ñòð. 7 |

PROPOSITION 3. Let p : MS -to N3 be a finite covering map of hyperbolic mani-

folds, such that ?rIM3 <J 1flN3. Then the order of the covering transformation group equals

V(M3)/v(N3).

Given the non-closed form of the hyperbolic volume function it is hard to be

more precise. However for a cusped orientable manifold analogy with Theorem

1 suggests the existence of some constant A, related to the minimal volume

for a suitable class of orbifolds, such that

(Here in contrast to the 2-dimensional case we include the orientation revers-

ing isometries.) Given that v( M4) = 2.0298 ..., the following calculation shows

that A> 3.9 ....

EXAMPLE: The full isometry group I(M4 ) is a dihedral group of order 8.

Proof: An isometry induces an automorphism of 1ft. Conversely such an au-

tomorphism can be realised as a homotopy equivalence, well-defined up to

inner automorphisms. By Mostow's rigidity theorem this homotopy equiva-

lence can be deformed to an isometry, showing that I(M4 ) =:! Out (1rlM). In [12]

W. Magnus proved directly that the outer automorphism group is dihedral.

This rather informal discussion suggests the following tentative definition:

the finite group G is a 3-dimensional Hurwitz group if and only if there is an

epimorphism <p : r 4 ""'* G.

p˜

EXAMPLE: If p is an odd prime, 5, then reduction modulo p defines

homomorphisms

<p â€¢ r4 ""'* {PSL2(Fp ), p == l(mod 3)

p ˜ l(mod 3).

PSL2(F p 3),

.

n,

t ˜ 1,

In both cases <p is an epimorphism, since if q = pt is a prime power with

GDa. (!e

a.nd q =1= 2t (t > 1) or 32 , then PSL 2 (F f ) is genera.ted by nd

where egenerates F g over Fp , see [6], Theorem 2.8.4.

Elementary properties of 3-dimensional Hurwitz groups: i) and ii) are iden-

tical with the corresponding properties in dimension 2. However, since r 4

maps onto the infinite cyclic group Z, property iii) must be modified. For

example no abelian group of rank greater than or equal to 2 can be Hurwitz,

and the positive answer to the Smith conjecture imposes severe restraints on

170 Thomas: Finite groups of hyperbolic isometries

the solvable groups which can arise. The divisibility condition iv) must also

be weakened: for any prime number i. ˜ 5, the example above shows the ex-

i.stence of Hurwitz groups G with l f IGI. Thus PSL 2 (F 7 ) has order 23 .3.7 and

PSL 2(':19) has order 22 .3 2 .5.19. The question of which sporadic simple groups

(if any) can be homomorphic images of r 4 would seem to be interesting.

We now turn to the problem of realising an arbitrary finite group G first

as a group of hyperbolic isometries, and then as the full group I(M3) for

some manifold M3. We are able to prove the weak analogue of Theorem

2 - this corresponds to the existence of a Fuchsian group with preassigned

signature. However as mentioned previously there are problems in extending

the notion of "maximal" from Fuchsian to I{leinian groups. We also outline

two proofs: the more geometric applies to a wider class of groups, but the

algebraic includes an estimate of volume and hence of the index [I(M3) : An].

THEOREM 4. The alternating group An can be realised as a group of hyperbolic

isometries for infinitely many values ofn.

(˜

First proof (n = odd): Consider the Pretzel knot of type illustrated

below.

d

(1m-1m)

-------

(m-2˜m˜1m)

(2 m-2 m)

â€¢

â€¢ -------

˜

)

(m-Smm-2l8 (m-2mm

(2mSlR

(lm2lR

[I (˜

(

â€¢

â€¢ -------

(mrt-12) (mm-l m-2)

-------

(mm-lI)

Thomas: Finite groups of hyperbolic isometrics

Here m ˜ 5 and m =2n - 1. With each crossing associate the rnarked 3-cycle,

and use it to map a generator of 1rK into A 2n - 1â€¢ This gives a honlomorphism of

1(]( onto A2n - h and it remains to show that the complement of K is hyperbolic.

It is easy to see that K is not a torus knot, for example by showing that the

centre of "IrK is trivial. On the other hand 8 3 \1< has a branched double cover,

which is Seifert fibred. The only incompressible tori bound neighbourhoods of

exceptional fibres over points of 8 2 , and on applying Z/2 their images in 8 3 \]<

turn out to be singular. Hence K cannot be a satellite knot, and Thurston's

theorem provides a hyperbolic structure. (I am grateful to W.B.R. Lickorish

for explaining some of these properties of Pretzel knots to me.)

There is a discussion of the case m = 5, i.e. of the Pretzel knot (3,3,3),

described as a modest example in [18] page 621. As in the case of the figure

of eight knot Riley constructs an explicit excellent representation, this time

over Z(1]), where", is a root of 3 + (h(yÂ»2 = 0, h(y) =y - 3y2 + 2y3 - 4y4 + y5 _ y6.

Of course by Reid's theorem the index of the image of 'irK in PSL 2 (Z(T/Â» must

be infinite.

Second proof (n =16t + 1, 16t + 6, 16t + 8): According to W. Magnus [12, p.153]

Drillick proves in his thesis, that for the stated values of n, An is a homomor-

phic image of PSL2(Z(iÂ». If L is the Whitehead link, illustrated below, then

there is an excellent representation of 'lrL in PSL 2 (Z(iÂ», such that the image

has index 12.

Indeed S3 \ L corresponds to the manifold M 125 in Weeks' list, and has volume

equal to 3.6638 .... The fundamental group f 125 must project onto An, since

the order n! is always greater than 12. The volume can also be estimated

as follows - we include the calculation in view of its potential application to

other discrete groups r.

Let 0 be the ring of integers in the imaginary quadratic field k. Write Die

for the discriminant of k over Q and (Ie for the Riemann zeta function of

k. Then the volume of the quotient orbifold defined by r = PSL 2 (O) equals

172 Thomas: Finite groups of hyperbolic isometries

IDI;13 / 2(1;{2) I 41r2 â€¢ With ID1 3 / 2 =

Ie = Q(i) the discriminant equals 4, so that 8.

The value of (1:(2) is not known in closed form, but

II1/(1- (-1 )p-2) =0.9159 ...

(k(2) = (Q(2)L(2), where Â£(2) =

, P

Since (Q(2) = ˜2, the orbifold volume equals 8 . ˜2 . 4;2 . (0.9159) = 0.3053 ....

The subgroup r125 has index 12 in PSL2(l(iÂ», so the volume of S3 \ L equals

3.6638 ... agreeing with the tabulated value.

Remark: C. Adams has shown recently that the threee smallest possible limit

volumes for hyperbolic 3-orbifolds are 0.3053 ..., 0.4444 ... and 0.4579 ..., see Ab-

stracts presented to the American Math. Soc., October 1989, no. 851-57-15.

Note that M 125 possesses 8 symmetries, and 3.6638 ... /8 = 0.4579 .... Confining

our attention to manifolds, V(M125 )/2 equal˜ the volume of the non-orientable

manifold Ml' triangulated by 2 ideal tetrahedra, and having 2 symmetries.

The considerations suggest that An(n == 1, 6 or 8( mod 16Â» is a subgroup of very

small index in the full isometry group o( the covering manifold M of M 125

given by the second proof.

4. AN EXTENSION OF THEOREM 4 TO CLOSED MANI-

FOLDS

If M is a hyperbolic manifold with one or more cusps, then M can be closed

by means of Dehn surgeries, each of which is associated with a pair (r, s) of

coprime integers. We choose a basis m,l for 'KiT = HIT, where T is a toral

cross-section of the cusp, cut this off, and glue in a solid torus D2 x S1 in such

a way as to kill the homotopy class l'm r â€¢ Then it is known that for almost

all pairs (r, s) the resulting manifold with one fewer cusps is still hyperbolic.

In the case of M 4 = S3 \ /(, where !( is the figure of eight knot, the only bad

pairs are (4,1) and (3,1), and we denote the resulting closed manifold by M(r,,).

Furthermore there is a formula for approximating v(M(r,,Â»), which in the case

of the figure of eight knot is

72r 2 s 2 + 144s 4 )1r4

(r 4

2v'311" 2 -

=v(M4 ) - r 2 + 1282 + 4/3 3(r 2 + 1282 )4

v(M(r,.Â» - â€¢..â€¢

see [14, Â§6).

As in the previous section let l/J : PSL 2 (l(w)) -* PSL 2 (F,) be reduction modulo

p for some prime p == l(mod 3), and let the image of w be u, a priInitive cube

root of 1 in Fpo The fundamental group of the closed manifold is

173

Thomas: Finite groups of hyperbolic isometries

(˜l s(4u ˜˜) + r ),

The image of the relator l'mr in PSL2(Fp ) equals (-It+.+t

and in order to factorise 4J through we must satisfy the pair of equa-

1rtM(r,3)

tions (modulo p):

=0, and u 2 +U + 1 =o.

s(4u + 2) + r

Assuming that s ˜ O(modp), these can be rewritten as

= -(28 + r)/48, r 2 = -128 2 , see [15].

u

THEOREM 5. Let p == 1(mod 3) and f > 0 be an arbitrary positive real number.

There exists a closed hyperbolic manifold M with Iv(M)--v(M4 )1 < f, ˜uch that PSL2(F p )

is a homomorphic image of1r 1 M. Hence there exists a closed mainfold if with PSL2(Fp ) ˜

I(M), such that v(M) is arbitrarily close to p(p2 - 1) (1.0149 ...).

