Gromov-Lawson conjecture

If $( M , g )$ is a metric of positive scalar curvature (cf. also Metric) on a compact spin manifold (cf. also Spinor structure), results of A. Lichnerowicz [a4] show that there are no harmonic spinors; consequently, the $\hat{A}$-genus of $M$ vanishes. M. Gromov and H.B. Lawson [a2], [a3] showed that if a manifold $M _ { 1 }$ can be obtained from a manifold $M _ { 2 }$ which admits a metric of positive scalar curvature, by surgeries in codimension at least $3$, then $M _ { 1 }$ admits a metric of positive scalar curvature. They wondered if this might be the only obstruction to the existence of a metric of positive scalar curvature in the spinor context if the dimension $m$ was at least $5$. (This restriction is necessary to ensure that certain surgery arguments work.) S. Stolz [a9] showed this was the case in the simply-connected setting: if $M$ is a simply-connected spin manifold of dimension at least $5$ (cf. also Simply-connected domain), then $M$ admits a metric of positive scalar curvature if and only if the $\hat{A}$-genus of $M$ vanishes. This invariant takes values in $\bf Z$ if $m \equiv 0$ modulo $8$, in $\mathbf{Z}_{2}$ if $m \equiv 1,2$ modulo $8$, in $2 \mathbf{Z}$ if $m \equiv 4$ modulo $8$, and vanishes if $m \equiv 3,5,6,7$ modulo $8$.

The situation is more complicated in the presence of a fundamental group $\pi$. Let $\mathcal{Z} _ { m} ^{\pi }$ be the Grothendieck group of finitely generated $\mathbf{Z}_{2}$-graded modules over the Clifford algebra $\operatorname { Clif } ({\bf R} ^ { m } )$ which have a $\pi$ action commuting with the $\operatorname { Clif } ({\bf R} ^ { m } )$ action. The inclusion $i$ of $\operatorname { Clif } ({\bf R} ^ { m } )$ in $\operatorname{Clif}( \mathbf R ^ { m + 1 } )$ induces a dual pull-back $i ^ { * }$ from $\mathcal{Z} _ { m + 1 } ^ { \pi }$ to $\mathcal{Z} _ { m} ^{\pi }$. The real $K$-theory groups of $\mathbf{R} \pi$ are given by:

$$\begin{equation*} K O _ { m } ( {\bf R} \pi ) = {\cal Z} _ { m } ^ { \pi } / i ^ { * } {\cal Z} _ { m + 1 } ^ { \pi }. \end{equation*}$$

J. Rosenberg [a6] defined a $K$-theory-valued invariant $\alpha$ taking values in this group which generalizes the $\hat{A}$-genus. It was conjectured that this might provide a complete description of the obstruction to the existence of a metric of positive scalar curvature; this refined conjecture became known as the Gromov–Lawson–Rosenberg conjecture.

The conjecture was established for spherical space form groups [a1] and for finite Abelian groups of rank at most $2$ and odd order [a8]. It has also been established for a (short) list of infinite groups, including free groups, free Abelian groups, and fundamental groups of orientable surfaces [a5]. S. Schick [a7] has shown that the conjecture, in the form due to J. Rosenberg, is false by exhibiting a compact spin manifold with fundamental group $\mathbf{Z}\oplus\mathbf{Z}\oplus\mathbf{Z}\oplus\mathbf{Z}\oplus\mathbf{Z}_3$ which does not admit a metric of positive scalar curvature but for which the $\alpha$ invariant vanishes. It is not known (1998) if the conjecture holds for finite fundamental groups.

References

[a1]	B. Botvinnik, P. Gilkey, S. Stolz, "The Gromov–Lawson–Rosenberg conjecture for groups with periodic cohomology" J. Diff. Geom. , 46 (1997) pp. 374–405
[a2]	M. Gromov, H.B. Lawson, "Spin and scalar curvature in the presence of a fundamental group I" Ann. of Math. , 111 (1980) pp. 209–230
[a3]	M. Gromov, H.B. Lawson, "The classification of simply connected manifolds of positive scalar curvature" Ann. of Math. , 111 (1980) pp. 423–434
[a4]	A. Lichnerowicz, "Spineurs harmoniques" C.R. Acad. Sci. Paris , 257 (1963) pp. 7–9
[a5]	J. Rosenberg, S. Stolz, "A "stable" version of the Gromov–Lawson conjecture" Contemp. Math. , 181 (1995) pp. 405–418
[a6]	J. Rosenberg, "$C ^ { * }$-algebras, positive scalar curvature, and the Novikov conjecture" Publ. Math. IHES , 58 (1983) pp. 197–212
[a7]	T. Schick, "A counterexample to the (unstable) Gromov–Lawson–Rosenberg conjecture" Topology , 37 (1998) pp. 1165–1168
[a8]	R. Schultz, "Positive scalar curvature and odd order Abelian fundamental groups" Proc. Amer. Math. Soc. , 125 : 3 (1997) pp. 907–915
[a9]	S. Stolz, "Simply connected manifolds of positive scalar curvature" Ann. of Math. , 136 (1992) pp. 511–540