View source for Gromov-Lawson conjecture

<!--This article has been texified automatically. Since there was no Nroff source code for this article, 
the semi-automatic procedure described at https://encyclopediaofmath.org/wiki/User:Maximilian_Janisch/latexlist
was used.
If the TeX and formula formatting is correct and if all png images have been replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category.

Out of 47 formulas, 46 were replaced by TEX code.-->

{{TEX|semi-auto}}{{TEX|done}}
If $( M , g )$ is a metric of positive [[Scalar curvature|scalar curvature]] (cf. also [[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 $\hat{A}$-genus of $M$ vanishes. M. Gromov and H.B. Lawson [[#References|[a2]]], [[#References|[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 [[#References|[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|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|fundamental group]] $\pi$. Let $\mathcal{Z} _ { m} ^{\pi }$ be the [[Grothendieck group|Grothendieck group]] of finitely generated $\mathbf{Z}_{2}$-graded modules over the [[Clifford algebra|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 [[#References|[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 [[#References|[a1]]] and for finite Abelian groups of rank at most $2$ 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 $\mathbf{Z}\oplus\mathbf{Z}\oplus\mathbf{Z}\oplus\mathbf{Z}_3$ 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====
<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,   "$C ^ { * }$-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>