Proof: Assume first the p == l(mod 12), so that -1 is a quadratic residue. In

order to replace the equation r 2 = -128 2 by a linear equat˜on it will be enough

to show that 3 is a quadratic residue modulo p. Writing p = 3k+ 1 the quadratic

reciprocity law implies that (3k˜1) = (3k/l) = 1, and it follows that -12 is a

square in these cases. If on the other hand p == 7(mod 12), the argument is

similar, and uses the fact that both -1 and 3 are now non-quadratic residues

modulo p. The theorem is now clear, since we can represent the solutions of

a pair of linear equations (mod p) by (r, s), where rand s are arbitrarily large.

Additional comments - with d2 =-12 the condition (r, s) = 1 for a Dehn surgery

implies that the smallest relevent pairs are (Â±d,l) and (1, Â±d-1 ). We tabulate

these for the first few primes:

p

7

13

19

31

37

42

An asterisk against an entry indicates that it corresponds to a manifold of

smallest possible volume for this construction. Note that for p =7, the pairs

(4,1), (3,1) are excluded by the hyperbolic condition, and that the pair (2,1)

leading to the minimal volume obtainable from M 4 in this way fails to satisfy

the equation v = Â±4s.

174 Thomas: Finite groups of hyperbolic isometries

Those prime numbers p such that p-12 is a prefect square would seeln to form

an interesting subclass. Thus we have

COROLLARY 6. Let p == 1( mod 3), p = 12+d2 , d odd and not divisible by 3. Then

PSL2(Fp ) is a homomorphic image of7rt(M(d,lÂ» with

As pointed out in the introduction this method can be greatly generalised,

and will serve to realise a large class of finite simple groups quite explicitly as

hyperbolic isometry groups. A possible alternative method of attack would

be to use Thurston's condition (stated in terms of pseudo-Anosov diffeomor-

phisms) for a surface bundle over 8 1 to be hyperbolic. "Vertical" isometries

of the total space would then be defined by isometries of the fibre compatible

with the structural diffeomorphism. Clearly this approach raises interesting

mathematical problems, but is harder than the one used here.

175

Thomas: Finite groups of hyperbolic isometries

REFERENCES

1. C. Adams, The non-compact hyperbolic 3-manifold of minimal volume,

preprint 1989, Williams College, MA 01267.

2. M. Conder, Generators for alternating and symmetric groups, J. London

Math. Soc. (1980) 22, 75-86.

3. J. Conway, et aI., Atlas of finite groups, Clarendon Press (Oxford) 1985.

4. A. Drillick, The Picard group, Ph.D. Thesis (1971), NYU.

5. H. Glover, D. Sjerve, Representing PSL 2 (p) on a Riemann surface of

least genus, L'Enseignement Math. (1985) 31, 305-325.

6. D. Gorenstein, Finite groups, Harper & Row (NY), 1968.

7. L. Greenberg, Maximal groups and signatures, Annals of Math. Studies,

no.79 (1973), 207-226.

8. A. Hurwitz, Uber algebraische Gebilde mit eindeutigen Transformatio-

nen in sich, Math. Annalen (1893) 41, 403-442.

9. S. Kobayashi, Transformation groups in differential geometry (EM 70),

Springer-Verlag (Heidelberg) 1972.

10. A. Macbeth, Generators of the linear fractional groups, Proc. Symp.

Pure Math. (1969) 12, 14-32.

11. W. Magnus, Untersuchungen liber einige unendliche diskontinuierliche

Gruppen, Math. Armalen (1931) 105,52-74.

12. W. Magnus, Non-Euclidean tesselations and their groups, Academic

Press (NY) 1974.

13. G. Malle, Hurwitz groups and G 2 (q), preprint 1989, Heidelberg Univer-

sity, D-6900.

14. W. Neumann, D. Zagier, Volumes of hyperbolic 3-manifolds, Topology

(1985) 24, 307-332.

15. A. Nicas, An infinite family of non-Haken hyperbolic 3-manifolds with

vanishing Whitehead groups, Math. Proc. Camb. Phil. Soc. (1986)

99, 239-246.

16. A. Reid, Arithmeticity of knot complements, preprint 1989, Ohio State

University, 43210.

17. R. Riley, A quadratic parabolic knot, Math. Proc. Camb. Phil. Soc.

(1975) 77, 281-288.

Thomas: Finite groups of hyperbolic isometries

176

18. R. Riley, Applications of a computer implementation of Poincare's the-

orem on fundamental polyhedra, Math. of Computation (1983) 40,

607-632.

19. W. Thurston, Geometry and Topology of 3.. manifolds, Princeton Uni-

versity notes (1978).

20. C.T.C. Wall, Geometric structures on compact complex analytic sur-

faces, Topology (1986) 25, 119-153.

21. J. Weeks, Updated manifold census, dated 16.1.1989.

22. A. Woldar, Genus actions on surfaces, preprint 1986., Villanova Univer-

sity, PA 19085.

Cambridge University & ETH Zurich,

January 1990.

Pin Structures on Low-dimensional Manifolds

by

R. C. Kirbyl and L. R. Taylor l

Â§o. Introduction

Pin structures on vector bundles are the natural generalization of Spin struc-

tures to the case of non-oriented bundles. Spin(n) is the central Z/2Z extension

(or double cover) of SO(n) and Pin-en) and Pin+(n) are two different central

extensions of O(n), although they are topologically the same. The obstruction to

putting a Spin structure on a bundle e(= Rn ˜ E ˜ B) is w2(e)eH2(B; Z/2Z);

for Pin+ it is still W2(e), and for Pin- it is W2(e) + w˜(e). In all three cases, the

eis acted on by Hl(B; Z/2Z) and if we choose a structure,

set of structures on

this choice and the action sets up a one-to-one correspondence between the set of

structures and the cohomology group.

Perhaps the most useful characterization (Lemma 1.7) of PinÂ± structures is

ecorrespond eEB eand

that Pin- structures on to Spin structures on Pin+ to

det

eEB ewhere eis

Spin structures on 3 det det the determinant line bundle. This is

eEB

useful for a variety of "descent" theorems of the type: a PinÂ± structure on 11

ewhen

descends to a Pin+ (or Pin- or Spin) structure on dimf/ = 1 or 2 and

various conditions on TJ are satisfied.

For example, if TJ is a trivialized line bundle, then PinÂ± structures descend to

˜ (Corollary 1.12),.which enables us to define PinÂ± bordism groups. In the Spin

e, eEB

case, Spin structures on two of 11 and 1] determine a Spin structure on the

eEB eorientable,

third. This fails, for example, for Pin- structures on 1] and 1] and

but versions of it hold in some cases (Corollary 1.15), adding to the intricacies of

the subject.

Another kind of descent theorem puts a PinÂ± structure on a submanifold which

is dual to a characteristic class. Thus, if V m - l is dual to Wl (TM) and Mm is PinÂ±,

then V rtl V gets a PinÂ± structure and we have a homomorphism of bordism groups

(Theorem 2.5),

that proved useful in [K-T]. Or, if Fm-2 is the obstruction to extending a Pin-

structure on Mm - F over M, then F gets a Pin- structure if M is oriented

(Lemma 6.2) or M is not orientable but FrtlV has a trivialized normal bundle in

V (Theorem 6.9). These results give generalizations of the Guillou-Marin formula

[G-M], Theorem 6.3,

=F Â· F - sign M (mod 16)

2{1(F)

Partially supported by the N.S.F.

1

Kirby & Taylor: Pin structures on low-dimensional manifolds

178

to any characterized pair (M 4 ,F2 ) with no condition on H}(AI 4 ; Z/2Z).

Here, f3(F) is the Z/8Z Brown invariant of a Z/4Z quadratic enhancement

of the Z/2Z intersection form on H}(F; Z/2Z); given a Pin- structure on F, the

enhancement counts half-twists, mod 4, in imbedded circles representing elements

of H 1 (F; Z/2ZZ). This is developed in Â§3, where it is shown that

{3 : nfin- Z/8Z

--+

gives the isomorphism in the following table.

n:

= Z/2Z nipin = 0 n;pin = Z

nfpin = Z/2Z pin

nfin- = Z/2Z nfin- = Z/8Z nfin- = 0 nfin- = 0

nfin+ = 0 nfin+ = Z/2Z nfin+ = Z/2Z nfin+ = Z/16Z

In Â§2 we calculate the 1 and 2 dimensional groups and show that the non-zero one

dimensional groups are generated by the circle with its Lie group framing, stie,

(note the Mobius band is a Pin+ boundary for Slie); Rp2 generates nfin-; the

Klein bottle, the twisted SÂ£ie bundle over S1, generates nfin+; and T(ie, the torus

with its Lie group framing generates n˜pin. By Â§5 enough technique exists to

calculate the remaining values and show that nfin+ is generated by the twisted

T 2 bundle over SI with Lie group framing on the fiber torus; nfin+ is generated .

by RP4. The Cappell-Shaneson fake Rp4 represents Â±9 E Z/16Z [Stolz]; the

Kummer surface represents 8 E Z/16Z and in fact, a Spin 4-manifold bounds a

Pin+ 5-manifold iff its index is zero mod 32. The Kummer surface also generates

n4Spin â€¢

Section 4 contains a digression on Spin structures on 3-manifolds and a geo-

metric interpretation of Turaev's work [Tu] on trilinear intersection forms

This is used in calculating the JL-invariant: let JL(M, 8 1 ) be the J.l-invariant of }'1 3

with Spin structure 8 1 - The group HI (M 3 ; Z/2Z) acts on Spin structures, so let

a E HI (M 3 ; Z/2Z) determine 8 2 â€¢ Then a is dual to an imbedded surface F2 in

M which gains a Pin- structure from 8 1 and

(mod 16)

Four dimensional characteristic bordism n˜ is studied in Â§6 with generalizations

of [F-K] and [G-M]. We calculate, in Theorem 6.5, the J.l-invariant of circle bundles

over surfaces, S(1]), whose disk bundle, D(T/), has orientable total space. Fix a Spin

structure on S(1]), e. Then

= sign (D(1]) + 2Â· b(F)

J.l(S(11), 8) - Euler class(1]) (mod 16)

Kirby & Taylor: Pin structures on low-dimensional manifolds 179

e

where b(F) = 0 if the Spin structure extends across D(7J) and is f3 of a Pin-

e

structure on F induced on F from otherwise.

