Morris W. Hirsch

Immersions of manifolds

Immersions of an m-manifold in an n-manifold, n>m, are classified up to regular homotopy by the homotopy classes of sections of a vector bundle E associated to the tangent bundle of M.xa0 When N = Rn , the fibre of E is the Stiefel manifold of m-frames in n-space.

On imbedding differentiable manifolds in Euclidean space

Author(s): Hirsch, MW | Abstract: Assume n,k,m,q are positive integers. Let M^n denote a smooth differentiable n-manifold and R^k Euclidean k-space. (a) If M^n is open it imbeds smoothly in R^k, k=2n-1 (b) If M^n is open and parallelizable it immerses in R^n (c) Assume M^n is closed and (m-1)-connected, 1l 2m-n l n+1. If a neighborhood of the (n-m)-skeleton immerses in R^q, ag2n-2m, then the complement of a point of M^n imbeds smoothly in R^q.

Foliated Bundles, Invariant Measures and Flat Manifolds

Let e = (p, E, M) be a smooth (= C1) bundle. A foliation of e is a foliation i of E where leaves are transverse to the fibres and of complementary dimension. The leaves are required to be smooth, and their tangent planes must vary continuously on M but the foliation may be merely C0. Such foliated bundles occur in several geometrical situations. Suppose a manifold M has an affine connection of zero curvature, i.e., M is a flat manifold; then its tangent sphere bundle has a foliation. Or if a compact Riemannian manifold has negative sectional curvature, then its tangent sphere bundle is foliated. The normal sphere bundle of a leaf of a foliation is foliated. The purpose of this paper is to show that under certain assumptions about the holonomy homomorphism of a foliated bundle, there are strong restrictions on the homomorphisms induced by the bundle projection in real homology and cohomology. In some cases it follows that the bundle has a section. A typical application of our results is:

On the existence and classification of differentiable embeddings

ON THE EXISTENCE AND CLASSIFICATION DIFFERENTIABLE ANDRE OF EMBEDDINGS HAEFLICER and MORRIS W. HIRSCH (Receked 3 Jnnunry 1963) sl. INTRODUCTION LET A4 be a compact /c-connected differential to prove, under suitable restrictions on k and the Euclidean space R’“-“-’ (Theorem (2.3)), of embeddings of 1M in R *“-’ if 1M is orientable Theorems (2.1) and (2.2) which reduce the immersions, and then applying A particular THEOREM n-manifold without boundary. Our object is II, an existence theorem for embedding IV in and a classification theorem for isotopy classes (Theorem (2.4)). This is done by first proving embedding problems to questions involving the theory of immersions case of (2.3) is the following: (I. 1). If n > 4, M is embeddable Whitney class p- in R 2n- 1 if and only if its normal Stiefel- ’ vanishes. Massey [5, 6, 71 has shown power of 2. Thus we obtain: that if P-l # 0, then M is non-orientable TIIEOREM (I .2). If n > 4 and A4 is orientable, M is embedable in R2”-‘. This is also true if n = 3; see [4]. The case n = 4 is unsolved, connected. embeddable However, in R5. Smale has proved (unpublished) and ?I is a that every even if M is simply homotopy 4-sphere is It should be remarked that the existence Theorems (2.1) and (2.3) apply to both orientable and non-orientable manifolds, but the classification Theorems (2.2) and (2.4) apply only to orientable manifolds. (1.3). DEFINITIONS AND NOTATION. All manifolds boundary of a manifold X is 2X. considered here are differential. The We put X - IYX = int X. An immersion of an n-manifold X in Euclidean r-space R” is a differentiable map f: A’-+ R” of rank n everywhere. An embedding is an immersion which is l- 1. If f and g are immersions of X in X”, a regular homotopy connectingfto g is a differentiable homotopy F: X x I+ R” such that F, = f, F, = g, and each F, is an immersion. If in addition each F, is an embedding, then F is an isotopy.

