Curvature properties of exotic spheres

From Manifold Atlas
Jump to: navigation, search

The user responsible for this page is Joachim. No other user may edit this page at present.

This page has not been refereed. The information given here might be incomplete or provisional.

1 Introduction

This page is in the construction process!!!

A homotopy sphere of dimension n is an oriented closed smooth manifold which is homotopy equivalent to the standard sphere S^n. A homotopy sphere is called an exotic sphere if it not diffeoemorphic to a standard sphere. General information about homotopy spheres (and exotic spheres) is given on Exotic spheres. One prominent question concerning the geometry of exotic spheres is: Given an exotic sphere, are there Riemannian metrics which fulfil specific positivity criteria? One typically considers the following three types.

1 Homotopy spheres with positive sectional curvature

2 Homotopy spheres with positive Ricci curvature

3 Homotopy spheres with positive scalar curvature

Hitchin (based on results in [Lichnerowicz1963]) proved that the so-called \alpha-invariant of a closed spin manifold provides an obstruction to the existence of a metric of positive scalar curvature on it (cf. [Hitchin1974]). The \alpha-invariant for a closed n-dimensional spin manifold (compare Spin bordism Invariants) is given as follows: Let Spin(M) the principal Spin(n)-bundle of M, and let S be obtained by adjoining the real Clifford algebra Cl(n) to Spin(M) using the left multiplication of elements in Spin(n) on Cl(n). The Dirac operator D then acts on the space of sections \Gamma(S). The kernel of D is called the space of (real) harmonic spinors. In case n=1 \mod 8 the space of harmonic spinors canonically has the structure of a complex vector space, while in case n=2 \mod 8 the space of harmonic spinors canonically carries the structure of a quarternionic vector space. The space of harmonic spinors determines an element in KO_n, the \alpha-invariant; and in particular, if the \alpha-invariant is non-trivial, the operator D must have a non-trivial kernel.

The \alpha-invariant of a homotopy sphere can be computed explicitely by the following means; therefore note that KO_n is isomorphic to \Zz/2 for n = 1 or 2 \mod 8.

Proposition 7.1. The \alpha-invariant of a homotopy sphere \Sigma is given by

\displaystyle  \alpha(\Sigma)  = \left\{\begin{array}{ccl}  \dim_{\Cc}Ker(D) \mod 2 &\quad\quad& \text{if $\dim(\Sigma)= 1 \mod 8$}\\ \dim_{\Hh}Ker(D) \mod 2 && \text{if $\dim(\Sigma)= 2 \mod 8$}\\ 0 && \text{in all other cases} \end{array}\right.

Theorem 7.2. Let \Sigma be an n-dimensional homotopy sphere with n\ge 5 then \Sigma admits a metric of positive scalar curvature if and only if \alpha(M) is trivial.

The fact that \alpha-invariant is an obstruction to the existence of postive scalar curvature follows from the Bochner-Weitzenböck formula, which yields the formula D^2 = \nabla^*\nabla + \frac{1}{4}scal(M) for the Dirac operator D. Here \nabla^*\nabla denotes the connection Laplacian which is a non-negative operator. Hence, if the scalar curature function scal(M) is strictly positive the operator D cannot have a non-trivial kernel, thus the \alpha-invariant must be trivial.

On the other hand, Stolz in [Stolz1992] proved that a simply connected closed spin manifold of dimension n\ge5 admits a metric of positive scalar curvature if its \alpha-invariant vanishes. The proof uses the surgery results for scalar curvature obtained (independently) in [Gromov&Lawson1980] and [Schoen&Yau1979], as well as a quite involved calculation within stable homotopy theory.

2 References

Personal tools