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Diffeomorphisms of 1-connected manifolds

1967; American Mathematical Society; Volume: 128; Issue: 1 Linguagem: Inglês

10.1090/s0002-9947-1967-0212816-0

1088-6850

William Browder,

Mathematical Dynamics and Fractals

WILLIAM BROWDER(') 0. Introduction.In this note we shall describe how to adapt the methods of [1] to studying the group of pseudo-isotopy classes of diffeomorphisms of 1-connected manifolds.We shall use the notation and results of [1] freely.In the first four sections we shall discuss smooth manifolds and orientation preserving diffeomorphisms, but the results may be proved for piecewise linear (p.l.) manifolds and p.l. equivalences using the technique of [3] to adapt these proofs to the p.l. case.I am indebted to W.-C. Hsiang and J. Milnor for helpful comments.In §1 we consider the problem of when a given homotopy equivalence is homotopic to a diffeomorphism.We get a condition in terms of the normal bundle.Our theorem refines a result of Novikov [12].In §2 we consider the uniqueness question, and get results on pseudo-isotopy of diffeomorphisms.In §3 we use the results of §1 to construct diffeomorphisms which are homotopic to the identity but are not pseudo-isotopic to the identity, even piecewise linearly or topologically.The lowest dimensional example is on a 5-manifold, e.g., S2 x S3.Some similar examples have been obtained by Hodgson [9].In §4 we consider the group of diffeomorphisms 3>(M) (up to pseudo-isotopy) and deduce an exact sequence relating it to the group of tangential equivalences F(M).It follows that the kernel of the homomorphism of S¡(M) into F(M) is finite up to conjugacy.In §5 we deduce some facts about automorphisms of SnxSx.We show in particular that the group of pseudo-isotopy classes of p.l. automorphisms of Sn x S1 is Z2 + Z2 + Z2 for « ^ 5 (cf.[6]).If ¿* is a &-plane bundle we denote by F(f) the Thorn complex of £, T(£) = E(Ç)IEo(Ç), where E(¿¡) is the total space of the closed disk bundle of f, Zs0(f) is its boundary sphere bundle.We recall that if Mn is a smooth manifold embedded in Sn + k with normal bundle vk, k very large, then the natural collapsing map of Sn+k onto the Thorn complex T(v) = E(v)IE0(v) defines an element in irn + k(T(v)).Any bundle automorphism sends this element into another one obtained in this way.We will call the set of such elements the normal invariants of Mn, and denote the set by CM<=irn+k(T(v)).\fZ=XxI\J, Y, where/: Xx{0, 1} -> Y and if £ is a bundle over Z then there is a natural collapsing map q: T(i) -* T(0IT(£\ Y) and F(0/F(|| Y) = T((Ç\X) + e1) where e1 is the trivial line bundle.