Geodesic
Torsion Obstructions to Positive Scalar Curvature
Symmetry, integrability and geometry: methods and applications, Tome 20 (2024) Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We study obstructions to the existence of Riemannian metrics of positive scalar curvature on closed smooth manifolds arising from torsion classes in the integral homology of their fundamental groups. As an application, we construct new examples of manifolds which do not admit positive scalar curvature metrics, but whose Cartesian products admit such metrics.
Keywords: positive scalar curvature, toral manifold, enlargeability, $\mu$-bubble; group homology, Riemannian foliation, band width inequality. 
@article{SIGMA_2024_20_a68,
     author = {Misha Gromov and Bernhard Hanke},
     title = {Torsion {Obstructions} to {Positive} {Scalar} {Curvature}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2024},
     volume = {20},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2024_20_a68/}
}
TY  - JOUR
AU  - Misha Gromov
AU  - Bernhard Hanke
TI  - Torsion Obstructions to Positive Scalar Curvature
JO  - Symmetry, integrability and geometry: methods and applications
PY  - 2024
VL  - 20
UR  - http://geodesic.mathdoc.fr/item/SIGMA_2024_20_a68/
LA  - en
ID  - SIGMA_2024_20_a68
ER  - 
%0 Journal Article
%A Misha Gromov
%A Bernhard Hanke
%T Torsion Obstructions to Positive Scalar Curvature
%J Symmetry, integrability and geometry: methods and applications
%D 2024
%V 20
%U http://geodesic.mathdoc.fr/item/SIGMA_2024_20_a68/
%G en
%F SIGMA_2024_20_a68
Misha Gromov; Bernhard Hanke. Torsion Obstructions to Positive Scalar Curvature. Symmetry, integrability and geometry: methods and applications, Tome 20 (2024). http://geodesic.mathdoc.fr/item/SIGMA_2024_20_a68/

[1] Adem A., Milgram R.J., Cohomology of finite groups, Grundlehren Math. Wiss., 309, 2nd ed., Springer, Berlin, 2004 | DOI | MR | Zbl

[2] Baas N.A., “On bordism theory of manifolds with singularities”, Math. Scand., 33 (1973), 279–302 | DOI | MR

[3] Brunnbauer M., Hanke B., “Large and small group homology”, J. Topol., 3 (2010), 463–486, arXiv: 0902.0869 | DOI | MR | Zbl

[4] Cartan H., Eilenberg S., Homological algebra, Princeton University Press, Princeton, NJ, 1956 | MR | Zbl

[5] Cecchini S., “A long neck principle for Riemannian spin manifolds with positive scalar curvature”, Geom. Funct. Anal., 30 (2020), 1183–1223, arXiv: 2002.07131 | DOI | MR | Zbl

[6] Chodosh O., Li C., Liokumovich Y., “Classifying sufficiently connected PSC manifolds in 4 and 5 dimensions”, Geom. Topol., 27 (2023), 1635–1655, arXiv: 2105.07306 | DOI | MR | Zbl

[7] Chodosh O., Mantoulidis C., Schulze F., “Improved generic regularity of codimension-1 minimizing integral currents”, Ars Inven. Anal., 2024, 3, 16 pp., arXiv: 2306.13191 | MR

[8] Conner P.E., Floyd E.E., Differentiable periodic maps, Ergeb. Math. Grenzgeb., 33, Springer, Berlin, 1964 | MR | Zbl

[9] Führing S., “A smooth variation of Baas–Sullivan theory and positive scalar curvature”, Math. Z., 274 (2013), 1029–1046, arXiv: 1203.4370 | DOI | MR | Zbl

[10] Führing S., “Bordism and projective space bundles”, Münster J. Math., 15 (2022), 355–388, arXiv: 2006.15394 | DOI | MR | Zbl

[11] Ghys E., de la Harpe P. (eds.), Sur les groupes hyperboliques d'après Mikhael Gromov, Progr. Math., 83, Birkhäuser, Boston, MA, 1990 | DOI | MR

[12] Gromov M., “Positive curvature, macroscopic dimension, spectral gaps and higher signatures”, Functional Analysis on the Eve of the 21st Century (New Brunswick, NJ, 1993), v. II, Progr. Math., 132, Birkhäuser, Boston, MA, 1996, 1–213 | DOI | MR | Zbl

[13] Gromov M., “Metric inequalities with scalar curvature”, Geom. Funct. Anal., 28 (2018), 645–726, arXiv: 1710.04655 | DOI | MR | Zbl