The characteristic bordism groups are calculated geometrically in Â§7, in par-

ticular,

n˜ = Z/8Z E9 ZJ4Z E9 Z/2Z .

Just as Robertello was able to use Rochlin's Theorem to describe the Ad in-

variant of a knot [R], so we can use f3 : nfin- ----+ Z/8Z to give a Z/8Z invariant to

a characterized link L in a Spin' 3-manifold M with a given set of even longitudes

for L (Definition 8.1). This invariant is a concordance invariant (Corollary 8.4),

and if each component of L is torsion in H1(M; Z), then L has a natural choice of

even longitudes (Definition 8.5).

Section 9 contains a brief discussion of the topological case of some of our

4-manifold results. In particular, the formula above must now contain the triangu-

lation obstruction K( M) for an oriented, topological 4-manifold M 4 :

+8˜(M)

2f3(F) == FÂ· F - sign (M) (mod 16)

(recall that (M,F) is a characterized pair).

Â§1. Pin Structures and generalities on bundles

The purpose of this section is to define the Pin groups and to discuss the notion

of a Pin structure on a bundle.

Recall that rotations of R n are products of reflections across (n - 1)-planes

through the origin, an even number for orientation preserving rotations and an odd

number for orientation reversing rotations. These (n - 1)-planes are not oriented so

they can equally well be described by either unit normal vector. Indeed, if u is the

unit vector, and if x is any point in R n, then the reflection is given by x - 2(x Â· u)u.

'rhus an element of O(n) can be given as (Â±Vl)(Â±V2)Â·Â·Â· (Â±Vk) where each Vi is

H. unit vector in Rn and k is even for SO(n). Then elements of Pin(n), a double

eover of O(n), are obtained by choosing an orientation for the (n - I)-planes or

(\quivalently choosing one of the two unit normals, so that an element of Pinen) is

VlÂ·Â· Â·Vk; if k is even we get elements of Spin(n). With this intuitive description as

Illotivation, we proceed more formally to define Pin (see [ABS]).

Let V be a real vector space of dimension n with a positive definite inner prod-

net, ( , ). The Clifford algebra, ClitF(V), is the universal algebra generated by

VÂ· with the relations

2(v w) for Clitr+(V)

VW+WV= ' for Clitr(V)

-2(v, w)

=

If el, Â· Â· Â· en is an orthonormal basis for V, then the relations imply that eiej

(˜iej, i f:. j and eiei = Â±1 in ClifF(V). The elements e/ = eit ... eik' I =

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

{I 5 it < i 2 Â·Â·â€¢ < ik $; n} form a (e/eJ = 0, e/el = Â±l) basis for ClifF(V).

So dim ClifF(V) = 2n ; note that as vector spaces, ClifF(V) is isomorphic to the

exterior algebra generated by V, but the multiplications are different, e.g. eiei =

f= 0 = ei A eiÂ·

Â±1

Let PinÂ±(V) be the set of elements of ClifF(V) which can be written in the

form VtV2Â·Â·Â· Vk where each Vi is a unit vector in V; under multiplication, PinÂ±(V)

is a compact Lie group. Those elements Vt V2 â€¢ â€¢ â€¢ Vk E PinÂ±(V) for which k is even

form Spin(V).

Define a "transpose" e} = eikÂ·Â·Â· eit = (-1)k-l e / and an algebra homo-

morphism aCe]) = (-1)k e] = (-l)IIl eI and extend linearly to ClifF(V). We

have a Z/2Z-grading on ClifF(V): ClifF(V)o is the +1 eigenspace of a and

ClifF(V)t is the -1 eie;enspace. For w E ClifF(V), define an automorphism

pew): ClifrÂ±(V) -+ ClifF(V) by

for ClifF-(V)

wvwt

= { a(w)vw t

p(w)(v) for Cliff+(V)

We can define a norm in the Clifford algebra, N: ClifF -+ R+ by N(x) = a(x)x for

all x E ClifrÂ±(V). Then we can define Pin:J:(V) to be {w E ClifF(V) I p(w)(V) =

V and N(w) = 1 }. Hence if w E PinÂ±(v), pew) is an automorphism of V so p is a

representation p: PinÂ±(V) -+ O(V) and by restriction p: Spin(V) -+ SO(V).

It is easy to verify that p(w) acts on V by reflection across the hyperplane wJ..,

e.g. for Pin-(V),

i rf 1

i=1

If r and I are basepoints in the components of O(V), where r is reflection

et,

then p-t {r, I} = {Â±el' Â±1} and

across

-1{ I} for Pin+(V)

Z/2Z EB Z/2Z

c:::! {

for Pin-(V)

Z/4Z

p r, -

The Z/2Z = {-I, I} E PinÂ± is central and PinÂ±(V)/{Â±1} = O(V). IT n > 1,

this Z/2Z is the center of PinÂ±(V) and, since O(V) has a non-trivial center, for "

˜˜

n > 1, the Z/2Z central extensions PinÂ± -+ O(V) are non-trivial.

.˜;

Thus PinÂ±(V) is a double cover of O(V). As spaces, PinÂ±(V) = Spin(V) II

Spin(V) but the group structure is different in the two cases. We can think of -1 E .;

p-l(I) as rotation of V ( about any axis) by 21r and +1 E p-l(I) as the identity. :

More precisely, an arc in PinÂ±(V) from 1 to -1 maps by p to a loop in O(V) which I

generates 1r1 (O(V)); in fact, for 8 E [0,1r], the arc 8 --+ Â±el . (cos 8el + sin Be2) is

one such. Even better, we may think of PinÂ±. as scheme for distinguishing an odd

number of full twists from an even number.

We use PinÂ±(n) to denote PinÂ±(V) where V is Rn.

Kirby & Taylor: Pin structures on low-dimensional manifolds 181

Remark. The tangent bundle of Rp2, TRP 2, has a Pin-(2)-structure.

We can "see" the Pin-(2) structure on TRP 2 as follows: decompose Rp2 into a

2-cell, B2, and a Mobius band, MB, with core circle Rpl. Then TRP21MB can be

described using two coordinate charts, Ul and U2 , with local trivializations (el' ei),

in which el is parallel to Rpl and e2 is normal, and with transition function

U1 n U2 --+ Pin-(2) which sends the two components of U1 n U2 to 1 and e2.

Then TRP21aM B is a trivial R 2-bundle over 8 1 = aMB which is trivialized by the

transition function 1 and e˜ = -1. Nowel would be tangent to 51 but the e˜ = -1

adds a rotation by 21r as 8 1 = aMB is traversed. But this trivialization on TRp 2 1s1

is exactly the one which extends over the 2-cell B2. Thus Rp2 is Pin -. ˜ote that

this process fails if e˜ = +1, and, in fact, Rp2 does not support a Pin+ structure

(see Lemma 1.3 below).

We now review the theory of G bundles, for G a topological group, and the

theory of H structures on a G bundle. A principal G bundle is a space E with a

left G action, E x G --+ E such that no point in E is fixed by any non-identity

element of G. We let B = E/G be the orbit space and p: E ..... B be the projection.

We call B the base of the bundle and say that E is a bundle over B. We also

require a local triviality condition. Explicitly, we require a numerable cover, {Ui },

P

ri

of B and G maps ri:Ui X G --+ E such that the composite Ui x G˜E˜B is just

projection onto Ui followed by inclusion into B. Such a collection is called an atlas

for the bundle and it is convenient to describe bundles in terms of some atlas. The

functions rj1 0ri are G maps, Ui nUj x G ..... Ui nUj x G, which commute with

the projection. Hence they can be given as transition functions 9ij:Ui n Uj --+ G.

Note 9ii = id, 9;;1 = 9ji and 9ik = 9ijO 9jk on Ui n Uj n Uk. Conversely, given any

numerable cover of a space B and a set of maps satisfying these three conditions,

we can find a principal G bundle and an atlas for it so the base space is B and the

transitions functions are our given functions.

Suppose Eo and E 1 are two G bundles over B o and B 1 respectively. Let

f: Eo ˜ E 1 be a map. A bundle map covering f is a G map F: Eo --+ E l so that

PI 0 F = f 0 Po, where Pi is the projection in the i-th bundle. We say two bundles over

B are equivalent iff there exists a bundle map between them covering the identity.

