Table Of ContentFIXED POINT INDICES AND
LEFT INVARIANT FRAMINGS
J.C. BeckeT and R.E. Schultz
.i Introduction.
Given a Lie group ,G a standard result of differential geometry
states that G has a trivial tangent bundle. In fact, the left in-
variant vector fields of G induce a specific isomorphism ~ between
the tangent space of G and a product bundle; this map si called the
left invariant framing of .G If G is compact, the pair (G,~) accord-
fr
ingly determines a bordism class [G,~] in the bordism groups ~ of
stably framed manifolds [29]. By a classical theorem of Pontrjagin,
these groups are isomorphic to the stable homotopy groups of spheres,
which will be denoted by ~.
Of course, the first question to ask is whether any of these classes
are nonzero if G is positive dimensional (if G is finite, one gets
the order of G in ~0~Z). The answer is yes; in fact, for some time
it has been well-known that [SI,~] and [$3,~] represent the Hopf
maps in 71 and 3 7 respectively. Furthermore, it is straightforward
to check that
[G,£ ]G x [H,~ ]H = [GxH,~GxH],
and therefore the results of [32] imply T ,2 T ,3 S3xS ,3 and S3xS3xS 3
with their left invariant framings are all nonzero in ~.
Since the stable homotopy groups of spheres are fairly well under-
stood in low dimensions, it is natural to ask if one can describe [G,~]
for other low-dimensional exampleS. This problem was first studied by
.L Smith [27] and later by N. Ray, .B Steer, and .R Wood [26, 28, 35]
for simple groups of rank .2 It turns out that additional nonzero
elements are obtained for G = SU (3),Sp(2),G .2 In addition, Smith
proved that S0(3) represented twice the Hopf map in ~3 (compare ]4[
and Theorem (6.3)).
For the most part, however, the known results have not suggested
that infinitely many of the classes [G,~] are nontrivial when dim
G > .0 A strong negative result in this direction, due to Atiyah and
Smith [4], states that the Adams d- and -e invarian~ ~ [G,~] fre-
quently vanish. More generally, K.-h. Knapp has shown that [G,~] has
filtration ~ rank G in the Adams-Novikov spectral sequence for ~
Partially supported by NSF Grants MPS76-09180 and MPS MPS76-08794
respectively.
(we would like to thank Steer and Knapp for informing us of this result).
This and other, informal evidense suggests the following negative state-
ment:
CONJECTURE. For some positive interger ° r (hopefully o - r i0 < or so),
all compact Lie groups of rank o e r have [G,~] = .0
In this paper we shall study [G,~] using an idea of Th. BrScker,
relating this class to the fixed point index of a fiber preserving self-
map over G in the sense of A. Dold [12,13]. By definition, this fixed
point class lies in the unreduced stable cohomotopy of G (i.e.,
~s °(G) = {G+'S°})' and it corresponds to a class in nS~ because the fixed
point index vanishes off a neighborhood of the identity (see Section 3
for the details and a proof of BrScker's result). The multiplicative
structure of ~(G) is very helpful for studying this fixed point
index. There is a ring ~(G)--which maps onto the Burnside ring of G--
and a homomorphism :I U(G) ÷ ~(G) such that the desired fixed point
index is trivial to describe using I (this is done in Section 4).
In particular, if G acts transitively on a sphere, the index we want
is a polynomial in certain J-homomorphisms associated to G; the ex-
plicit formulas are given in (5.3) and (5.4). Using this information
and some standard homotopy-theoretic machinery, we are able to evaluate
[G,~] in some new cases as follows:
(I) [U(2n),~] = 0 for all n (4.7a)
(2) [SO(n),~] = 0 for n = 4, 6, 7, 8, 9
((4.8a), (7.3), (7.4), (7.13), (8.1))
(3) [Spin(n),£] = 0 for n = ,7 ,8 9
((7.13), (8.1))
(4) [SU(4),G] = ,rl £ ~15 (6.5)
(5) [Sp(3),~] = 03 + ~n C ~21 (9.1)
The elements in (4) and (5) are discussed in [32] and [19] respectively.
