Since Milnor’s discovery of exotic spheres [Mi], one of the most intriguing problems in Riemannian geometry has been whether there are exotic spheres with positive curvature. It is well known that there are exotic spheres that do not even admit metrics with positive scalar curvature [Hi]. On the other hand, there are many examples of exotic spheres with positive Ricci curvature (cf. [Ch1], [He], [Po], and [Na]) and this work recently culminated in [Wr] where it is shown that every exotic sphere that bounds a parallelizable manifold has a metric of positive Ricci curvature. This includes all exotic spheres in dimension 7. So far, however, no example of an exotic sphere with positive sectional curvature has been found. In fact, until now, only one example of an exotic sphere with nonnegative sectional curvature was known, the so-called Gromoll-Meyer sphere [GM] in dimension 7. As one of our main results we prove:

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Curvature and symmetry of Milnor spheres

In this paper we explore the geometry and topology of cohomogeneity one manifolds, i.e. manifolds with a group action whose principal orbits are hypersurfaces. We show that the principal group action of every principal SO(3) and SO(4) bundle over S^4 extends to a cohomogeneity one action. As a consequence we prove that every vector bundle and every sphere bundle over S^4 has a complete metric with non-negative curvature. It is well known that 15 of the 27 exotic spheres in dimension 7 can be written as S^3 bundles over S^4 in infinitely many ways, and hence we obtain infinitely many non-negatively curved metrics on these exotic spheres. A further consequence will be that there are infinitely many almost free actions by SO(3) on S^7, i.e. all isotropy groups are finite. These actions preserve the Hopf fibration S^3 -> S^7 -> S^4 but do not extend to the disc D^8. We also construct infinitely many such actions on the 15 exotic 7-spheres mentioned above.

Starting with Hirzebruch's A-genus we define a numerical invariantμ for certain (4k-1)-manifolds. We produce examples to show howμ can distinguish differentiable structures on certain topological manifolds.

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An invariant for certain smooth manifolds

Braid theory and knot theory are related to each other via two famous results due to Alexander and Markov. Alexander's theorem states that any knot or link can be put into braid form. Markov's theorem gives necessary and sufficient conditions to conclude that two braids represent the same knot or link. Thus one can use braid theory to study knot theory, and vice versa. In this book we generalize braid theory to dimension four. We develop the theory of surface braids and apply it to study surface links. Especially, the generalized Alexander and Markov theorems in dimension four are given. This book is the first place that contains a complete proof of the generalized Markov theorem. Surface links are also studied via the motion picture method, and some important techniques of this method are studied. For surface braids, various methods to describe them are introduced and developed: the motion picture method, the chart description, the braid monodromy, and the braid system. These tools are fundamental to understanding and computing invariants of surface braids and surface links. A table of knotted surfaces is included with a computation of Alexander polynomials. The braid techniques are extended to represent link homotopy classes. Library of Congress Cataloging-in-Publication D a t a Kamada, Seiichi, 1964Braid and knot theory in dimension four / Seiichi Kamada. p. cm. — (Mathematical surveys and monographs ; v. 95) Includes bibliographical references and index. ISBN 0-8218-2969-6 (alk. paper) 1. Braid theory. 2. Knot theory. I. Title. II. Mathematical surveys and monographs ; no. 95. QA612.23.K36 2002 514.224—dc21 2002018274 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, P. O. Box 6248, Providence, Rhode Island 02940-6248. Requests can also be made by e-mail to reprint-permissionOams.org. © 2002 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. @ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at URL: http://www.ams.org/ 10 9 8 7 6 5 4 3 2 1 07 06 05 04 03 02

Braid and Knot Theory in Dimension Four

The theory of foliations of codimension greater than one

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On Pontrjagin classes and homotopy types of manifolds

https://www.jstage.jst.go.jp/article/jmath1948/10/1/10_1_29/_pdf

Homeomorphy classification of total spaces of sphere bundles over spheres.

© Publications mathématiques de l’I.H.É.S., 1981, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » (http:// www.ihes.fr/IHES/Publications/Publications.html) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

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On transverse foliations

In this note we shall introduce a new structure, called a spinnable structure, on a differentiable manifold. Roughly speaking, a differentiable manifold is spinnable if it can spin around an axis as if the top spins. Such structure originated in connection with the study of foliations (Lawson [1], Tamura [4], [5]), and the existence of a codimension-one foliation follows from a spinnable structure of a special kind. Definition 1. An m-dimensional differentiable manifold Mm is called spinnable if there exists an (m-2)-dimensional submanifold X satisfying the following conditions : (i) The normal bundle of X is trivial. (ii) Let X x D2 be a tubular neighborhood of X, then C= Mm -Xx Int DZ is the total space of a fibre bundle over a circle . (iii) Let p : C->Ss be the projection of , then the diagram

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Itiro Tamura

Papers

On Pontrjagin classes and homotopy types of manifolds

Homeomorphy classification of total spaces of sphere bundles over spheres.

On transverse foliations

Spinnable structures on differentiable manifolds

Every odd dimensional homotopy sphere has a foliaton of codimension one