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Spin structure, so the result is true in dimension 1. The case in which the line
: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,В·В·В·, 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

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.вЂў
The other result relates double covers and the '11 2 вЂў 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:PinВ±(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') ˜ PinВ±(oE).
to be det ToE- This gives a natural function '11 2

Theorem 2.9. The PinВ± 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
˜
PinВ±(M) P=f(E')
1 1L
L1 2

'11 2
PinВ±(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
PinВ±(M) ˜ PinВ±(8E)
1 1ВЈ4
La
PinВ±(Mo) ˜ PinВ±(8Eo)
commutes, where ВЈ3 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.вЂў
В§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) +2В· x.y for
all x, y E HI (F; Z/2Z) (here 2В· 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 вЂў 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 В· 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, В·;'o.o˜ o.o
=
say W, and an embeddIng W c F so that oW L 3 вЂў 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) + 2В· x-yo вЂў
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 в‚¬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 В·вЂў.'. :.;l
...
',В·
.....

˜
Euler characteristic of our surface, the second result follows.
Hence, if f3(F) = 0, the manifold is an unoriented boundary, say of W вЂў 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.вЂў

= В±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
В§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) +2В· A(a, x).
Given a E V, define qa by qa(X)

= f3(q) + 2В· 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 в‚¬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.вЂў
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.вЂў
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 + вЂў
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]' вЂў
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 вЂў 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 вЂў
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).вЂў
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 PinВ±
structure on F, we can induce a Spin structure on F.
Lemma 3.13. The orientation double cover map induces homomorphisms
nPinВ± nSpin
˜ 01ВЈ2
11ВЈ2

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
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.вЂў
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).

В§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 В· 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.вЂў
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

,(x+y) ,(x)+,(y)+f(x,y)
=
= r 2 вЂў ,(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 вЂў (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.вЂў
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 ВЈ(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
В·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
В§6, so we carry out the discussion in В§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 вЂў 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 вЂў
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).вЂў
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) В·
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.вЂў
Finally, we have a function

{3: H2 (M; Z/2Z)
(4.8) Z/8Z В·
-+

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) В·
(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.вЂў
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 В§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).вЂў
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
В§5. Geometric calculations of n˜˜nВ± вЂў

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 PinВ± manifold M 3 , we will try to find a PinВ± bordism
to an orientable manifold which then PinВ± 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 PinВ± 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 PinВ± structure}
extending the one on (aY) X B2. We can form M x [0,1] UE by gluing (ay2) X B 2 .I ;.˜ В·
to C x B2 X 1 where C x B2 is the trivialized disk bundle to C above. Clearly the
PinВ± structure extends across the bordism, and the "top" is a new PinВ± manifold
"..
вЂў.˜
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 PinВ± 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 PinВ± 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.вЂў
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 PinВ± 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 вЂў 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.вЂў
Since nfpin ˜ Z generated by the Kummer surface this lemma calculates the
n: in nfinВ± and our next goal is to try to produce a PinВ± bordism
pin
image of
from any PinВ± 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 PinВ± 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 PinВ± structure on M to put a PinT structure on F and it is not
hard to check that we get a homomorphism nfinВ± -+ nfin T вЂў If F bounds in this
structure, one can easily see a PinВ± bordism to an new 4-manifold M I in which
the dual to WI has trivial normal bundle. This puts a PinВ± structure on VI. By
orienting Vi we get a Spin manifold and if VI bounds in this Spin structure, M 1
PinВ± 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 вЂў 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 'Вў'(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.вЂў
p'

В§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˜ вЂў
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 вЂў 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.вЂў

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 В· (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 вЂў
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.вЂў
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(",) (В±1 or 0 depending on X(TJВ»):
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.вЂў
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)В·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) В· 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) вЂў 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 вЂў 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 в‚¬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 вЂў
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-В·
,(cos(t)
1somorph'
, cos(t)' vve ue
phisms together to get an isomorphism between TJ EB det(TaE) and det(p) в‚¬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 в‚¬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.
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