Given a bundle over B, say E, with atlas Ui and 9ij, and a map f: Bo --+ B,

the pull-back of E along f is the bundle over B o with numerable cover f- l (Ui) and

transition functions 9ijO f. The pull-backs of equivalent bundles are equivalent. A

hundle map between Eo and E 1 covering f: B o --+ B 1 is equivalent to a bundle

(˜quiva1ence between Eo and the pull-back of E 1 along f. Hence we mostly discuss

t.he case of bundle equivalence.

Given any atlas for a bundle, say Ui; gij, and a subcover Vor of Ui we can restrict

the 9ij to get a new family of transition functions gap. Clearly these two atlases

l'(˜present the same bundle. Given two numerable covers, it is possible to find a third

Ilumerable cover which refines them both, so it is never any loss of generality when

Kirby & Taylor: Pin structures on low..dimensional mani˜olds

182

considering two bundles over the same base to assume the transition functions are

defined on a common cover.

A bundle equivalence between bundles given by transition functions gij and gi;

for the same cover is given by maps hi:Ui -+ G such that, for all i and j and all

u E Ui nUj, g˜j(u) = hi(U)gij(U) (hj(u))-l.

Given a continuous homomorphism 'ljJ: H -+ G, we can form a principal G

bundle from a principal H bundle by applying tP to any atlas for the H bundle.

If p: E --+ B is the H bundle, we let P1/J: E XH G -+ B denote the associated

G bundle. Equivalent H bundles go to equivalent G bundles. We say that a G

bundle, p: E --+ B, had an..,H structure provided that there exists an H bundle,

PI: E I -+ B so that the associated G bundle, (PI)",: E I XH G --i' B is equivalent to

the G bundle. More correctly one should say that we have a "p structure on our G

bundle, but we won't. An H structure for a G bundle, p: E --i' B consists of a pair:

an H bundle, PI: E 1 --+ B, and a G equivalence, 1 from (PI)tb: E 1 XH G --+ B to the

original G bundle, p: E --. B. Two structures PI: El -+ B, 1'1 and P2: E 2 --+ B, 12

on p: E -+ B are equivalent if there exists an equivalence of H bundles f: E 1 --+ E 2

such that, if ft/J denotes the corresponding equivalence of G bundles, 11 = 12 0 ft/J.

We assume the reader is familiar with this next result.

Theorem 1.1. For any topological group, G, there exists a space Bo such that

equivalence classes of G bundles over B are in 1-1 correspondence with homotopy

classes of maps B -+ BG. (A map B --i' Ba corresponding to a bundle is called a

classifying map for the bundle.) Given"p: H --+ G we get an induced map B"p: BH --.

Ba. If this map is not a fibration, we may make it into one without changing Ba

or the homotopy type of BH, so assume B1/J is a Hurewicz fibration. Given a G

bundle with a classifying map B -+ Ba, H structures on this bundle are in 1-1

correspondence with lifts of the classifying map for the G bundle to B H â€¢

Example. Let p: E --+ B be a trivial O(n) bundle, and suppose the atlas has one

open set, namely B, and one transition function, the identity. One SO(n) structure

on this bundle consists of the same transition function but thought of as taking

values in SOCn) together with the bundle equivalence which maps B to the identity

in O(n). Another SOCn) structure is obtained by using the same transition functions

but taking as the bundle equivalence a map B to O(n) which lands in the orientation

reversing component of O(n). Indeed any map B -+ O(n) gives an SO(n) structure

on our bundle. It is not difficult to see that any two maps into the same component

of O(n) give equivalent structures and that two maps into different components give

structures that are not equivalent as structures. Clearly the SO(n) bundle in all

cases is the same. One gets from here to the more traditional notion of orientation

for the associated vector bundle as follows. Since the transition functions are in

O(n), O(n) acts on the vector space fibre. But for matrices to act on a vector space

a basis needs to be chosen. This basis orients the SO(n) bundle: in the first case

Kirby & Taylor: Pin structures on low-dimensional manifolds 183

the equivalence orients the underlying O(n) bundle one way and in the second case

the equivalence orients the bundle the other way.

Finally recall that an O(n) bundle has an orientation iff the first Stiefel-

Whitney class, WI of the bundle vanishes. If there is an SOCn) structure then

HO (B; Z/2Z) acts in a simply transitive manner on the set of structures.

The Lie group Spin(n) comes equipped with a standard double cover map

Spin(n) --+ SO(n), and this is the map 1/J we mean when we speak of an SO(n)

bundle, or an oriented vector bundle, having a Spin structure. There is a fibration

sequence B Spin(n) --+ BSO(n) --+ K(Z/2Z,2), so the obstruction to the existence

of a Spin structure is a 2-dirnensional cohomology class which is known to be the

second Stiefel-Whitney class W2. If the set of Spin structures is non-empty, then

HI (B; Z/2Z) acts on it in a simply transitive manner.

The action can be seen explicitly as follows. Fix one Spin structure, say gij. An

element in H 1 (B; Z/2Z) can be represented by a Cech cocycle: i.e. a collection of

maps Cij:Ui nUj --+ Â±1 satisfying the same conditions as the transition functions for

a bundle. The new Spin structure consists of the transition functions gij Â·Cij with the

same SO(n) bundle equivalence, where we think of Â±1 as a subgroup of Spin(n) and

. denotes group multiplication. It is not hard to check that cohomologous cocycles

give equivalent structures.

We now explore the relation between Spin structures on an oriented vector

bundle and frarnings of that bundle. A framing of a bundle is the same thing as an

H structure where H is the trivial subgroup. Hence H is naturally a subgroup of

Spin(n) and an equivalence class offramings of a bundle gives rise to an equivalence

class of Spin structures. Consider first the case n = 1. Recall SO(I) is trivial and

Spin(l) = Z/2Z. Hence an SO(1) bundle already has a unique trivialization, and

hence a "canonical" Spin structure. There are often other Spin structures, but,

none of these come from framings. In case n = 2, 5pin(2) = 8 1 , 50(2) = 51 and

the map is the double cover. If an 50(2) bundle is trivial, frarnings are acted on

simply transitively by HI (B; Z). The corresponding Spin structures are equivalent

iff the class in HI (B; Z/2Z) is trivial. If B is a circle the bundle is trivial iff it has a

Spin structure and both Spin structures come from frarnings. The Spin structure

determines the framing up to an action by an even element in Z, so we often say

that the Spin structure determines an even framing. If n > 2 and B is still a circle,

then the bundle is framed iff it has a Spin structure and now framings and Spin

structures are in 1-1 correspondence.

Of course, given any Spin structure on a bundle over B, and any map f: 8 1 -+

n, we can pull the bundle back via f and apply the above discussion. Since Spin

st.ructures on the bundle are in 1-1 correspondence with HI (B; Z/2Z), which is

detected by mapping in circles, we can recover the Spin structure by describing

how the bundle is framed when restricted to each circle (with a little care if n = 1

or 2). Moreover, if an SO(n) bundle over a CW complex is trivial when restricted

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

to the 2-skeleton, then W2 vanishes, so the bundle has a Spin structure. H n -# 2

and the bundle has a Spin structure then, restricted to the 2- skeleton, it is trivial.

If n = 2 this last remark is false as the tangent bundle to 52 shows.

Finally, we need to discuss stabilization. All our groups come in families in-

dexed by the natural numbers and there are inclusions of one in the next. An

example is the family O(n) with O(n) -+ O(n + 1) by adding a 1 in the bottom

right, and all our other families have similar patterns. This is of course a special case

e,

of our general discussion of H structures on G bundles. Given a vector bundle,

eextend

and an oriented line bundle, â‚¬1 , the O(n) transition functions for naturally

to a set of O(n + 1) transition functions for ˜ E9 â‚¬1 using the above homomorphism,

eEB

and any of our structures on ˜ will extend naturally to a similar structure on â‚¬1 â€¢

e.

We call the structure on ˜ EB â‚¬1 the stabilization of the structure on

A particular case of great interest to us is the relation between tangent bundles

in a manifold with boundary. Suppose M is a codimension 0 subset of the boundary

of W. We can consider the tangent bundle of W, say Tw, restricted to M. It is

naturally identified with TM EB VMCW where v denotes the normal bundle. This

normal bundle is framed by the "inward-pointing" normal, so we can compare

structures on M with structures on W using stabilization.

Since both PinÂ±(n) are Lie groups and have homomorphisms'into O(n), the

above discussion applies.

Remarks. With this definition it is clear that, if there is a PinÂ± structure on a

eover a space B then HI (B; Z/2Z) acts on the set of PinÂ± structures in a

bundle

simply transitive manner. It is also clear that the obstruction to existence of such

a structure must be a 2-dimensional cohomology class in H2 (EO(n); Z/2Z) that

restricts to W2 E H2 (BSO(n); Z/2Z) and hence is either W2(˜) or W2(e) + w˜(e).

Here Wi denotes the i-th Stiefel-Whitney class of the bundle.

We sort out the obstructions next.

Lemma 1.2. Let A be a line bundle over a CW complex B. Then,\ has a Pin+

structure and A E9 A E9 ,\ has a Pin- structure.

Proof: Since Pin+(l) ˜ 0(1) is just a projection, Z/2Z (f) Z/2Z ˜ Z/2Z, there is

a group homomorphism, 0(1) ˜ Pin+(l), splitting the projection. H we compose

transition functions for Awith this homomorphism, we get a set of Pin+ transition

functions for A. Hwe have an equivalent 0(1) bundle, the two Pin+(1) bundles are

also equivalent.

