Geodesic
The Gromov–Lawson codimension 2 obstruction to positive scalar curvature and the C∗–index
Kubota, Yosuke1 ; Schick, Thomas2
1 Department of Mathematical Sciences, Shinshu University, Matsumoto, Japan
2 Mathematisches Institut, Universität Göttingen, Göttingen, Germany
Geometry & topology, Tome 25 (2021) no. 2, pp. 949-960.

Voir la notice de l'article provenant de la source Mathematical Sciences Publishers

Gromov and Lawson developed a codimension 2 index obstruction to positive scalar curvature for a closed spin manifold M, later refined by Hanke, Pape and Schick. Kubota has shown that this obstruction also can be obtained from the Rosenberg index of the ambient manifold M, which takes values in the K–theory of the maximal C∗ –algebra of the fundamental group of M, using relative index constructions.

In this note, we give a slightly simplified account of Kubota’s work and remark that it also applies to the signature operator, thus recovering the homotopy invariance of higher signatures of codimension 2 submanifolds of Higson, Schick and Xie.

DOI : 10.2140/gt.2021.25.949
Classification : 19K56, 46L80, 58J22
Keywords: higher index theory, positive scalar curvature, higher signature, $C^*$–index theory

Kubota, Yosuke 1 ; Schick, Thomas 2

1 Department of Mathematical Sciences, Shinshu University, Matsumoto, Japan
2 Mathematisches Institut, Universität Göttingen, Göttingen, Germany 
@article{GT_2021_25_2_a6,
     author = {Kubota, Yosuke and Schick, Thomas},
     title = {The {Gromov{\textendash}Lawson} codimension 2 obstruction to positive scalar curvature and the {C\ensuremath{*}{\textendash}index}},
     journal = {Geometry & topology},
     pages = {949--960},
     publisher = {mathdoc},
     volume = {25},
     number = {2},
     year = {2021},
     doi = {10.2140/gt.2021.25.949},
     url = {http://geodesic.mathdoc.fr/articles/10.2140/gt.2021.25.949/}
}
TY  - JOUR
AU  - Kubota, Yosuke
AU  - Schick, Thomas
TI  - The Gromov–Lawson codimension 2 obstruction to positive scalar curvature and the C∗–index
JO  - Geometry & topology
PY  - 2021
SP  - 949
EP  - 960
VL  - 25
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.2140/gt.2021.25.949/
DO  - 10.2140/gt.2021.25.949
ID  - GT_2021_25_2_a6
ER  - 
%0 Journal Article
%A Kubota, Yosuke
%A Schick, Thomas
%T The Gromov–Lawson codimension 2 obstruction to positive scalar curvature and the C∗–index
%J Geometry & topology
%D 2021
%P 949-960
%V 25
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.2140/gt.2021.25.949/
%R 10.2140/gt.2021.25.949
%F GT_2021_25_2_a6
Kubota, Yosuke; Schick, Thomas. The Gromov–Lawson codimension 2 obstruction to positive scalar curvature and the C∗–index. Geometry & topology, Tome 25 (2021) no. 2, pp. 949-960. doi : 10.2140/gt.2021.25.949. http://geodesic.mathdoc.fr/articles/10.2140/gt.2021.25.949/

[1] A Engel, Wrong way maps in uniformly finite homology and homology of groups, J. Homotopy Relat. Struct. 13 (2018) 423 | DOI

[2] M Gromov, H B Lawson Jr., Positive scalar curvature and the Dirac operator on complete Riemannian manifolds, Inst. Hautes Études Sci. Publ. Math. 58 (1983) 83 | DOI

[3] B Hanke, D Pape, T Schick, Codimension two index obstructions to positive scalar curvature, Ann. Inst. Fourier (Grenoble) 65 (2015) 2681 | DOI

[4] B Hanke, T Schick, Enlargeability and index theory, J. Differential Geom. 74 (2006) 293 | DOI

[5] N Higson, T Schick, Z Xie, C∗–algebraic higher signatures and an invariance theorem in codimension two, Geom. Topol. 22 (2018) 3671 | DOI

[6] Y Kubota, The relative Mishchenko–Fomenko higher index and almost flat bundles, II : Almost flat index pairing, preprint (2019)

[7] M Nitsche, T Schick, R Zeidler, Transfer maps in generalized group homology via submanifolds, preprint (2019)

[8] J Rosenberg, C∗–algebras, positive scalar curvature, and the Novikov conjecture, Inst. Hautes Études Sci. Publ. Math. 58 (1983) 197 | DOI

[9] T Schick, The topology of positive scalar curvature, from: "Proceedings of the International Congress of Mathematicians, II" (editors S Y Jang, Y R Kim, D W Lee, I Yie), Kyung Moon Sa (2014) 1285

Cité par Sources :