punctured SI x 52 with x(B- 2n l s 2) = -2n. We form

a 5-dimensional bordism to the boundary connected sum, i.e. in M 4 X 1 C M 4 xl,

choose a 4-ball of the form B 2 x B 2 where B2 x 0 C p2 - (V n F) and p x B 2

is a normal plane of 71 over p, and identify B2 X B2 with B-2nls˜ where 5˜ is a

hemisphere of 52.

The new boundary to our !-bordism, which we shall denote (M, V, F, 1]) now

has a trivial normal bundle 71.

Since I3(F2) = 0, F Pin- bounds a 3-manifold N 3 , so we add N 3 x B 2 to

M x 1 along the normal bundle TJ to F, F X B 2 , where it does not matter how

we trivialize 71. The Pin- structure on M - F extends over the complement of N

(using the Pin- Correspondence Theorem, 6.9, and the Pin- structure on N), so

the new boundary to our !-bordism consists of a Pin- manifold M with empty F2.

Since'4-dimensional Pin- bordism, nf in -, is zero, we can complete our !-bordism

by gluing on to M x 1 a 5-dimensional Pin - manifold. â€¢

Remark. It is worth comparing this argument with the argument in [F-I(J showing

that if (M 4 , P2) is a characteristic pair with M 4 and F 2 orientable and with

sign(M 4 ) = 0 and F.F = 0, then (M, F) is characteristically bordant to zero.

The arguments would have been formally identical if we had also assumed that the

Spin structure on F, obtained from the Pin- Correspondence Theorem, bounded

ni

in 2-dimensional Spin bordism, pin = Z/2Z (corresponding to f3(F) = 0 above).

However, it is possible to show that n˜har = Z EB Z without the extra assumption

on F, and this ZJ2Z improvement leads to Rochlin's Theorem (see (F-K], [[{i}, ...).

Further Remark. The image of the Guillou-Marin bordism in this theory can be

determined as follows. The group is Z E9 Z generated by (54, RP2) and (C p2 , S2 ).

Both f3 and 'Â¥ vanish on (CP2, S2), but W2 is non-zero. On (S4, RP2), W2 evaluates

o(the nonnal bundle comes from a line bundle): f3 is either 1 or -1 depending on

which embedding one chooses. Moreover, '11 is either 1 or -1 (the same sign as (3).

Â§8. New knot invariants.

The goal here is to describe some generalizations of the usual Ad invariant of

a knot (or some links) due to Robertello, [R].

We fix the following data. We have a 3-manifold M 3 with a fixed Spin structure

and a link L: JJ-51 -+ M 3 â€¢ Since M is Spin, w2(M) = 0 and we require that [L] E

I

HI (M; Z/2Z) is also 0, hence dual to w2(M). We next require a characterization of

Â·"'''40,.â€¢

˜. Â·˜.7Â·'''' - -.,. ......., ....... ,˜.,.

Kirby & Taylor: Pin structures on low-dimensional manifolds 233

the pair, (M, L): i.e. a Spin structure on M - L which extends across no component

of L. We call such a characterization even iff the Pin- structure induced on each

component of L by Lemma 6.2 is the structure which bounds. We say the link is

even iff it has an even characterization.

One way to check if a link is even is the following. Each component of L has a

normal bundle, and the even framing of this normal bundle picks out a mod 2 lon-

gitude on the peripheral torus. The link is even iff the sum of these even longitudes

is 0 in H 1 (M - L; Z/2Z)

a are even: the Hopf link in S3 is an

Remark. Not all links which represent

example where any structure which extends across no component of L induces the

Lie group Spin structure on the two circles. We shall see later that a necessary and

sufficient condition for a link in S3 to be even is that each component of the link

should link the other components evenly. This is Robertello's condition, [RJ.

Definition. A link, L, in M 3 with a fixed Spin structure on M and a fixed Spin

structure on M - L which extends across no component of L and induces the

bounding Pin- structure on each component of L is called a characterized link.