Transition functions for 3A are given by taking transition functions for A and

composing with the homomorphism 0(1) ˜ 0(3) which sends Â±1 to the matrix

Â±1 Â±0 0)

(o 0 . It is easy to check that this homomorphism lifts through a

1

o 0 Â±1

Kirby & Taylor: Pin structures on low-dimensional manifolds 185

homomorphism 0(1) ˜ Pin-(3). If we have an equivalent 0(1) bundle, the two

Pin-(3) bundles are also equivalent.â€¢

Addendum to 1.2. Notice that we have proved a bit more. The homomorphisms

we chose are not unique, but can be chosen once and for all. Hence a line bundle

has a "canonical" Pin+ structure and 3 times a line bundle has a "canonical" Pin-

structure.

Remark. There are two choices for the homomorphisms above. If we choose the

other then the two "canonical" Pin+ structures on a line bundle differ by the action

of WI of the line bundle, with a similar remark for the Pin-case.

Lemma 1.3. The obstruction to lifting an O(n)-bundle to a Pin+(n)-bundle is

eEf ehas

W2, and to a Pin- (n )-bundle is W2 + w˜. If ) A = trivial bundle, then a

Pin- structure iff ,\ has a Pin+ structure.

Proof: A line bundle has a Pin+ structure by Lemma 1.2, so W2 = 0, but there are

examples, e.g. the canonical bundle over Rp2, for which wi f. O. Hence W2 is the

obstruction to a bundle having a Pin+ structure.

For 3 times a line bundle, W2 = w˜, so we can find examples, e.g. 3 times the

w;

canonical bundle over Rp2, for which W2 + w˜ = 0 but W2 f. O. Hence W2 + is

the obstruction to having a Pin- structure.

The remaining claim is an easy characteristic class calculation.â€¢

The fact that the tangent bundle and normal bundles have different structures

can lead to some confusion. In the rest of this paper, when we say a manifold has

a PinÂ± structure, we mean that the tangent bundle to the manifold has a PinÂ±

structure. As an example of the possibilities of confusion, the Pin bordism theory

calculated by Anderson, Brown and Peterson, [ABP2], is Pin- bordism. They do

the calculation by computing the stable homotopy of a Thom spectrum, which as

usual is the Thorn spectrum for the normal bundles of the manifolds. The key fact

that makes their calculation work is that W2 vanishes, but this is W2 of the normal

bundle, so the tangent bundle has a Pin- structure and we call this Pin- bordism.

We remark that a PinÂ± structure is equivalent to a stable PinÂ± structure and

similarly for Spin. This can be seen by observing that

PinÂ±(n +1)

PinÂ±(n) --+

! !

O(n + 1)

O(n) ---+

commutes and is a pull-back of groups, with a similar diagram in the Spin case.

In order to be able to carefully discuss structures on bundles, we introduce the

f()llowing notation and definitions. Given a vector bundle, ˜, let PinÂ±(e) denote the

eis

˜Â·iÂ«Â·t of PinÂ± structures on it. If an oriented vector bundle, let Spin(e) denote

Kirby & Taylor: Pin structures on low-dimensional manifolds

186

the set of Spin structures on it. Throughout this paper we will be writing down

functions between sets of PinÂ± or Spin structures. All these sets, if non-empty are

acted on, simply transitively, by HI (B; Z/2Z) where B is the base of the bundle.

Definition 1.4. We say that a function between two sets of structures on bundles

over bases B I and B 2 respectively is natural provided there is a homomorphism

HI (B 1 ; Z/2Z) ˜ HI (B 2 ; Z/2Z) so that the resulting map is equivariant.

One example of this concept is the following construction.

j: el

˜ e2 be a bundle map covering f: B 1 --+ B 2 â€¢ Given

Construction 1.5. Let

a cover and transition functions for B 2 and e2, we can use 1 and j to construct a

e1.

1

cover and transition functions for B and This construction induces a natural

function

j*: PinÂ±(e2) --+ PinÂ±(el)

j

with a similar map for Spin structures if we use to pull back the orientation.

There are two examples of this construction we will use frequently. The first

is to consider an open subset U C M of a manifold M: here the derivative of

the inclusion is a bundle map so Construction 1.5 gives us a natural restriction

of structures. The second is to consider a codimerision 0 immersion between two

manifolds, say f: N ˜ M. Again the derivative is a bundle map so we get a natural

restriction of structures.

We can also formally discuss stabilization.

ebe

a vector bundle, and let e1 be a trivial line bundle, both

Lemma 1.6. Let

over a connected space B. There are natural one to one correspondences

.4 e1 )

Sr(e): 1'inÂ±(e) --+ 1'inÂ±(e ffi .

1=1

eis oriented there is a natural one to one correspondence

If

el ˜ e2, there is another bundle map (I EB .$ 1): el EB .67 e1˜

" r r

Given a bundle map f:

&=1 1=1

r

e2 ED 1

The obvious squares involving these bundle maps and the stabilization

.61 e .

1=1

maps commute.

We would like a result that relates PinÂ± structures on bundles to the geometry

of the bundle restricted over the I-skeleton mimicking the framing condition for the

en

Spin case. We settle for the next result. Let be an n-plane bundle over a CW-

ebe en.

complex X, and let det the determinant bundle of

Kirby & Taylor: Pin structures on low-dimensionul Inanifolds IM7

Lemma 1.7. There exist natural bijections

+ 1) <let e)

\II4k+l (e): Pin -(e) ˜ Spin(e EB (4k

W4k+3(e): Pin+(e) -+ Spince EB (4k +3) det e)

\II4k+2Ce): PinÂ±(e) -+ Pin+(e EB (4k +2) det e)

'!J4k(e): PinÂ±(e) ˜ PinÂ±(e EB (4k) det e)

q,tk(e): Spin(e) ˜ Spin(e EB (4k) det e) Â·

and

el el

˜ e2 defines a bundle map det ˜ det e2. Using this map

A bundle map j:

between determinant bundles, all the squares involving the \II maps commute.

Proof: It follows from Lemma 1.3 that the existence of a structure of the correct

eis eEBr e.

sort on equivalent to the existence of a structure of the correct sort on det

Let us begin by recalling the transition functions for the various bundles. There

are homomorphisms 6r : O(n) ˜ O(n +r) defined by sending an n X n matrix A to

the (n + r) X (n + r) matrix which is A in the first m X m locations, det A in the

remaining r diagonal locations, and zero elsewhere.

e,

If Ui, 9ij:Ui nUj --. O(n) is a family of transition functions for then br 09ij

eEB e.

is a family of transition functions for r det

eto

Next, we describe a function from the set of structures on the set of

e67 e.

structures on r det

Begin with the case in which e has a Pin- structure with transition functions

Gij: Uj nUj -+ Pin - (n) lifting the given set gij into O(n). Pick an element e in

the Clifford algebra for R n EB R 1 so that e2 = -1 and e maps to reflection through

R n under the canonical map to O( n + 1). There are two such choices but choose

one once and for all. Define Hij into Pion-en + 1) by Hij(u) = i(Gij(u) Â· Xij(U)

where i denotes the natural inclusion of Pin-en) into Pin-en + 1) and Xij(U) is e

if det gij ( u) = -1 and 1 otherwise.

It is clear that the Hij land in Spin(n + 1), but what needs to be checked

is that they are a set of transition functions for our bundle. Clearly they lift the

+1) bundle, so we need to c˜nsider

transition functiQns for the underlying SO(n

the cocycle relation. This says that Hij(u)Hjk(U)Hki(U) = 1. If we replace the

H's by G's, we do have the relation, so let us compute Hij(u)Hjk(u)Hki(u) =

Gij(U)Xij(U)Gjk(U)Xjk(U)Gld(U)Xki(U). Any x commutes past a G if the x associ-

a.ted to the G is 1 and it goes past with a sign switch if the x associated to the G is

(). Also note that either none or two of the x's in our product are e. We leave it to

t.he reader to work through the cases to see that the cocycle relation always holds

and to note that the key point is that e2 = -1.

ehas a Pin+ structure, and let Gij continue

Next, consider the case in which

1.0 denote the transition functions. Let el, e2 and e3 denote elements in the Pin+

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

188

Clifford algebra for Rn Ea R 3 : each ei covers reflection in a hyperplane perpendic-

ular to one of the three standard basis vectors for the R3 factor. Define Hij as

above except replace e by ele2ea. The proof goes just as before after we note that

(ele2ea)2 = -1.

emay

For the case in which r = 2 and have either a Pin+ or a Pin- structure,

choose el and e2; note that (et e2)2 = -1 and proceed as above.

eit

The last natural bijection is also easy. If gij are transition functions for

is easy to choose the cover so that there are lifts Gij of our functions to Pin-(n)

(or Pin+(n) if the reader prefers), but the cocycle relation may not be satisfied.

We can define new functions Hij into Spin(4n) by just juxtaposing 4 copies of Gij

thought of as acting on four copies of the same space. These functions can easily

be checked to satisfy the cocycle condition.

Now that we have defined our functions, the results of the theorem are easy.

The reader should check that the functions we defined are HI ( ; Z/2Z) equivariant

and hence induce 1-1 transfonnations. ˜

Remark 1.8. We did make a choice in the proof of 1.7. The choice was global

and so the lemma holds, but it is interesting to contemplate the effect of making

the other choice. It is not too hard to work out that if we continue to use 1, but

replace e by -e, the new Spin structure will differ from the old one by the action

of Wl(e). The same result holds if we switch an odd number of the el, e2, e3 in the

Pin+ case or an one of et, e2 in the r = 2 case.

