Some Group Actions on Homotopy Spheres of Dimension Seven and Fifteen

Abstract

0. Introduction. This paper is based on the simple observation that every 4-plane bundle over S4 admits a natural action of 50(3) by bundle maps and that every 8-plane bundle over 58 admits a natural action of the compact Lie group G2 by bundle maps. (These actions are easy to see once one remembers that SO (3) is the group of automorphisms of the quaternions and that G2 is the group of automorphisms of the Cayley numbers.) The study of these actions is closely connected to several well-known phenomena 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 54, [26]. Milnor proved that if the Euler class of such a bundle is a generator of H4(S4; Z), then its total space is homeomorphic to 57. 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] introduced a refinement of this invariant, called the it-invariant. Using the ,u-invariant, they proved that the sphere bundle M7 (Milnor's notation) is a generator for the group of homotopy 7-spheres. These constructions also work for 7-sphere bundles over 58, [32], and the manifold M15 is a generator for bP16, the group of homotopy 15-spheres which bound wN-manifolds. It is a routine matter to check that Milnor's arguments and their subsequent refinements work G-equivariantly where G 5SO(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,,+1, 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 it-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