Of course, [U(2),~] = 0 is trivial because ~4 = 0. More significantly,
B. Steer has informed us that he had calculated [SU(4),~] using the
methods of [28], and G. al-Sabti has informed us that he had evaluated
[SO(4),~] independently. The above calculations use a variety of
techniques and are presented mainly in Section 6-9. In principle, our
methods allow an evaluation of [SU(5),~]; however, this would require
some relatively formidable calculations in the BP(2 ) Adams spectral
sequence, and therefore we shall merely outline the approach in Section
i0. Work done by Steer seems to suggest that [SU(5),~] = .0 At any
rate, we can easily add one further calculation to the list:
(6) [U(5),~] = 0 (10.4).
Our methods by themselves lead to no new information on other
natural framings of .G However, it is possible that further informa-
tion can be obtained by combining our results with the methods of Ray,
Steer, and Wood.
Acknowledgements: We would like to thank .G al-Sabti, N. Ray, .B Steer,
G. Walker, and R. Wood for conversations and correspondence dealing with
their work related to the computation of [G,~]. We would also like to
thank A. Dold for showing us the outline Br~cker's proof (which is dif-
ferent from the one presented here.)
.2 Fixed point index.
We suppose given a smooth fiber bundle p: E ÷ B with B and E
always assumed to be compact smooth manifolds and :f E + E a fiber
preserving continuous map. Let F denote the fiber. Then, as in
[12,13] there is a fixed point transfer T(f) which is an S-map
(2.1) z(f): + ÷ E B .+
The fixed point transfer has the following properties.
(2.2) the composite ]+B(~ Tp*>~(E+) )+E(~>*)f(T
is multiplication by the Lefschetz number A(f) f~_o [: F + F, the re-
striction of f to fiber.
(2.3) If f,f': E ÷ E are fiber preserving maps wi~f = 'f through
fiber preserving maps then T(f) T(f'). =
(2.3) Given smooth fiber bundles PI: E1 ÷ B and P2:E2 ÷ ,B
fiber preserving maps fl: E1 ÷ El' f2:E2 ÷ E2' and a fiber homotop¥
equivalence :g 1 2 ÷ E E such that f2 g = gfl then
gT(fl) = T(f2)
(2.4) Given p: E + B and fiber preserving map :f E + E and
a smooth map h: X ÷ B we have the induced diagram
h
X B-
and fiber preserving mar f: E ÷ E K b f(x,e) = (x,f(e)), Then
(2.5) (Product property) Given Pl: El ÷ B and
P2: E2 ÷ B together with fiber prescrying maps fl: E1 ÷ B1
and :2f E2 + B2 we have the product Pl x P2: ElXEx ÷ BIXB2
and fiber preserving map fl x f2: 1 1 2 x + x E E E E .2
Then z(flxf2 ) = ~(fl ) x z(f2 .)
(2.6) (Excision property) Suppose that p: E ÷ B is the union
of tamely intersecting subbundles PI: E1 ÷ B and P2: E2 ÷ B
(i.e. 1 2 g E E is a smooth subbundle). Let El2 1 = 2 ~ E E and let
PI2: El2 ÷ B denote the restriction of p. Let if: 1 ÷ E E,
i2: 2 ÷ E E, and i12: El2 ÷ E denote the inclusions. Suppose that
:f E ÷ E is a fiber preserving map such that f(Ej) c Ej, j = i, 2.
Let fl: E1 ÷ El' f2: E2 ÷ E2' and f12:El2 ÷ El2 denote the
restriction of .f Then
r(f) = T(f I) + ~(f2 ) - T(fl2).
(2.7) If f: E ÷ E is fixed point free then ~(f) = .0
These properties are established in [5], [12], and [13] with the
exception of the excision property whose proof will be omitted. How-
ever, compare [12, (2.7)].
Let h denote a reduced multiplicative cohomology theory. The
fixed point index of a fiber preserving map :f E ÷ E over B is
(2.8) I(f) = ~(f)*(1) ~ h°(B+).