For later use, we need a version of Lemma 1.7 in which the line bundles are

merely isomorphic to. the determinant bundle. To be able to describe the effect of

changing our choices, we need the following discussion.

There is a well-known operation on' an oriented vector bundle known as "re-

versing the orientation". Explicitly, suppose that we have transition functions,

gij, defined into SO(n) based on a numerable cover {Ui}. Then we choose maps

hi:Ui ˜ O(n) - SO(n) and let the bundle with the "opposite orientation" have

transition functions hi o gij o hjl and use the maps hi to get the O(n) equivalence,

with the original bundle. The choice of the hi is far from unique, but any two choices '

yield equivalent SOCn) bundles. In the same fashion, given a Spin(n) bundle, we

can consider the opposite Spin structure. Proceed just as above using Spinen) for '

SO(n) and Pin+(n) or Pin-(n) for O(n). .:

Note that a Spin structure and its opposite are equivalent Pin+ or Pin- :;

structures. Conversely, given a PinÂ± structure on a vector bundle which happens to

be orientable, then there are two compatible Spin structures which are the opposites'

of each other. We summarize the above discussion as

eis

Lemma 1.9. If an oriented vector bundle, then there is a natural one to one

correspondence, called reversing the spin structure,

Spin( -e)

Re: Spin(e) -+

Kirby & Taylor: Pin structures on low-dimensional manifolds 189

-e ewith

where denotes the orientation reversed. We have that 'Re0'R- e is the

identity. Finally, given a bundle map j as in Construction 1.5, the obvious square

commutes.

Proof: We described the transfonnation above, and it is not hard to see that it

is HI (; Z/2Z) equivariant. It is also easy to check that the conlposition fonnula

holds.â€¢

In practice, we can rarely identify our bundles with the accuracy demanded

by Lemma 1.7 or Lemma 1.6, so we discuss the effect of a bundle automorphism

r

eEB

on the sets of structures. Suppose we have a bundle X = .E9 A, where A is a

1=1

line bundle. We will study the case A is trivial (so called "stabilization") and the

e.

case A is isomorphic_ to det Let, be a bundle automorphism of X which is the

r

eand

sum of the identity on some automorphism of .$ A. The transition functions

.=1

r

for Ea A are either the identity or minus the identity, both of which are central in

i=1

O( r) so , is equivalent to a collection of maps ,: B --+ O(r), where B is the base of

the bundle. The bundle automorphism induces a natural automorphism of PinÂ±

structures on x, described in the proof of

Lemma 1.10. Let the base of the bundle, B, be path connected. The map induced

,*, is the identity if, lands in SOCr). Otherwise it

by , on structures, denoted

reverses the Spin structure in the Spin case and acts via Wl(e) in the PinÂ± case if

..\ is trivial and by r . WI (e) if A is isomorphic to det e.

e(eitherSpin

Proof: To fix notation, choose transition functions for a structure on

or PinÂ±). Pick transition functions for A using the same cover. If A is trivial, take

the identity for the transition functions and if A is the determinant bundle take the

e.

determinant of the transition functions for The new structure induced by , has

transition functions .y(U)Oij(U}y-l(U) where 0ij denotes the old transition functions

and .y(u) denotes a lift of I(U) to PinÂ±(r) and then into PinÂ±(n +r) where ehas

dimension n. There may be no continuous choice of 7', but since the two lifts yield

the same conjugation, the new transition functions remain continuous. The element

Oij( U )PinÂ±(n + r) has the fonn x with x involving only the first n basis vectors in

the Clifford algebra if det Oij(u) = 1 or if A is trivial: otherwise xe n+l .. Â· en+r with

x as before.

Recall i'x = (_l)a(x)a("Y)x-y and -)ie n +l ... en +r = (-1Yl'("Y)(r-l)e n +l ... en +r 7'

where it on PinÂ± is the restriction of the mod 2 grading from the Clifford algebra

a.nd it on O(r) is 1 iff the element is in BO(r). The result now follows for PinÂ±

structures. The result for Spin structures is now clear. H I takes values in SO(r)

t.hen the bundle map preserves the orientation and the underlying Pin- structure,

hence the Spinstructure. H I takes values in O(r) - SO(r), compose the map

iuduced by , with the reverse Spin structure map. The reverse Spin structure map

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

is induced by any constant map B --t OCr) - SO(r). Hence the composite of these

two maps is induced by a map B -+ SO(r) and hence is the i,dentity.â€¢

There are a couple of further compatibility questions involving the functions we

eand

have been discussing. Given an SOCn) bundle an oriented trivial line bundle

-e

l and an isomorphism ED reI ˜

e , we get a natural SOCn + r) bundle e EI1 re

1

-(e EB reI).

Lemma 1.11. With the above identifications, stabilization followed by reversing

the Spin structure agrees with reversing the Spin structure and then stabilizing:

i.e. n.effirEOS:(e) = s:( -e)oR(e).

Proof: Left to the reader.â€¢

Let Mrn be PinÂ± and let vrn-l be a codimension 1 manifold of M with normal

line bundle v. We wish to apply Lemma 1.7 to the problem of constructing a

"natural" structure on V. IT there is a natural map from structures on M to

structures on V, we say that V inherits a structure from the structure on M. Of

course, the homomorphism HI (M; Z/2Z) --. HI (V; Z/2Z) implicit in the use of

"natural" is just the one induced by the inclusion.

Corollary 1.12. If v is trivialized then V inherits a PinÂ± structure from a PinÂ±

structure on M. If M and V are oriented then V inherits a Spin structure from a

Spin structure on M. \

Proof: When v is trivialized the result follows from Lemma 1.6. If M and V a.re

oriented, then we can trivialize (i.e. orient) v so that the orientation on Tv EB v

agrees with the orientation on TMlv .â€¢

A case much like Corollary 1.12 occurs when M is a manifold with boundary,

V = 8M. In this case, the normal bundle, v, is trivialized by the geometry, namely

the preferred direction is inward. Just as in Corollary 1.12, we put v last getting

TMlaM = TaM EI1 v. On orientations this gives the convention "inward normal last"

which we adopt for orienting boundaries. Furthermore, a Spin or PinÂ± structure

on M now induces one on 8M, so we have a bordism theory of Spin manifolds and

of PinÂ± manifolds.

In the Spin case, the inverse in the bordism group is formed by taking the

manifold, M, with Spin structure on TM, and reversing the Spin structure. In

either the Pin+ or the Pin- case, the inverse in bordism is formed by acting on the

given structure by wI(M). Having to switch the PinÂ± structure to form the inverse

is what prevents n;in:i: from being a Z/2Z vector space like ordinary unoriented

bordism. The explicit formula for the inverse does imply

Corollary 1.13. The image ofn˜pin(x) in n˜inÂ± (X) has exponent 2 for any C"YV

complex X, or even any spectrum.

Kirby & Taylor: Pin structures on low-dimensional manifolds 191

The "inward normal last" rule has some consequences. Suppose we have a

manifold with boundary M, 8M, and a structure on M X R I â€¢ We can first restrict

to the boundary, which is (aM) x R 1 , and then do the codimension 1 restriction, or

else we can do the codimension 1 restriction to M and then restrict to the boundary.

Lemma 1.14. The two natural functions described above,

differ by the action ofWt(M). The same map between Spin structures reverses the

Spin structure.

Proof: By considering restriction maps it is easy to see that it suffices to prove the

result for M = (aM) x [0,00), and here the functions are bijections. Consider the

inverse from structures on 8M to structures on aM x R I X [0, 00). The two different

functions differ by a bundle automorphism which interchanges the last two trivial

factors. By Lemma 1.10, this has the effect claimed.â€¢

In the not necessarily trivial case we also have a "restriction of structure" result.

Corollary 1.15. If v is not necessarily trivial, then V inherits a structure from

one on M in three of the four cases below:

Pin+ Pin-

V orientable

Spin None

v = detTM

V not necessarily orientable

v = detTv

3

= TMlv has a Pin+ structure, so TM Ea det TM

Proof: In the northwest case, Tvffiv

3 3 4

But TM EB det TM Iv = Tv EB v EB EB det TM Iv = Tv EB det TM Iv

has a Spin structure.

so Tv and hence V acquires a Spin structure. However, there is a choice in the

above equation: we have had to identify v with det TM Iv. When we say that the v

and det TM are equal, we mean that we have fixed a choice.

A similar argument works in the southeast case: Tv E9 det Tv is naturally

oriented, so an identification of v with det Tv gives Tv ED v = TM Iv. Since M has

a Pin- structure, V gets a Pin-structure.

In the southwest case, consider E C M, a tubular neighborhood of V. Since v

and det Tv are identified, and since Tv Ea det Tv is naturally oriented, E is oriented

and hence the Pin+ structure reduces uniquely to a Spin structure. From here the

argument is the same as in the last paragraph.

Lastly, consider the northeast case. If we let V = Rp5 C Rp6 = M, we see

that M has a Pin- structure; v and det TM are isomorphic; V is orientable but

does not have any Spin structures at all.â€¢

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

Remark. If we just assume that the line bundles in the table are isomorphic, which

is surely the more usual situation, then we no longer get a well-defined structure.

The new structure is obtained from the old one by first reversing orientation in the

Spin case, and then acting by Wl(V). A similar remark applies to Corollary 1.12.

Â§2. Pin- structures on low-dimensional manifolds and further generalities.