Given a characterized link, (M, L), we define a class i E HI (M - L; Z/2Z): 1

is the class which acts on the fixed Spin structure on M - L to get the one which is

the restriction of the one on M. The class i is defined by the characterization and

conversely a characterization is defined by a choice of class i E HI (M - L; Z/2Z) so

that, under the coboundary map, the image of i in H2 (M, M - L; Z/2Z) hits each

generator. (Recall that by the Thorn isomorphism theorem, H2 (M, M - L; Z/2Z)

is a sum of Z/2Z's, one for each component of L.)

Let E be the total space of an open disk bundle for the normal bundle of L, and

let S be the total space of the corresponding sphere bundle. Note S is a disjoint

union of a peripheral torus for each component of L. The class '1 is dual to an

embedded surface F C M - E and 8F n S is a longitude in the peripheral torus

of each component of L. Let l denote this set of longitudes. We will call.e a set of

even longitudes. We will call F a spanning surface for the characterized link.

The set of even longitudes is not well-defined from just the characterized link. It

is clear that two surfaces dual to the same / must induce the same mod 2 longitudes.

But if we act on one component of L by an even integer, we can find a new sutface

dual to 1 which has the same longitudes on the other components and the new

longitude on our given component differs from the old one via action by this even

integer. Hence the characteristic structure only picks out the mod 2 longitudes and

any set of integral classes which are longitudes and which reduce correctly mod 2

can be a set of even longitudes. Moreover, any set of even longitudes is induced by

an embedded surface.

Since M is oriented, the normal bundle to any embedded surface, F, is isomor-

phic to the determinant bundle associated to the tangent bundle of F. The total

.

. ..."..

Kirby & Taylor: Pin structures on low-dimensional manifolds

234

space of the determinant bundle to the tangent bundle is naturally oriented. The

total space to the normal bundle to F is M is oriented by the orientation on M.

Choose the isomorphism between the normal bundle to F in M and the determi-

nant bundle to the tangent bundle of F so that, under the induced diffeomorphism

between the total spaces, the two orientations agree. Under these identifications,

Corollary 1.15 picks out a Pin- structure on F from the Spin structure on M. We

apply this to an F which is a spanning surface for our link. Of course we could apply

the same result but use the Spin structure on M - L. It is not hard to check that

the two structures on F differ under the action of wl(F) since this is the restriction

of / to F. Hence it is not too crucial which structure we use but to fix things we

use the structure on M.

We can restrict this structure on F to a component of L. If we put the Spin

structure on F coming from that on M - L it is easy to see that we get the bound-

ing Pin- structure on each component of L. Hence this also holds for the Pin-

structure on F coming from the one on M. Hence, a spanning surface for a char-

acterized link has an induced Pin- structure which extends to the corresponding

closed surface uniquely.

Our link invariant is a mod 8 integer which depends on the characterized link

and the set of even longitudes.

Definition 8.1. Given a characterized link, (M, L), and a set of even longitudes,

f, pick a spanning surface F for L which induces the given set of longitudes. Then

define

f3( L, f, M) = (3(F)

where F is F with a disk added to each component of L; the Pin- structure is

extended over each disk; and f3 is the usual Brown invariant applied to a closed

surface with a Pin-structure.

Remarks.

i) Notice that unlike Robertello's invariant, our invariant does not require that

the link be oriented.

ii) It follows from the proof of Theorem 4.3 that a knot is even iff it is mod 2

trivial.

iii) IT each component of L represents 0 in HI (M; Z/2Z) then the mod 2 linking

number of a component of L with the rest of the link is defined. If F is

an embedded surface in M with boundary L, the longitude picked out for a

component of L is even iff the mod 2 linking number of that component of L

with the rest of the link is O.

iv) IT M is an oriented Z/2Z homology 3 sphere, then it has a unique Spin structure

and there is a unique ˜ay to characterize an even link L.

v) Let M be an integral homology 3 sphere containing a link L. Orient each

component of the link. Let fi be the linking number of the ith component of L

Kirby & Taylor: Pin structures on low-dimensional manifolds 235

with the rest of the link. Each component of L has a preferred longitude, the

one with self-linking 0, so li also denotes a longitude. The link L is even iff

each ti is even. Robertello's Arf invariant is equal to f3( L, -l, M), where the

Spin structure and characterization are unique and t is the set of longitudes

obtained by using -ti on each component. Notice that ti depends on how the

link is oriented.