Embeddings and compressions of polyhedra and smooth manifolds

Topology Vol. 4 pp. 361-369. EMBEDDINGS Pergamon Press, 1966. Printed in Great Britain AND COMPRESSIONS OF POLYHEDRA SMOOTH MANIFOLDS? AND MORRIS W. HIRSZH (Receiued 23 July 1965)

ANOSOV MAPS, POLYCYCLIC GROUPS AND HOMOLOGY

1. INTRODUCTION IT IS OFTEN desirable to compress a subset X of a manifold V into a submanifold Y’ by an isotopy of V. For example, V might be a Euclidean space R” and V’ might be p, with k much smaller than n. If X has some special properties ensuring that such a compression is always possible, then we conclude that X embeds in ti. In this article the problem of compressing X into the boundary of an s-cell is studied, where s = dim V. As applications several theorems are proved about embedding polyhedra and smooth manifolds in Eucli- dean space. The main theorems say that a compact polyhedron or smooth submanifold XC V compresses into the boundary of an s-cell provided (1) there exists a “Dehn cone” on X, and (2) V-X is highly connected. A Dehn cone on XC V is an embedding X x I c V with X x 0 = X together with a null homotopy of X x 1 in Y - X x 0. If V-X is sufhci- ently connected, a regular neighborhood theorem of Hudson and Zeeman, together with an engulfing theorem due to Zeeman and the author, provides an s-cell E c V with X c i?E. If V = R’, this implies the compressibility of X into R* -I or Ss -‘. First the piecewise linear (=PL) theory is developed, then the smooth case is reduced to the PL case. In the applications we take X c R4+’ and try to compress X into R4. There are three steps : (1) extend X to X x I c RQ+’ ; (2) choose the extension so that X x 1 3: 0 in RQ+’ - X; (3) prove P+’ - X is sufficiently connected for the Enguhing Theorem to apply. Step (1) is difficult in the PL case, so in the applications it is assumed as part of the hypothesis. In the smooth case, however, (1) is equivalent to the existence of a normal vector field on the smooth submanifold X. Step (2) is accomplished through algebraic topology, sometimes by luck-as when X is a smooth homology sphere-more usually by just assuming that the obstructions to a t This work was supported by the National Science Foundation grant GP-4035

On non-linear cell bundles

Many manifolds that do not admit Anosov diffeomorphisms are constructed. For example: the Cartesian product of the Klein bottle and a torus.

On piecewise regular n-knots

Annals of Mathematics On Non-Linear Cell Bundles Author(s): Morris W. Hirsch Source: Annals of Mathematics, Second Series, Vol. 84, No. 3 (Nov., 1966), pp. 373-385 Published by: Annals of Mathematics Stable URL: http://www.jstor.org/stable/1970452 . Accessed: 10/11/2014 23:00 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. Annals of Mathematics is collaborating with JSTOR to digitize, preserve and extend access to Annals of Mathematics. http://www.jstor.org This content downloaded from 128.104.46.206 on Mon, 10 Nov 2014 23:00:42 PM All use subject to JSTOR Terms and Conditions

The Imbedding of Bounding Manifolds in Euclidean Space

Annals of Mathematics On Piecewise Regular n-Knots Author(s): Morris W. Hirsch and Lee P. Neuwirth Source: Annals of Mathematics, Second Series, Vol. 80, No. 3 (Nov., 1964), pp. 594-612 Published by: Annals of Mathematics Stable URL: http://www.jstor.org/stable/1970665 Accessed: 24-08-2015 16:29 UTC Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at http://www.jstor.org/page/ info/about/policies/terms.jsp JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. Annals of Mathematics is collaborating with JSTOR to digitize, preserve and extend access to Annals of Mathematics. http://www.jstor.org This content downloaded from 128.104.46.206 on Mon, 24 Aug 2015 16:29:08 UTC All use subject to JSTOR Terms and Conditions