We begin this section by recalling some well-known characteristic class formu-

las. Every I-dimensional manifold is orientable and has Spin and PinÂ± structures.

It is easy to parlay this into a proof that n˜pin ˜ Z and GfinÂ± ˜ Z/2Z, with

the isomorphism being given by the number of points (for Spin) and the number

of points mod 2 for PinÂ±. Using the Wu relations, [M-S, p. 132], we see that

every surface and every 3-manifold has a Pin- structure, and hence oriented 2

and 3-manifolds have Spin structures. We can also say that a 2 or 3-manifold has

a Pin+ structure iff w˜ = O. For surfaces this translates into having even Euler

characteristic or into being an unoriented boundary.

We next give a more detailed discussion of structures on SI. The tangent

bundle to S1 is trivial and I-dimensional, hence a trivialization is the same thing

as an orientation. Since HI (51; Z/2Z) ˜ Z/2Z, there are two Spin structures on

the circle. Since the tangent bundle to 8 1 does not extend to a non-zero vector

field over the 2-disk, the two Spin structures on an oriented Slcan be described as

follows: one of them is the Spin structure coming from the framing given by the

orientation (this is called the Lie group framing or the Lie group Spin structure)

and the other one is the one induced by the unique Spin structure on the 2-disk

restricted to SI.

Theorem 2.1. The group n˜pin ˜ Z/2Z, generated by the Lie group Spin struc-

ture on the circle; nfin- ˜ Z/2Z and the natural map n˜pin --+ nfin- is an

isomorphism; ni in + = o.

Proof: Since the 2-disk has an orientation reversing involution, the restriction of

this involution to the boundary gives an equivalence between 51 with Lie group Spin

structure and 8 1 with the orientation reversed and the Lie group Spin structure.

Hence n˜pin and Orin:!: are each 0 or Z/2Z. Suppose 8 1 is the boundary of an

oriented surface F. It is easy to check that all Spin structures on F induce the same

Spin structure on S1. If we let F denote FU B2 then F also has a Spin structure,

and it is easy to see that any Spin structure on F extends (uniquely) to one on F.

In particular, the Spin structure induced on S1 is the one which extends over the

2-disk, so 51 with the Lie group Spin structure does not bound.

The proof for the Pin - case is ident.ical because any surface has a Pin - struc-

ture.

In the Pin+ case however, Rp2 does not have a Pin+ structure. On the other

hand, Rp2 - int B2 (which is the Mobius band) does have a Pin+ structure. The

Kirby & Taylor: Pin structures on low-dimensional manifolds 193

induced Pin+ structure on the boundary must therefore be one which does not

extend over the 2-disk, and hence the circle with the Lie group Pin+ structure

does bound.â€¢

In dimension 4, the generic manifold supports neither a Spin nor a PinÂ±

structure. A substitute which works fairly well is to consider a 4-manifold with a

submanifold dual to W2 or W2 +wi. We will also have need to consider submanifolds

dual to WI. A general discussion of these concepts does not seem out of place here.

Let M be a paracompact manifold, with or without boundary. Let a be a

cohomology class in Hi (M; Z/2Z). We say that a codimension i submanifold of

M, say W eM, is dual to a iff the embedding of W in M is proper and the boundary

of M intersects W precisely in the boundary of W. The fundamental class of W

aw;

is a class iIi H˜Â·!:i(W, Z/2Z), where H'Â·I. denotes homology with locally finite

chains. With the conditions we have imposed on our embedding, this class maps

under the inclusion to an element in H˜˜Â·i(M,aM; Z/2Z). Under Poincare duality,

H˜˜Â·i(M, aM; Z/2Z) is isomorphic to Hi (M; Z/2Z) and we require that the image

of the fundamental class of W map under this isomorphism to a. Specifically, in

H˜˜Â·i(M,aM;Z/2Z),we have the equation an [M,oM] = i*[W,aW].

A cohomology class in HR (B; A), is given by a homotopy class of maps, B --+

[Â«A,n), where ]((A,n) is the Eilenberg-MacLane space with 7r n ˜ A. IT TO(n)

denotes the Thorn space of the universal bundle over BO(n), then the Thorn class

gives a map TO(n) --+ K(Z/2Z,n). IT M is a manifold, the Pontrjagin-Thom

construction shows that a E Hn (M; Z/2Z) is dual to a submanifold iff the map

M -+ K(Z/2Z,n) representing a lifts to a map M -+ TO(n). Similar remarks

hold if A = Z with BO(n) replaces by BSO(n). The submanifold, V, is obtained

by transversality, so the normal bundle is identified with the universal bundle over

BO(n) or BSO(n) and the Thorn class pulls back to a. Hence there is a map

(M,M - V) -+ (TO(n),*) which is a monomorphism on H n (;Z/2Z) by excision.

The Thorn isomorphism theorem shows Hn (M, M - V; Z/2Z) ˜ HO (V; Z/2Z) so

Hn (M, M - V; Z/2Z) is naturally isomorphic to a direct product of Z/2Z's and

the Thom class in Hn (TO(n), *; Z/2Z) restricts to the product of the generators.

It follows that a restricted to M - V is o. It also follows that a restricted to V is

the Euler class of the normal bundle.

Since TO(I) = Rp oo = K(Z/2Z, I) all I-dimensional mod 2 cohomology

classes have dual submanifolds. Since TSO(I) = S1 = K(Z,I) all I-dimensional

integral homology classes have dual submanifolds with oriented normal bundles.

This holds even if M is not orientable, in which case the submanifold need not be

orientable either. Since TSO(2) = Cpoo = !(Z,2), any 2-dimensional integral

cohomology class has a dual submanifold with oriented normal bundle. A case of

interest to us is TO(2). The map TO(2) -+ [Â«Z/2Z, 2) is not an equivalence, and

not all 2-dimensional mod 2 cohomology classes have duals. As long as the manifold

has dimension ˜ 4, duals can be constructed directly, but these techniques fail in di-

Kirby & Taylor: Pin structures on low-dimensional manifolds

194

mensions 5 or more. A more detailed analysis of the map TO(2) ˜ K(Z/2Z,2) also

shows the same thing: there are no obstructions to doing the lift until one gets to di-

mension 5 and then there are. It is amusing to note that the obstruction to realizing

a class a in a 5-manifold is Sq2 SqI a+aS ql a E H 5 (M; Z/2Z) / Sql(H 4 (M; Z/2Z)):

in particular, if M is not orientable, then any class can be realized.'

In our case we want to consider duals to WI, w2 and W2 + w˜. We begin with

WI' This is an example for which the above discussion shows that we always have

a dual, say V m- I C Mm. We want to use the fact that we have a dual to WI. The

first question we want to consider is when is an arbitrary codimension 1 submanifold

dual to WI. The answer is supplied by

Lemma 2.2. A codimension 1 submanifold V C M is dual to wl(M) iff there

exists an orientation on M - V which does not extend across any component of V.

The set of such orientations is acted on simply transitively by H O (M; Z/2Z).

Remark. We say that an orientation on N - X does not extend across X if there

is no orientation on N which restricts to the given one on N - X. We can take

N = (M - V) U YO and X = Yo, where Vo is a component of V. By varying Vo over

the path components of V we get a definition of an orientation on M - V which does

not extend across any component (= path component) of V. A similar definition

applies to the case of a Spin or PinÂ± structure on M - V which does not extend

across any component of V.

Proof: Suppose that M - V is orientable and fix an orientation. If Vi denotes

the normal bundle to the component Vi of V, let (D(Vi), S(ViÂ») represent the disk

sphere bundle pair. Each S(Vi) is oriented by our fixed orientation on M - V

since M - -Ll D(Vi) C M - V is a codimension 0 submanifold (hence oriented) and ,

JL B(Vi) can be naturally added as a boundary. Define b E H 1 (M, M - V; Z/2Z) ˜

(fJHl (D(Vi)' S(Vi); Z/2Z) ˜ ffiZ/2Z on each summand as 1 if the orientation on

S(Vi) extends across D(Vi) and -1 if it does not. The class b hits wI(M) in

HI (M; Z/2Z). This can be easily checked by considering any embedded circle

in M and making it transverse to the Vi's subject to the further condition that if it

intersects ˜ at a point then it just enters S(Vi) at one point and runs downs a fibre

and out the other end. The tangent bundle of M restricted to this circle is oriented

iff it crosses the Vi in an even number of points iff (i*(b),j*[SI]) = 1, where i*(b) is

the image of b in HI (M; Z/2Z) and j*[Sl] is .the image of the fundamental class of

the circle in HI (M;Z/2Z). Since wl(M) also has this property, i*(b) = wl(M) as

claimed. If we act on this orientation by c E HO (M - V; Z/2Z), the new element

in HI (M, M - V; Z/2Z) is just b + 6*(c), where 6*(c) is the image of c under the

coboundary HO (M - V; Z/2Z) -4 HI (M,M - V; Z/2Z).

Now suppose that M - V has an orientation which does not extend across any

component of V. The bfor this orientation has a -1 in each summand, and is hence

the image of the Thom class. Therefore V is dual to wI(M).

Kirby & Taylor: Pin structures on low-dilnensional manifolds ]95

Next suppose that V is dual to wl(M). Then wl(M) reBtri(˜ts () to M -

V, and hence M - V is orientable. Fix one such orientation and consider the

corresponding b. Since both b and the image of the Thorn class hit, 11)1, we can find

c E HO (M - V; Z/2Z) so that b+ 8*(c) is the image of the Thorn class. If we alter

the given orientation on M - V by c, we get a new one which does not extend across

any component of V .â€¢

There is also a "descent of structure" result here.

