A Note on the Existence of Exotic 7-spheres
نویسنده
چکیده
Here’s the game plan: in Section 2 we’ll examine some properties of smooth 8-manifolds. Then in Section 3 we shall consider the disk bundle of a certain vector bundle (to be constructed in Section 4), the total space of which has boundary homeomorphic to the 7-sphere. Assuming it’s in fact diffeomorphic, we can construct a smooth 8-manifold from the disk bundle that fails to support the properties from Section 2, giving us a contradiction.
منابع مشابه
Sasakian Geometry and Einstein Metrics on Spheres
This paper is based on a talk presented by the first author at the Short Program on Riemannian Geometry that took place at the Centre de Recherche Mathématiques, Université de Montréal, during the period June 28-July 16, 2004. It is a report on our joint work with János Kollár [BGK03] concerning the existence of an abundance of Einstein metrics on odd dimensional spheres, including exotic spher...
متن کاملCurvature and Symmetry of Milnor Spheres
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...
متن کاملMilnor’s Construction of Exotic 7-spheres
In this paper, I will provide a detailed explanation of Milnor’s construction of exotic 7-spheres. The candidate manifolds will be constructed as total spaces of S3 bundles over S4, denoted Mh,l. The subset of these candidates satisfying the condition h + l = ±1 will be shown to be topological spheres by Morse Theory. A subset of these that do not satisfy (h−l)2 ≡ 1 (mod 7) will be shown to not...
متن کاملPositive Ricci Curvature
We discuss the Sasakian geometry of odd dimensional homotopy spheres. In particular, we give a completely new proof of the existence of metrics of positive Ricci curvature on exotic spheres that can be realized as the boundary of a parallelizable manifold. Furthermore, it is shown that on such homotopy spheres Σ the moduli space of Sasakian structures has infinitely many positive components det...
متن کاملAn Infinite Family of Gromoll-meyer Spheres
We construct a new infinite family of models of exotic 7-spheres. These models are direct generalizations of the Gromoll-Meyer sphere. From their symmetries, geodesics and submanifolds half of them are closer to the standard 7-sphere than any other known model for an exotic 7-sphere.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2007