It is not yet clear that our invariant really only depends on the characterizations

and the even longitudes.

Theorem 8.2. Let L be a link in a 3-manifold M. Suppose M has a Spin structure

and that L is characterized: Let t be a collection ofeven longitudes. Then (1(L, l, M)

is well-defined. Let W 4 be an oriented bordism between M 1 and M 2 â€¢ Let Li C Mi,

i = 1,2 be characterized links. Let FeW be a properly embedded surface with

F n Mi = Li. Suppose W - F bas a Spin structure which extends across no

component of F and which gives a Spin bordism between the two structures on

Mi - L j , i = 1,2, given by the characterizations.

The normal bundle to F in W has a section over every non-closed component

of F so pick one. This choice selects a longitude for each component of each link.

Suppose the longitudes picked out for each Li, say ti, are even. The surface F

receives a Pin- structure by Lemma 6.2. With this structure, each component of

8F bounds and hence F has a f3 invariant. H we orient W so that M 1 receives the

reverse Spin structure then tbe following. formula bolds.

aSpin

Proof: We begin by discussing some constructions and results involving 3-

2

manifold N and a spanning surface, V for a characterized link, L. To begin, given

e: V 2 C N 3 , define V C N x [0,1] as the image of e x I, where I: V --+ [0,1/2] is any

map with 1- 1 (0) = aVe If N has a Spin structure, N x [0, 1] receives one. The class

represented by [V, L] in H 2 (N x [0, 1], N x 0 JL N x 1; Z/2Z) ˜ HI (N x 0; Z/2Z)

is the same as that represented by [L] in HI (N x 0; Z/2Z). Hence it represents

o. Since w2(N X [0,1]) is also trivial, there is a Spin structure on N x [0, 1] - V

which does not extend across any component of V. Such structures are acted on

simply transitively by HI (N; Z/2Z), so it is easy t˜ construct a unique such Spin

structure which restricts to the initial one on N X 1.

We proceed to identify the Spin structure induced on N X 0 - L. Let X =

V x [0, 1] and embed two copies of V in the boundary so that 8X = V U V where

the union is along 8V thought of as 8V x 1/2. First observe that we can em-

bed X in N x [0, 1] so that ax is V c N x 0 union V x 1 = V. Since X

has codimension 1, the Poincare dual to W is a I-dimensional cohomology class

Kirby & Taylor: Pin structures on low-dimensional manifolds

236

x E HI (N X [0,1] - V; Z/2Z). Suppose we take the Spin structure on N x [0,1]

and restrict it to N x [0,1] - V and then act on it by x. This is a Spin structure on

N x [0,1]- V which extends across no component of V and which is the original

one on N x 1. On N x 0- L it can be described as the one obtained by taking the

given Spin structure on N x 0, restricting it, and then acting on it by the restriction

Â°and

of x. But the restriction of x is just the Poincare dual of FeN x so it is

the Spin structure which characterizes the link. By Lemma 6.2, there is a preferred

Pin- structure on V, which is easily checked to be the same as the one we put on

it in Â§4. The above Spin structure on N x [0,1] - V will be called the standard

characterization of the pair (N x [0, 1], V).

With this general discussion behind us, let us turn to the situation described

in the second part of the theorem. Recall W 4 is an oriented bordism between M I

and M 2 ; L 1 C M I and L 2 C M 2 are characterized links; F2 C W be a properly

embedded surface with FnMi = Li; and W -F has a Spin structure which extends

across no component of F and which gives a Spin bordism between the structures

on Mi - Li. Define sets of even longitudes f,j as in the statement of the theorem.