We have I(f) = A(f) + I(f) and we call I(f) the reduced fixed point
index of f. We will mainly be interested in the case where h is
(reduced) stable cohomotopy theory Ws"
3. Group action!.
Suppose that G is a compact Lie group acting smoothly on a com-
pact smooth manifold M. We then have an associated fiber preserving
map G x M ~ ~ G x M
where ~(g,x) = (g,gx) and P is projection onto the first factor.
We denote the fixed point index of ~ in ~(G) by IG(M).
The following properties are easily derived from the properties
of the transfer listed in section .2
(3.1) If f:M ÷ ' M is an equivariant homotopy equivalence then
IG(M ) -- IG(M, .)
(3.2) If h:G' + G is a homomorphism and M is a G-manifold
then h~(IG(M)) = I G,(M), where ,G acts on M through h.
(3.3) If M. is a G.-manifold, i = 1,2, then
-- 1 1
IGIXG 2(Mlx~42) = IGI(MI) x IG2(M2).
(3.4) If M is the union of tamely intersecting G-invariant sub-
manifolds 1 M and 2 M (i.e. Mlf~ 2 is a submanifold) then
IG(M) = IG(M I) + IG(M z) - I¢(_MI n M2).
Now consider the action of G on itself through left multiplica-
tion (which has a fixed point index IG(G ) g T°(G+)). Fix an orienta-
tion ~ of G and let ~ denote the left invariant framing of the
tangent bundle of G. We then have the element [G,~,~] s ~s°(SN),
N = dim G, and we will relate this element to IG(G ). Choose a coordi-
nate neighborhood U of the identity element of G and identify U
with N R in an orientation preserving way. Then G/G-U N = S and we
have an exact sequence
0÷ o N)22
The following result is due to T. Brbcker.
(3.5) THEOREM. jc~([g,a,Z]) = IG(G)-
Proof. We first recall the construction of the fixed point index
of a fiber preserving map
f
as given in [ 5, section 9]. Let ~:E ÷ B s x R be an embedding
homotopic to P and let B denote its normal bundle. Let ~ denote
the bundle of tangents along the fiber. Then I(f) is represented by
+ B AS "s ~#~> EB (l,f) 2> (E2)~I(~) d# ~> E~ ~ E~ A s S s. 2> S
Here 2 E is the fiber square, d:E ÷ 2 E is the diagonal embedding,
~I is projection onto the first factor, and (l,f) is a bundle map
covering (l,f):E + E .2
Now in the situation at hand,
GxG .... ~ 2> GxG
the above sequence of maps can be simplified slightly. Let ~:G s ÷ R
be an embedding with normal bundle v. Let T denote the tangent
bundle of G. Then IG(G ) is represented by
SA+G s I~G+AGv ~G+AGv d#~G~G+AsS ~sS
where ~ (g,v~) = (gg,v~) and once again, d:G ÷ GxG is the diagonal
embedding.
Recall that the left invariant framing ~:T(G) ÷ GxR N of G is
given by ~(vg) = (g'(R~-l)*(vg))'-I ~ where R G: ~ G denotes right
g-IRN
multiplication by g . Here we ident~fy with the tangent
• .
space eT )G( in an orientation preserving way.
We will take as a coordinate neighborhood U of the identity el-
ement the image of an c-disk DE(T (G)) e under the exponential map
ExPe:Te(G ) ÷ G relative to a G-invariant metric (we then identify
Dc(Te (G)) with Te(G) N = R by radial extension).
Now consider
G+AS s IZ~# .~ G+AG v _ ~ ~ G+AG v
sNAGV~ ~ G ~-I@ k T~_~ G + AS s ss. +
The commutativity of the right hand square is a result of the commuta-
tive diagram
n
GXEv .... ~>GxE v
IExPeXl I 'd
Te(G)xE v ~-I~I~ET~E v
where d'(vg,wg) = (Expg(Vg),Wg).
Tracing the upper sequence of maps yields IG(G ) whereas tracing
the lower sequence yields j~([G,~,~]).
.4 The ring U(G).
