View source for Gromov-Lawson conjecture

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120070/g1200701.png" /> is a metric of positive [[Scalar curvature|scalar curvature]] (cf. also [[Metric|Metric]]) on a compact spin manifold (cf. also [[Spinor structure|Spinor structure]]), results of A. Lichnerowicz [[#References|[a4]]] show that there are no harmonic spinors; consequently, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120070/g1200702.png" />-genus of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120070/g1200703.png" /> vanishes. M. Gromov and H.B. Lawson [[#References|[a2]]], [[#References|[a3]]] showed that if a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120070/g1200704.png" /> can be obtained from a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120070/g1200705.png" /> which admits a metric of positive scalar curvature, by surgeries in codimension at least <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120070/g1200706.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120070/g1200707.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120070/g1200708.png" /> was at least <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120070/g1200709.png" />. (This restriction is necessary to ensure that certain surgery arguments work.) S. Stolz [[#References|[a9]]] showed this was the case in the simply-connected setting: if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120070/g12007010.png" /> is a simply-connected spin manifold of dimension at least <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120070/g12007011.png" /> (cf. also [[Simply-connected domain|Simply-connected domain]]), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120070/g12007012.png" /> admits a metric of positive scalar curvature if and only if the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120070/g12007013.png" />-genus of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120070/g12007014.png" /> vanishes. This invariant takes values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120070/g12007015.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120070/g12007016.png" /> modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120070/g12007017.png" />, in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120070/g12007018.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120070/g12007019.png" /> modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120070/g12007020.png" />, in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120070/g12007021.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120070/g12007022.png" /> modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120070/g12007023.png" />, and vanishes if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120070/g12007024.png" /> modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120070/g12007025.png" />.

The situation is more complicated in the presence of a [[Fundamental group|fundamental group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120070/g12007026.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120070/g12007027.png" /> be the [[Grothendieck group|Grothendieck group]] of finitely generated <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120070/g12007028.png" />-graded modules over the [[Clifford algebra|Clifford algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120070/g12007029.png" /> which have a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120070/g12007030.png" /> action commuting with the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120070/g12007031.png" /> action. The inclusion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120070/g12007032.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120070/g12007033.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120070/g12007034.png" /> induces a dual pull-back <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120070/g12007035.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120070/g12007036.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120070/g12007037.png" />. The real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120070/g12007038.png" />-theory groups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120070/g12007039.png" /> are given by:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120070/g12007040.png" /></td> </tr></table>

J. Rosenberg [[#References|[a6]]] defined a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120070/g12007041.png" />-theory-valued invariant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120070/g12007042.png" /> taking values in this group which generalizes the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120070/g12007043.png" />-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 [[#References|[a1]]] and for finite Abelian groups of rank at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120070/g12007044.png" /> and odd order [[#References|[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 [[#References|[a5]]]. S. Schick [[#References|[a7]]] has shown that the conjecture, in the form due to J. Rosenberg, is false by exhibiting a compact spin manifold with fundamental group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120070/g12007045.png" /> which does not admit a metric of positive scalar curvature but for which the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120070/g12007046.png" /> invariant vanishes. It is not known (1998) if the conjecture holds for finite fundamental groups.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  B. Botvinnik,   P. Gilkey,   S. Stolz,   "The Gromov–Lawson–Rosenberg conjecture for groups with periodic cohomology"  ''J. Diff. Geom.'' , '''46'''  (1997)  pp. 374–405</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  M. Gromov,   H.B. Lawson,   "The classification of simply connected manifolds of positive scalar curvature"  ''Ann. of Math.'' , '''111'''  (1980)  pp. 423–434</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  A. Lichnerowicz,   "Spineurs harmoniques"  ''C.R. Acad. Sci. Paris'' , '''257'''  (1963)  pp. 7–9</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  J. Rosenberg,   S. Stolz,   "A  "stable"  version of the Gromov–Lawson conjecture"  ''Contemp. Math.'' , '''181'''  (1995)  pp. 405–418</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  J. Rosenberg,   "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120070/g12007047.png" />-algebras, positive scalar curvature, and the Novikov conjecture"  ''Publ. Math. IHES'' , '''58'''  (1983)  pp. 197–212</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  T. Schick,   "A counterexample to the (unstable) Gromov–Lawson–Rosenberg conjecture"  ''Topology'' , '''37'''  (1998)  pp. 1165–1168</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  R. Schultz,   "Positive scalar curvature and odd order Abelian fundamental groups"  ''Proc. Amer. Math. Soc.'' , '''125''' :  3  (1997)  pp. 907–915</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  S. Stolz,   "Simply connected manifolds of positive scalar curvature"  ''Ann. of Math.'' , '''136'''  (1992)  pp. 511–540</TD></TR></table>