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Some Group Actions on Homotopy Spheres of Dimension Seven and Fifteen Some Group Actions on Homotopy Spheres of Dimension Seven and Fifteen Michael W. Davis American Journal of Mathematics, Vol. 104, No. 1. (Feb., 1982), pp. 59-90. Stable URL: http://links.jstor.org/sici?sici=0002-9327%28198202%29104%3A1%3C59%3ASGAOHS%3E2.0.CO%3B2-Y American Journal of Mathematics is currently published by The Johns Hopkins University Press. Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/about/terms.html. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/journals/jhup.html. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. The JSTOR Archive is a trusted digital repository providing for long-term preservation and access to leading academic journals and scholarly literature from around the world. The Archive is supported by libraries, scholarly societies, publishers, and foundations. It is an initiative of JSTOR, a not-for-profit organization with a mission to help the scholarly community take advantage of advances in technology. For more information regarding JSTOR, please contact [email protected]. http://www.jstor.org Thu Oct 4 14:39:29 2007 SOME GROUP ACTIONS ON HOMOTOPY SPHERES OF DIMENSION SEVEN AND FIFTEEN 0. Introduction. This paper is based on the simple observation that every 4-plane bundle over s4admits a natural action of SO(3) by bundle maps and that every 8-plane bundle over S' admits a natural action of the compact Lie group G2by bundle maps. (These actions are easy to see once one remembers that SO(3) is the group of automorphisms of the quater- nions and that G2is the group of automorphisms of the Cayley numbers.) The study of these actions is closely connected to several well-known phe- nomena in differential topology and compact transformation groups. The most obvious connection is to Milnor's original construction of exotic 7-spheres as 3-sphere bundles over s4, 1261. Milnor proved that if the Euler class of such a bundle is a generator of H~(s~;Z), then its total space is homeomorphic to s'. He also defined a numerical invariant of the diffeomorphism type and used it to detect an exotic differential structure on some of these sphere bundles. Subsequently, Eells and Kuiper [12] in- troduced a refinement of this invariant, called the p-invariant. Using the p-invariant, they proved that the sphere bundle M: (Milnor's notation) is a generator for the group of homotopy 7-spheres. These constructions also work for 7-sphere bundles overs8, [32], and the manifold M:' is a gener- ator for bPI6,the group of homotopy 15-spheres which bound T-mani- folds. It is a routine matter to check that Milnor's arguments and their subsequent refinements work G-equivariantly where G = SO(3) or G2, and we shall do this in Sections 2 and 3. In particular, each sphere bundle with the correct Euler class is G-homeomorphic to an orthogonal action on S2" + l , where 2n + 1 = 7 or 15. Moreover, distinct sphere bundles have distinct oriented G-diffeomorphism types. The proof of the second fact uses an equivariant version of the p-invariant. As originally defined this invariant takes values in Q/Z. However, it is well-known that in the presence of a G-action, with S' C G, its value in Q is well-defined. Using Manuscript received May 16, 1980. *Partially supported by NSF Grant MCS 79-04715. American Journal of Mathematics, Vol. 104, No. 1, pp. 059-090 0002-9327/82/1041-0059 $01.50 Copyright O 1982 by The Johns Hopkins University Press 60 MICHAEL W. DAVIS this fact we shall show in Theorem 3.3, that the G-manifold (M:"", G) generates the infinite cyclic group (under equivariant connected sum) of oriented G-diffeomorphism types of smooth G-actions on homotopy (2rz + 1)-spheres all of which are G-homeomorphic to a standard linear action on s2"+'. A second connection is to the group c~.~of knotted 3-spheres in S' and to the work of Montgomery-Yang [28], [29] on semi-free circle actions on homotopy 7-spheres with fixed point set the standard 3-sphere. Let C,(SO(~))be the abelian group of oriented equivariant diffeomorphism classes of such actions. The results of Montgomery-Yang together with those of Levine [24] show that this group is isomorphic to z6.j.The iso- morphism is defined by sending (C7, SO(2)) E 6,(~0(2))to its "orbit knot" (C/S0(2), C~O(~)).According to Haefliger [16], [17], the group c~.