The set of equivariant homotopy classes of compact smooth G-mani-
folds, for fixed ,G can be given the structure of a semi-ring under
the operations of disjoint union as addition and cartesian product as
multiplication. Let ~(G) denote the ring completion of this semi-ring
modulo the ideal generated by elements of the form
(4.0) [M] - [M1] - [M2] + [M 1 N M2]
when M is the union of tamely intersecting G-invariant smooth submani-
folds 1 N and M .2
According to properties (3.1),~3.3) and (3.4) we have a ring homo-
morphism.
(4.1) Index: U(G) + ~(G +) by [M] ÷ IG(M).
If H is a subgroup of G let (H) denote its conjugacy class
and if M is a compact smooth G-manifold let M(H ) denote the sub-
space of M consisting of points x whose isotropy subgroup x G is
in (H). Let ~(H) denote the one point compactification of M(H)"
The triangulation theorem of C.T. Yang [36] or the existence of
o
G-invariant Morse functions [34] implies that M(H)/G has the homotopy
type of a finite CW Gomplex.
The ring U(G) is very similar to the Burnside ring ~(G) of
tom Dieck; in fact, there is a ring epimorphism from U(G) to ~(G),
taking the U-class of a manifold to its Burnside class. The follow-
ing result, which will play a central role in our work, parallels the
additive decomposition of ~(G) in [I0,§ 2]:
(4.2) THEOREM. The ring ~(G) is free abelian on the ,h.0m£geneous
spaces {[G/H]I (H)<G}. Furthermore, the class of a smooth G-manifold '
satisfies the identit K
(4.3) ]M[ = ~(H) x(M(H)/G' ~) [G/HI, the sum taken over all conjugacy
classes of isotropx subgroups.
Proof It suffices to prove (4.3) for ]M[ because the maps
~H:[ ] M + ×(~(H)/G, ~) are well defined additive morphisms U(G) + ,Z
and the induced map ~ into ~(H)Z has the above-mentioned free abelian
group as its image. In fact, there is an additive map
~: Image ~ ÷ U(G) taking ~n(H ) G/H to ~n(H ) [G/H], and (4.3)
reduces to the assertion ]M[ = ~[M].
If M is zero-dimensional, then (4.3) is true essentially by a
counting argument, so we preceed by induction on dim M. Construct a
smooth G-handle decomposition of M using an invariant Morse function
[34,1oc.cit.], and proceed by induction on the number of G-handles
(the case with no G-handles being trivial). Therefore we assume that
M = N U ~, where ~ is a smooth G-handle GXHD(V)xD(W ) (with rounded
corners), N A ~ is G-diffeomorphic to GXHS(V)xlxD(W) (rounded
corners again), and (4.3*) holds for N. But (4.3") holds for
because it is G-homotopy equivalent to G/H, and (4.3~) also holds for
N g ~ because (i) N N • is G-homotopy equivalent to GXHS(V ) (ii)
the induction on demension shows that (4.3*) holds for GXHS(V .) It
follows that ~¢[P] = ]P[ for P = N, ~, N n ~, and the identity
~}[M] = ]M[ is a consequence of this and (4.0).
(4.4) COROLLARY. If G acts freely on ' M then ]M[ = x(M/G)[G].
(4.5) THEOREM. If H is a proper closed subgroup of G and
dim G/H > 0, then x(G/H)[H,~,~] = .0
Proof. Let H act on the left of G and consider the element
[G] s ~(H). By the above corollary [G] = X(G/H) [H]. Applying the
index homomorphism (4.6) IH(G).= x(G/H)IH(H).
Let X:H ÷ G denote the inclusion and let U be a coordinate neigh-
borhood of the identity element of G. Since H is proper and dimG/H>0,
we have a homotopy factorization
H ~G
G-U
Now
)G(HI = ~*(IG(G))= .))G(GI(*i~-~
Observe that i~(IG(G)) = 0 since the fiber preserving map
GxG ~-~GxG ~
is fixed point free over G-U. Consequently IH(G) = 0. Then from
(4.6) and theorem (5.5) we have
0 = x(G/H) IH(H ) = x(G/H)j*([H,~,~])
Finally, since j* is a monomorphism, x(G/H) [H,~,~] = .0
Hereafter we abbreviate [G,~,~] to [G,~]. As a first application
of these results we have the following.
