Curvature properties of exotic spheres

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1 Introduction

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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/var/www/vhost/, 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

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