~ is infinite cyclic. Restricting the SO(3)-action on M: to the subgroup SO(2) c SO (3) gives an action of SO(2) on M:. This action is readily seen to be semi-free and to have fixed point set s3.We shall show in Section 7 that the orbit knot of this action is actually a generator for zh.%nd hence, that (M:, SO(2)) is a generator for 6,(~0(2)).Thus, each of the Mont- gomery-Yang examples actually come from the restriction of a SO (3)-action. A third connection is to the construction of Gromoll-Meyer [IS] of a metric of non-negative sectional curvature on a exotic 7-sphere. They pro- duced an example showing that M: admits such a metric. In their ex- ample the group SO(3) X O(2) acts as isometries. The SO(3)-action on M: coincides with the one which we have been discussing above. (In fact, this paper grew out of an effort to explain the SO(3)-action of Gromoll- Meyer.) The O(2)-action also appears in a more general context. We shall show in Section 1 that every 4-plane bundle over s4(or every 8-plane bundle over s') admits a canonical action of GL(2, R) by bundle maps which commute with the action of SO(3) (or of Gz)The action of the sub- group O(2) C GL(2, R) on M: also coincides with the Gromoll-Meyer 0(2)-action. Actually the various connections these actions make with the above- mentioned constructions are only peripheral concerns of this paper. Our primary concern is to investigate the role these actions on sphere bundles play in the general theory of biaxial actions on homotopy spheres devel- oped in [2], [3], 151, [6], [7], [ll], 1131, [14], [19], [20]. The word "biaxial" refers to a smooth action of a group G C O(n) on a (212 + rn)-manifold such that the action is modeled on (i.e., locally isomorphic to) the linear GROUP ACTIONS ON HOMOTOPY SPHERES 61 G-action on the unit sphere in R" @ R" @ R"'+' via two times the stan- dard representation of O(n) plus the (m + 1)-dimensional trivial repre- sentation. If G is transitive on the Stiefel manifold V,,,z,then the orbit space B of such an action on a homotypy sphere C2"+"' is a contractible (nz + 3)-manifold with boundary and the fixed point set B, is a Z/2Z- homology m-sphere embedded in aB. Thus, the study of such actions is closely related to the study of knots in codimension 2. If G = U(n) C 0(2n), 12 r 2, or G = SU(12), 17 2 3, then the orbit space of C4"+"' is a contractible (m + 4)-manifold and the fixed point set is an integral homology m-sphere. Thus, such actions are related to knots in codimen- sion 3. For G equal to O(n), with n 2 2, SO(n), n 2 4, or Spin(7) C 0(8), any biaxial G-action on a homotopy sphere C2"+"' has the following nice property: 1) (C, G) is equivalent to a pullback of its linear model. This implies that 2) C equivariantly bounds a parallelizable manifold, and that 3) C is determined by its "orbit triple" (B, aB. B,). These facts are proved in 161, 1191, 1201. Since SO(3) is transitive on V3,2 and G2 is transitive on V7,2, one might expect that similar results hold in these cases; however, the proofs break down. For G = SO(3) or G2, our G-actions on sphere bundles are easily seen to be biaxial. However, for these actions all three of the above properties fail to hold. Moreover, if the sphere bundle is not the actual Hopf-bundle these actions do not extend to biaxial O(n)-actions, n = 3 or 7. There are similar nice results for biaxial actions of U(rz), n 2 2, and SU(n), 12 2 4; in particular, for such actions properties 1) and 2) hold. However, the corresponding results are false for SU(3). In fact, in Section 7 we shall show that the action of SU(3) C G2 on the sphere bundle M:' is a counterexample to both these properties. For G = SO(3) or G2, 1) can be replaced by the following property: 1)' (C, G) is equivalent to a pullback of the natural G-action on the quaternionic projective plane if G = S0(3), or on the Cayley projective plane if G = G2.
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