Proposition 2.3. Given Mrn, the Poincare dual to WI (M) is an orientable (m -1 )-

dimensional manifold vm-l. There is an orientation on M - V which does not

extend across any component of V and this orients the boundary of a tubular

neighborhood ofV. This boundary is a double cover oiV and the covering transla-

tion is an orientation preserving free involution. In particular, V is oriented. Recall

that a E HO (M; Z/2Z) acts simply transitively on the orientations of M - V whicb

do not extend across any component oiV. Hence a acts on the set of orientations of

V by taking the image of a in HO (V; Z/2Z) and letting this class act as it usually

does.

Remark. If V has more components than M, not all orientations on V can arise

from this construction.

Proof: Suppose there is a loop ,\ in V which reverses orientation in V. If the normal

line bundle v to V in M is trivial when restricted to '\, then ,\ reverses orientation in

=1 (mod 2); but ,\.V = 0 since v is trivial over A, a contradiction.

M also, so ,\.V

=0

If viA is nontrivial, then A preserves orientation in M so A.V (mod 2); but

'x.V = 1 since v is nontrivial, again a contradiction. So orientation reversing loops

A cannot exist.

Another proof that V is orientable: As we saw above Wl(V) = i*(Wl(M), where

i: V c M. Since TMlv = Tv E9 v, it follows easily from the Whitney sum formula

that Wl (V) = O.

We now continue with the proof of the proposition. Let E he a tubular neigh-

borhood of V and recall that HI (E, BE; Z/2Z) is HO (V; Z/2Z) by the Thom iso-

morphism theorem. By Lemma 2.2 each component of BE can be oriented so that

the orientation does not extend across E. Clearly BE is a double cover of V classified

by i*(Wl(M)). Since V is orientable, the covering translation must be orientation

preserving and we can orient V so that the projection map is degree 1. It is easy to

check the effect of changing the orientation on M - V which does not extend across

any component of V .â€¢

We continue this discussion for the 2-dimensionaJ cohomology classes W2 and

'(02 + w˜. Again we need a lemma which enables us to tell if a codimension 2

submanifold is dual to one of these classes. We have

Theorem 2.4. Let M be a paracompact manifold, with or without boundary. Let

F be a codimension 2 submanifold of M with finitely many components and with

Kirby & Taylor: Pin structures on low-dimensional manifolds

196

aM n F = aF. Then F is dual to W2 + w˜ iff there is a Pin- structure on M - F

which does not extend across any component of F. Furthermore HI (M; Z/2Z) acts

simply transitively on the set of Pin- structures which do not extend across any

component of F. There are similar results for Pin+ structures and Spin structures.

Proof: The proof is rather similar to the proof of the previous result. First, let

F be a codimension 2 submanifold of M with i: F -+ M denoting the inclusion.

Let (D(Vi)' S(ViÂ») denote the disk, sphere bundle tubular neighborhoods to the

components of F. Suppose M - F has a Pin- structure. (The proof for Pin+ or

Spin structures is sufficiently similar that we leave it to the reader.) From Lemma

1.6, each S(Vi) inherits a Pin- structure. Define b E H2 (M, M - F; Z/2Z) ˜

ffiH2 (D(Vi), S(Vi); Z/2Z) ˜ ffJZ/2Z on each summand as 1 if the Pin- structure

on S(Vi) extends across D(Vi) and -1 if it does not. The class b hits w2(M) in

H 2 (M;Z/2Z). To see this, let j:N -+ M be an embedded surface which either

misses an Fi or hits it in a collection of fibre disks. As before (i*(b),j*[N]) is 1 if

TMIN has a Pin- structure and is -1 if it does not, since a bundle over a surface

with a Pin- structure over N - Jl D2 such that the Pin- structure does not

extend over the disks has a Pin- structure iff there are an even number of such

disks. Since w2(M) has the same property, i*(b) = w2(M).

Now HI (M - F; Z/2Z) acts simply transitively on the Pin- structures on

M - F and, for c E HI (M - F; Z/2Z), the new b one gets is b + <5*(c). The proof

is now sufficiently close to the finish of the proof of Lemma 2.2 that we leave it to

the reader to finish.â€¢

There is also a "descent of structure" result in this case, but it is sufficiently

complicated that we postpone the discussion until Â§6.

There are two cases in which we can show a "descent of structure" result for

PinÂ± structures. As above, given M we can find a submanifold V dual to wl(M).

We can then form V rtl V which is the submanifold obtained by making V transverse

to itself. If v denotes the normal bundle to V in M, then the normal bundle to V rh V

in V is naturally identified with vi v rh v and hence the normal bundle to V rtl V in

M is naturally identified with vlvrtlvffJvlvrhv' Since V is orientable, 2.3, vlvrtlv is

isomorphic to det TMlvrtlv' Hence by Lemma 1.7, a PinÂ± structure on M induces

one on VrhV after we identify vlvrtlv with detTMl vrnv ' If we choose the other

identification, the structure on V m changes by twice WI (M) restricted to V m

V V:

i.e. the final structure on V rh V is independent of the identification.

Theorem 2.5. The function above

,is a natural function using the map, HI (M; Z/2Z) -+ HI (VrtlV; Z/2Z), induced by

the inclusion. If VI rh VI is another choice then there is a dual to WI, W C M x [0, 1]

which is V at one end and VI at the other, so that W rtl W can be constructed

Kirby & Taylor: Pin structures on low..dhncilsiollul munifolds 197

[rlu,ll

as a PinT bordism between the two PinT structures. TJJt1 rnA]' inducnlf tI,

homomorphism of bordism theories

for any CW complex or spectrum X.

Proof: The naturality result follows easily from the naturality result in Lelnma 1.7.

The first bordism result follows easily once we recall that TO(t) ˜ JÂ«Z/2Z, 1) so

I-dimensional cohomology classes in M are the same as codimension 1 submanifolds

up to bordism in M X [0,1]. The bordism result is also not hard to prove.â€¢

For another example of "descent of structure" , we consider the following: given

any manifold, Mm , the dual to WI (M) is a codimension 1 submanifold vm-I. Since

V is orientable, Proposition 2.3, we are in the northwest situation of Corollary 1.15

and V receives a pair of Spin structures. Let (n˜in+)o denote the subgroup of

n˜in+ consisting of those elements so that the two Spin structures on V are bordant.

It is not hard to see that if the two structures are bordant for one representative in

n˜in+, then they are for any representative. Moreover, it is easy to check that the

induced map is a homomorphism:

Lemma 2.6. There is a well-defined homomozphism

]. Pin

[nw1Â· (n m +) 0 ˜ nm-l

Spin

Remark. It is not difficult to see that (n˜in+)o contains the kernel of the map

[nw˜] since any such element has a representative for which the normal bundle to

V is trivial. For such a V, we see a Spin bordism of 2 Â· V to zero, so V and - V

represent the same element in Spin bordism. Moreover, the cohomology class by

which we need to change the Spin structure is the zero class.

We conclude this section with some results we will need later which state that

different ways of inducing structures are the same.

The first relates structures (Spin or Pin:!:) and immersions. Given an immer-

sion f: N ˜ M the derivative gives a bundle map between the tangent bundles and

so we can use it to pull structures on M back to N. The induced map on structures,

denoted f*, is natural in the technical sense defined earlier. Suppose we have an

embedding M o X Rl C M. Let No = f-l(Mo) and note that there is an embedding

No x Rl C N so that f restricted to No X Rl is 9 X id where g: No ˜ M o is also

an immersion.

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

Lemma 2.7. The following diagram commutes

,. 'PinÂ±(M)

PinÂ±(N) -..-...+

ls˜

sNl

g.

----.

PinÂ±(No) 'PinÂ±(Mo )

where we orient R 1 and Lemma 1.6 gives us the natural map 8 Mas the composite

s

'PinÂ±(M) -. PinÂ±(Mo X R t )---+'PinÂ±(Mo) with a similar definition for S'tv. There

is a similar result for Spin structures.

Proof: We can easily reduce to the case M = M o X Rl. The required result can

now be checked by choosing transition functions on M o and extending to transition

functions for all the other bundles in sight, The two bundle we want to be isomorphic

will be identical.â€¢

The next result relates double covers and Pin+ structures. Let M be a manifold

with a Spin structure, and let x: 1rI(M) ˜ Z/2Z be a homomorphism (equivalently,

x E HI (M; Z/2ZÂ». Let E be the total space of the induced line bundle over M.

By Lemma 1.7, E has a natural Pin+ structure induced from the Spin structure

on M. Hence BE receives a Pin+ structure. Furthermore, BE is orientable and

we orient it by requiring the covering map 11": {)E -+ M to be degree 1. The Pin+;,i

structure and the orientation give a Spin structure on 8E. We can also use the oj r.

immersion 7r to pull the Spin structure on M back to one on BE. ;J

˜

˜

Lemma 2.8. The two Spin structures on 8E are the same.

.:˜

.J

Proof: Begin with the I-dimensional case. Here we are discussing Spin structures

1

on the circle. Suppose that the line bundle is non-trivial. Thinking of the circle

.˜

as the boundary of E, we see that it has the Lie Spin structure from Theorem 2.1. .'˜

Thinking of it as the connected double cover we also see that it has the Lie group

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