Gromov-Lawson conjecture

If 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 -genus of vanishes. M. Gromov and H.B. Lawson [a2], [a3] showed that if a manifold can be obtained from a manifold which admits a metric of positive scalar curvature, by surgeries in codimension at least , then 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 was at least . (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 is a simply-connected spin manifold of dimension at least (cf. also Simply-connected domain), then admits a metric of positive scalar curvature if and only if the -genus of vanishes. This invariant takes values in if modulo , in if modulo , in if modulo , and vanishes if modulo .

The situation is more complicated in the presence of a fundamental group . Let be the Grothendieck group of finitely generated -graded modules over the Clifford algebra which have a action commuting with the action. The inclusion of in induces a dual pull-back from to . The real -theory groups of are given by:

J. Rosenberg [a6] defined a -theory-valued invariant taking values in this group which generalizes the -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 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 which does not admit a metric of positive scalar curvature but for which the 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, "-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