Let Fi C Mi be a spanning surface for L i â€¢ Inside W = M 1 X [-1, 0] U W U M 2 X

[0,1] embed F = F U F U F , where F is defined with function I: F 1 ˜ [-1/2,0]

I

I 2

and still 1- (0) = 8Ft â€¢ There is a Spin structure on W - F which extends across

1

x

no component of F. It is just the union of the standard characterization of M 1

[-1,0], F , the given Spin structure on W - F and the standard characterization

1

of M 2 x [0,1],F2 â€¢

By Lemma 6.2 again, there is a preferred Pin- structure on F, which agrees

with the usual ones on F I and F 2 â€¢ In particular, F also receives a Pin- structure

which only depends on W, not on the choice of F I or F2 â€¢ However, from F 1 and

F2 , we see that the Pin- structure induced on each component of each link is the

bounding one. Moreover, f3(p) = f3(F) + f3(F2 ) - f3(F1 ).

By construction, F."F is 0, so 6.4 says that

where the J.l invariants arise because 6.4 only applies to closed manifolds.

Apply this to the case W = M x [0, 1], F = L x [0, 1] embedded as a product.

Since we may use different spanning surfaces at the top and bottom, this shows f3 is

well-defined. The formula in the theorem now follows from the formula immediately

above.â€¢

The next thing we wish to discuss is how our invariant depends on the longi-

tudes. Given two different sets of even longitudes, i and i', for a characterized link

L C M 3 , there is a set of integers, one for each component of L defined as follows.

The integer for the ith component acts on the longitude for f, t6 give the longitude

for i'. Since both these longitudes are even, so is this integer.

Kirby & Taylor: Pin structures on low-dimensional manifolds 237

Theorem 8.3. Let L C M 3 be a characterized link with two sets of even longitudes

R. and it. Let 2r be the sum of the integers which act on the longitudes i to give

the longitudes i'. Then

+r (mod 8) .

f3(L,i',M) = f3(L,i,M)

Proof: Given F1 , a spanning surface for the longitude i, we can construct a spanning

surface for i' as follows. Take a neighborhood of the peripheral torus, which will

have the form W = T2 X [0,1]. Inside W embed a surface V which intersects

T2 X 0 in the longitude R., which intersects T2 x 1 in the longitude i', which has no

boundary in the interior of W; and which induces the zero map H 2 (V, av; Z/2Z) --+

H2 (W, aw; Z/2Z). The Spin structure on M restricts to one on W which is easily

described: it is the stabilization of one on T2 and this can be described as the

Â°on Â°on

one which has enhancement the longitude and the meridian. Since the

Pin- structure induced from Corollary 1.15 is local, we see that F 2 = V U F1 has

invariant the invariant for F1 plus the invariant for V. We further see that the

invariant for V only depends on the surface and the Spin structure in W. But

these are independent of the link and so we can calculate the difference of the f3's

using the unknot.

Furthermore, we see that the effect of successive changes is additive, so we only

need to see how to go from the 0 longitude to the 2 longitude, and the 2 longitude

is given by the Mobius band, which inherits a Pin- structure. This Pin- structure

extends uniquely to one on Rp2 and this Rp2 has (3 invariant +1.

Remark. Even in the case of links in S3, the longitudes used enter into the answer.

It is just in this case that there is a unique set of longitudes given by using an

orientable spanning surface.

Unfortunately, in general there is no natural choice of longitudes so it seems

simplest to incorporate them into the definition. The drawback comes in discussing

notions like link concordance. In order to assert that our invariant is a link concor-

dance invariant, we need to describe to what extent a link concordance allows us

to transport our structure for one link to another. Recall that a link concordance

between L o C M and L l C M is an embedding of (Jl.. SI) x [0,1] C M x [0,1] with

is (1L SI) X i being Li for i = 0,1. Suppose L o is an even link with f o a set of even

longitudes. There is a unique way to extend this framing of the normal bundle to

L o in M to a framing of the normal bundle of (Jl.. SI) x [0,1] in M x [0,1]. Hence

the concordance picks out a set of longitudes for L 1 which we will denote by R.I.

There is a unique way to extend a characterization of L o to a Spin structure on

M x [0,1] - (1L SI) x [0,1] and hence to M - L 1 â€¢

Corollary 8.4. Let L o and L l be concordant links in M. Suppose L o is charac-

terized and that i o is a set of even framings. Then the transport of framings and

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

Spin structures described above gives a cbaracterization of L 1 and Â£1 is a set of

even framings. Furtbermore P( L o,f o, M) = f3( L 1 , Â£1 , M).