(4.7) THEOREM.
)a( [U (2n) ,£] = 0
(b) 2 [SO(Zn),£] = 0
Proof. U(n) ~ SO(2n) and x(SO(2n)/ U(n)) = 2 n-I Therefore
2n-l[u(n),Z] 0. On the other hand we have U(n) ~ SU(n+I) by
=
A
[~ Ate0d I I -
and SU(n+I)/U(n) = CP n. Since ×(CP n) = n+l, we also have
(n+l) [u(n),d~] = .O
For statement )b( we have SO(2n) ~ SO(2n+l) and
×(SO(2n+l)/SO(2n)) = .2
Of course, the homogeneous spaces G/H with ×(G/H) ~ 0 have
been classified in principle by Borel and de Siebenthal [8], and one
can derive additional results resembling (4.7) from their work. For
our purposes, the following particular examples are necessary:
(4.8) THEOREM.
(a) [so(4),£] = 0
(b) 3[Spin(9),(cid:127)] = 0
Proof. Take the well kno~hinclusions SO(4) ~ G 2, Spin(9)~ F4,
which have x(G2/SO(4) ) = x(F4/Spin(9))= .3 Thus 3[SO(4),Z] =
3[Spin(9),~] = .0 But dim SO(4) = 6 and 76 is 2-torsion [32], and
therefore [SO(4),~] must be zero.
Problem. What are the kernel and image of the map IG: U(G) + ~r°(G)?
In particular, is G I always onto? Since ~s °(G) ~ Z@ finite group,
the kernel must be quite large in general.
.5 Relation with the J-homomorphism.
Given a subgroup H of G we have a "restriction" map
i*:U(G) ~ U(H).
There is also an "induction" map
i, : it(H) + LEG)
by i,([M]) = [GXHM .] Here GXHM is the quotient of GxM obtained by
identifying (g,x) with (gh-l,hx), h c H, and G acts on GXHM through
the first factor. The composite
U(G) i* f'~- II(H) i~ -~"i U(G)
is given by
(5.1) i,i*([M]) = [G/H] ]M[
since the map MHXG + G/H x M given by [g,x] ÷ ([g],gx) is a
01
G-diffeomorphism.
Formula (5.I) can be restated in the following form which is
often useful for computational purposes:
(5.1A) Suppose that i*[M] = ~(L)<H C(L)[H/L] in II(H). Then
[G/HI • [M] = [(L)<H CL[G/L]"
Suppose now that V is an (orthogonal) G-module of dimension n.
The action of G on the unit sphere S(V) of V,
G x S(V) + S(V)
determines, by the Hopf construction, a map
Jv:G n A n. ÷ S S
We may view JV as an element of ~°(G). On the other hand there is
the fixed point index I G(S(V)) c ~s °(G).
I G(S(V)) = V n even
JV' n odd.
The proof, which amounts to a straightforward comparison of the
two maps, will be omitted.
We write JR s "g°(SO(n)), JC ° c n (U(n)) and JH c ~~Os (Sp(n))
for the classical J-homomorphisms (i.e. the Hopf construction applied
to the standard action).
(5.3) THEOREM
(a) [SO(2n),~] = JRn(z - JR )n-I
(b) [SO(2n + 1),£] = JRn(z - JR )n
(c) [U(n),£] = Jc n
(d) [SU(n),£] CJ n-1
=
(e) [Sp(n),N] = JH n
Proof. eW first take up the case of U(n). Consider the composite
(U(n) i* > (U(n-k)) i. ~ (U(n)),
0 ~- k < n, which we apply to U (n) /U (n-1) = S n-1 The action of
U(n-l) on S 2n-I obtained by restriction has 2 isotropy subgroups
U(n-k-l) and U(n-l). In the first case
x(~Zn-1
(U(n_k_l))/U(n-l),~) = x(s2k,pt.) = 1
In the second case
x z~-1 (~ (U(n_l))/U(n-l),~) = x(S 2k-l) = .0
Thus, by theorem (4.2)
i*([s2n-1]) = [SZ(n-k)-l],
