# Curvature properties of exotic spheres

## 1 Introduction


## 3 Homotopy spheres with positive scalar curvature

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

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

Proposition 7.1. The $\alpha$$\alpha$-invariant of a homotopy sphere $\Sigma$$\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$$\Sigma$ be an $n$$n$-dimensional homotopy sphere with $n\ge 5$$n\ge 5$ then $\Sigma$$\Sigma$ admits a metric of positive scalar curvature if and only if $\alpha(M)$$\alpha(M)$ is trivial.

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

On the other hand, Stolz in [Stolz1992] proved that a simply connected closed spin manifold of dimension $n\ge5$$n\ge5$ admits a metric of positive scalar curvature if its $\alpha$$\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.