Proof: The proof follows immediately from Theorem 8.2 and the fact that (ll

8 1 ) X [0, 1], when capped off with disks, is a union of 8 2 's and so has f3 invariant

o.â€¢

We do know one scheme to remove the longitudes which works in many cases.

Suppose that each component of the link represents a torsion class in HI (M; Z).

Each component has a self-linking and by Lemma 4.1 the framings, hence longitudes

are in one to one correspondence with rational numbers whose equivalence class in

Q/Z is the self-linking number. There is a unique such number, say qi for the ith

component, so that qi represents an even framing and 0 ˜ qi < 2. We say that this

is the minimal even longitude. To calculate linking numbers it is necessary to orient

the two elements one wants to link, but the answer for self-linking is independent

of orientation.

Definition 8.5. Let L be a link in M so that each component of L represents a

torsion class in HI (M; Z). Suppose L is characterized. Define

= f3(L,i,M)

P(L,M)

where /, is the set of even longitudes such that each one is minimal.

/3 is a concordance invariant.

Remark. It is not hard to check that

As we remarked above, (3 and /3 (if it is defined) do not depend on the orien-

tation of the link. H we reverse the orientation of M, and also reverse the Spin

structure on M and on M - L, it is not hard to check that the new Pin- structure

on F is the old one acted on by WI (F) so the new invariant is minus the old one.

The .remaining point to ponder is the dependence on the two Spin structures.

To do this properly would require a relative version of the (3 function 4.8. It does

not seem worth the trouble.

Remark. We leave it to the reader to work out the details of starting 'with a

characteristic structure on M 3 with the link as a dual to W2 + w˜ (i.e. represents 0

in HI (M; Z/2Z)).

Â§9. Topological versions.

There is a topological version of this entire theory. Just as Spin(n) is the

double cover of SO(n) and PinÂ±(n) are the double covers of O(n), we can consider

the double covers of STop(n) and Top(n). We get a group TopSpin(n) and two

groups TopPinÂ±(n). A Top(n) bundle with a TopPinÂ±(n) structure and an O(n)

structure is equivalent to a PinÂ±(n) bundle.

Kirby & Taylor: Pin structures on low-dimensional manifolds 239

Any manifold of dimension $ 3 has a unique smooth structure, so there is no

difference between the smooth and the toplogical theory in dimensions 3 and less.

The 3-dimensional bordism groups might be different because the bounding objects

are 4-dimensional, but we shall see that even in bordism there is no difference.

We turn to dimension 4. First recall that the triangulation obstruction (strictly

speaking, the stable triangulation obstruction) is a 4-dimensional cohomology class

so evaluation gives a homomorphism, which we will denote ˜, from any topological

bordism group to Z/2Z. Since every 3-manifold has a unique smooth structure,

the triangulation obstruction is also defined for 4-manifolds with boundary. Every

connected 4-manifold M 4 has a smooth structure on M - pt, and any two such

structures extend to a smoothing of M x [0,1] - pt x [0,1].

Some of our constructions require us to study submanifolds of M. In particular,

the definition of characteristic requires a submanifold dual to WI and a submanifold

dual to W2 + w˜. We require that these submanifolds be locally-fiat and hence,

by [Q], these submanifolds have normal vector bundles. Of course we continue to

require that they intersect transversely. Hence we can smooth a neighborhood of

these submanifolds. The complement of these smooth neighborhoods, say U, is a

manifold with boundary, which may not be smooth. If we remove a point from the

interior of each component of U, we can smooth the result. With this trick, it is not

difficult to construct topological versions of all our "descent of structure" theorems.

In particular, the [nw˜], [nWl] and R maps we defined into low-dimensional PinÂ±

bordism all factor through the corresponding topological bordism theories.

Theorem 9.1. Let Smooth-bordism. denoten;pin, n;in:l:, n˜, or the Freedman-

Kirby or Guillou-Marin bordism theories. Let Top - bordism* denote the topolog-

ical version. The natural map

Smooth - bordism3 Top - bordism3

-+

is an isomorphism.

