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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" />.
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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:
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If $( M , g )$ 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 $\hat{A}$-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 $8$.
  
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.
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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:
  
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.
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\begin{equation*} K O _ { m } ( {\bf R} \pi ) = {\cal Z} _ { m } ^ { \pi } / i ^ { * } {\cal Z} _ { m + 1 } ^ { \pi }. \end{equation*}
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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.
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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 <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 $\alpha$ invariant vanishes. It is not known (1998) if the conjecture holds for finite fundamental groups.
  
 
====References====
 
====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>
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<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>

Revision as of 17:00, 1 July 2020

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 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
How to Cite This Entry:
Gromov-Lawson conjecture. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Gromov-Lawson_conjecture&oldid=50375
This article was adapted from an original article by P.B. Gilkey (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article