Internat. J. Math. & Math. Sci.
VOL. 19 NO. 4 (1996) 825-827
825
EXOTIC STRUCTURES ON QUOTIENT SPACES OF -ACTIONS
L. ASTEY, E. MICHA and G. PASTOR
Centro de Investigac6n del IPN
Apartado Postal 14-740
M6xico 07000 D F
and
Instituto Tecnol6gico Aut6nomo de M6xico
Rio Hondo No
San Angel, M6xico 01000 D F
(Received May 4, 1994)
ABSTRACT. A correct version of some results by A Rigas regarding S acnons on S
the symplecnc group Sp2 with quotients exotic seven-spheres is presented
S and on
KEY WORDS AND PHRASES: Exotic spheres, principal bundles, group actions
1991 AMS SUBJECT CLASSIFICATION CODES: 57R55, 57S25
1.
INTRODUCTION
The present note is a result of our interest in finding exotic structures on 7-dimensional manifolds (cf
Guest and Micha [3], Astey, Micha and Pastor ]) and its purpose is to correct some mistakes that occur
in a paper by A Rigas [6] Our contribution is simply to provide the correct statement and a different
proof of the key corollary that appears on page 76 of Rigas [6], but we take the opportunity to restate
S and on the
several results of the paper which refer to the existence of free ,5,3 actions on S
symplectic group Sp2 with quotients exotic seven-spheres, which also appear incorrectly stated in that
paper
2. MAIN RESULTS
We begin by recalling some definitions and notation of Rigas [6] Principal S bundles over S
are classified by 7r3S which is naturally isomorphic to the group of integers Z Let P denote the
total space of the bundle corresponding to the integer n Similarly, the principal S’3 bundles over
We shall denote by E the total space of the bundle corresponding to
are classified by 7r6S’3
The
here is such that E1 RP2 Let /3, denote the pull-back of
E 7r6S’3
isomorphism
Z19.
’7
S
S Then, as a principal S’3 bundle,/, is classified by the composition
under the Hopf map
s
BS
fn denotes the map of degree n, and the rightmost arrow is the inclusion of the bottom cell
THEOREM. The bundles/5, and E,(,_I)/9. are isomorphic as principal S bundles over S
where
This theorem is the correct version of the corollary on page 76 of Rigas [6] The mistake leading to
the incorrect statement in Rigas [6] occurs in the calculation of the map f, o h, where the author fails to
826
I,
A,".;I’I.IY, !., MICIIA ANI)(I
IA.NI()R
terate correctly a formula of Hlton
[4] An alternative proof using a dfferent bundle decomposition s
presented n 3 below
It follows from the theorem that
and the trlwal bundle 5,7 5" are isomorphic only if n _= 0, 1, 9 or 16 mod 24
(a)
5’7 are isomorphic only ifn 2 or 23 mod 24
(b) zh,, and the canomcal bundle 5’p_,
In particular,
s not a trivial bundle This renders 4 of Rigas [6] invalid The theorem also allows us
to recnfiy the statements of two mportant results of Rigas [6] as follows
COROLLARY. There exist free acnons of 5" on 5"7 5": wth quotient the exotic seven-spheres
of Eells-Kuiper lnvariants 16, 40 and 48
COROLLARY. There exist free actions of S on 5’p_, with quotient the exotic seven-spheres of
Eells-Kmper invariants 2, 26, 34 and 42
3. PROOF OF THE THEOREM
As s shown in Rgas [6], 5" can be decomposed into two solid ton U 5"x D and
V D
S such that the restriction of the bundle/5,, to each toms s trivial Moreover, the transition
map
,,
.
Auv S’3 S
S
s gven by
Auv (Z, y)
xn l(yx 1)n ly
(n
1),
xyx-y
where the group structure of unit quaternions is understood on S Since the commutator
generates 76S (Hilton and Roitberg [5]) and since A factors through S 6, the theorem is a consequence
of the following result
PROPOSITION. The map A S x 5
5"3 given by A(x, y) x ’- (yx-)- ly- (,- is
(
)/2
homotopic to (xyx yWe first prove the following lemma
LEMMA. The maps xkyZx ky-Z and (xyx-ly-) kz are homotopic
PROOF. Consider the following commutative diagram
S xS
S
S xS
S
S
S6
where a(x,y)- (xk,yl), fl(x,y): xyx-ly -1, 7(X): x kl, p is the projection that collapses the 3skeleton, fkz is a map of degree kl, and w is the generator of 6S 3 But since S is an H-space,
o w Therefore,
homotopy compositions are biadditive (Whitehead [7], p 479), so w o fk
xkylx-ky
-
oa
o
(xyx-y-1) kt
We now prove the proposition by induction on n Let c xyx-ly -1 If we take k
the lemma we obtain xy- x- y cyxy- ix- Hence,
c- ycy-
(yxy- X- )ycy-
y(xy-lx-ly)cy -1
yc -1 cy -1
1,
y- c
thats, cy
Assume nowthat x(yx-1)y
c k()
Clearly, k(1)
1 But now
1 and
in
EXOTIC STRUCTURES ON QUOTIENT SPACES OF Sa-ACTIONS
827
-
(z,yz-,,y -)yx,,- (yx
Cny(xn-1 (yZ-1)n-ly-(n-1))y-1
c’yc(,-)y-1
cn+k(n-1).
Therefore, k(n)
[2]
[3]
[4]
[5]
n
+ k(n 1), that is, k(n) n(n + 1)/2
This proves the proposition
REFERENCES
ASTEY, L, MICHA, E and PASTOR, G., Diffeomorphism type of Eschenburg spaces, to appear
in Differenttal Geometry and tts Applications.
EELLS, J and KUIPER, N., An invariant for certain smooth manifolds, Ann. Mat. Pure Appl. 61}
(1962), 93-110
GUEST, M. and MICHA, E, Detecting exotic structures via the Pontrjagin-Thom construction,
Mathematika 41 (1994), 145-148.
HILTON, P, Suspension theorems and the generalized Hopf Invariant, Proc. London Math. Soc., 3
(1951), 462-492.
HILTON, P and ROITBERG, J, On principal S3-bundles over spheres, Ann. Math. 91} (1969), 91107
[6] RIGAS, A., S3-Bundles and exotic actions, Bull. Soc. Math. France 112 (1984), 69-92.
[7] WHITEHEAD, G, Elements of Homotopy Theory, Springer-Verlag, 1978.