It

Smooth - bordism4 Top - bordism4-+Z/2Z 0

-+ -+

is exact.

Proof: The E s manifold, [F], is a Spin manifold with non-trivial triangulation

obstruction. Suppose M 3 is a 3-manifold with one of our structures which is a

topological boundary. Let W 4 be a boundary with the necessary structure. Smooth

neighborhoods of any submanifolds that are part of the structure. This gives a new

4-manifold with boundary U 4 â€¢ If the triangulation obstruction for a component of

U is non-zero, we may form the connected sum with the E s manifold. Hence we

may assume that U has vanishing triangulation obstruction. By [L-S] we can add

some 8 2 x 8 2 's to U and actually smooth it. The manifold W can now be smoothed

Kirby & Taylor: Pin structures on low-dimensional manifolds

240

so that all submanifolds that are part of the structure are smooth. Hence M 3 is

already a smooth boundary.

The E s manifold has any of our structures, so the map Top-bordism4 ˜ Z/2Z

given by the triangulation obstruction is onto.

Suppose that it vanishes. We can smooth neighborhoods of any submanifolds,

so let U be the complement. Each component of U has a triangulation obstruction

and the sum of all of them is O. We can add Es's and - E s 's so that each component

has vanishing triangulation obstruction and the new manifold is bordant to the old.

Now we can add some S2 x 5 2 's to each component of U to get a smooth manifold

with smooth submanifolds bordant to our original one.â€¢

Theorem 9.2. The topological bordism groups have the following values. OTopSpin

˜ z; nropPin- ˜ Z/2Z; OrOpPin+ ˜ Z/8Z Ea Z/2Z; and n[op-! ˜ Z/8Z $ Z/4Z EB

Z/2Z e1 Z/2Z. The triangulation obstruction map is split in all cases except the

Spin case: the smooth to topological forgetful map is monic in all cases except

the TopPin+ case where it has kernel Z/2Z. The triangulation obstruction map

is split onto for the topological versions of the Freedman-Kirby and Guil1ou-Marin

theories and the smooth versions inject.

Proof: The TopPin- case is easy from the exact sequence above. The TopSpin

case is well-known but also easy. The Es manifold has non-trivial triangulation

obstruction and twice it has index 16 and hence generates nfpin.

There is a [nw˜] homomorphism from nropPin+ to nfin- ˜ Z/8Z which is

onto. Consider the manifold M = E S #S2 X RP2. The oriented double cover

of M is Spin and has index 16, hence is bordant to a generator of the smooth

Spin bordism group. It is not hard to see that the total space of the non-trivial

line bundle over M has a Pin+ structure, so the !{ummer surface is a TopPin+

boundary. Hence there is a Z/2Z in the kernel of the forgetful map and the [nw˜]

map shows that this is all of the kernel. Furthermore, E s represents an element of

order 2 with non-trivial triangulation obstruction.

Or

The homomorphisms used to compute n˜ factor through nJop-!, so -! ˜

op

n˜ E9 Z/2Z.

Likewise, the homomorphisms we use to compute smooth Freedman-Kirby or

Guillou-Marin bordism factor through the topological versions.â€¢

Corollary 9.3. Let M 4 be an oriented topological 4-manifold, and suppose we

have a characteristic structure on the pair (M, F). The following formula holds:

2Â· (J(F) = F.F - sign(M) +8Â· K(M) (mod 16)

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

on (M, F) via the topological version of the Pin-Structure Correspondence, 6.2.

Kirby & Taylor: Pin structures on low-dimensional manifolds 241

Proof: Generators for the topological Guillou-Marin group consist of the smooth

generators, for which the formula holds, and the E g manifold, for which the formula

is easily checked.â€¢

Remark. The above formula shows that the generator of H 2 ( ; Z) of Freedman's

Chern manifold, [F, p. 378], is not the image of a locally-fiat embedded 8 2 â€¢

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Department of Mathematics

Department of Mathematics

University of California, Berkeley University of Notre Dame

Berkeley, California 94720 Notre Dame, Indiana 46556

UW 8102787 X

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