Curvature properties of exotic spheres

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1 Introduction

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A homotopy sphere of dimension $n$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}n$ is an oriented closed smooth manifold which is homotopy equivalent to the standard sphere $S^n$$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.

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

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.

2 References

• [Gromov&Lawson1980] M. Gromov and H. B. Lawson, Spin and scalar curvature in the presence of a fundamental group. I, Ann. of Math. (2) 111 (1980), no.2, 209–230. MR569070 (81g:53022) Zbl 0445.53025
• [Hitchin1974] N. Hitchin, Harmonic spinors, Advances in Math. 14 (1974), 1–55. MR0358873 (50 #11332) Zbl 0284.58016
• [Lichnerowicz1963] A. Lichnerowicz, Spineurs harmoniques, C. R. Acad. Sci. Paris 257 (1963), 7–9. MR0156292 (27 #6218) Zbl 0714.53041
• [Schoen&Yau1979] R. Schoen and S. T. Yau, On the structure of manifolds with positive scalar curvature, Manuscripta Math. 28 (1979), no.1-3, 159–183. MR535700 (80k:53064) Zbl 0423.53032
• [Stolz1992] S. Stolz, Simply connected manifolds of positive scalar curvature, Ann. of Math. (2) 136 (1992), no.3, 511–540. MR1189863 (93i:57033) Zbl 0784.53029