[14] Gromov M., Four lectures on scalar curvature, Perspectives in Scalar Curvature, 1, World Scientific Publishing, Hackensack, NJ, 2023, 514 pp., arXiv: 1908.10612 | DOI | MR | Zbl

[15] Gromov M., Lawson Jr. H.B., “The classification of simply connected manifolds of positive scalar curvature”, Ann. of Math., 111 (1980), 423–434 | DOI | MR | Zbl

[16] Gromov M., Lawson Jr. H.B., “Positive scalar curvature and the Dirac operator on complete Riemannian manifolds”, Inst. Hautes Études Sci. Publ. Math., 1983, 83–196 | DOI | MR | Zbl

[17] Guo H., Xie Z., Yu G., “Quantitative K-theory, positive scalar curvature, and bandwidth”, Perspectives in scalar curvature, v. 2, World Scientific Publishing, Hackensack, NJ, 2023, 763–798, arXiv: 2010.01749 | DOI | MR

[18] Hanke B., “Positive scalar curvature on manifolds with odd order abelian fundamental groups”, Geom. Topol., 25 (2021), 497–546, arXiv: 1908.00944 | DOI | MR | Zbl

[19] Hitchin N., “Harmonic spinors”, Adv. Math., 14 (1974), 1–55 | DOI | MR | Zbl

[20] Joachim M., “Toral classes and the Gromov–Lawson–Rosenberg conjecture for elementary abelian 2-groups”, Arch. Math. (Basel), 83 (2004), 461–466 | DOI | MR | Zbl

[21] Kahn J., Markovic V., “Immersing almost geodesic surfaces in a closed hyperbolic three manifold”, Ann. of Math., 175 (2012), 1127–1190, arXiv: 0910.5501 | DOI | MR | Zbl

[22] Lohkamp J., Minimal smoothings of area minimizing cones, arXiv: 1810.03157

[23] Margolis H.R., “Eilenberg–Mac Lane spectra”, Proc. Amer. Math. Soc., 43 (1974), 409–415 | DOI | MR | Zbl

[24] Matthey M., Oyono-Oyono H., Pitsch W., “Homotopy invariance of higher signatures and 3-manifold groups”, Bull. Soc. Math. France, 136 (2008), 1–25, arXiv: math.AT/0412363 | DOI | MR | Zbl

[25] Milnor J., “A unique decomposition theorem for 3-manifolds”, Amer. J. Math., 84 (1962), 1–7 | DOI | MR | Zbl

[26] Morgan F., Pansu P., “A list of open problems in differential geometry”, São Paulo J. Math. Sci., 15 (2021), 305–321 | DOI | MR | Zbl

[27] Räde D., “Scalar and mean curvature comparison via $\mu $-bubbles”, Calc. Var. Partial Differential Equations, 62 (2023), 187, 39 pp., arXiv: 2104.10120 | DOI | MR | Zbl

[28] Rosenberg J., “$C^\ast$-algebras, positive scalar curvature, and the Novikov conjecture. III”, Topology, 25 (1986), 319–336 | DOI | MR | Zbl

[29] Rosenberg J., Stolz S., “A “stable” version of the Gromov–Lawson conjecture”, The Čech Centennial, Contemp. Math., 181, American Mathematical Society, Providence, RI, 1995, 405–418, arXiv: dg-ga/9407002 | DOI | MR | Zbl

[30] Rosenberg J., Stolz S., “Metrics of positive scalar curvature and connections with surgery”, Surveys on Surgery Theor, v. 2, Ann. of Math. Stud., 149, Princeton University Press, Princeton, NJ, 2001, 353–386 | DOI | MR | Zbl

[31] Schick T., “A counterexample to the (unstable) Gromov–Lawson–Rosenberg conjecture”, Topology, 37 (1998), 1165–1168, arXiv: math/0403063 | DOI | MR | Zbl

[32] Schick T., Zenobi V.F., “Positive scalar curvature due to the cokernel of the classifying map”, SIGMA, 16 (2020), 129, 12 pp., arXiv: 2006.15965 | DOI | MR | Zbl

[33] Schoen R., Yau S.-T., “Existence of incompressible minimal surfaces and the topology of three-dimensional manifolds with nonnegative scalar curvature”, Ann. of Math., 110 (1979), 127–142 | DOI | MR | Zbl

[34] Schoen R., Yau S.-T., “On the structure of manifolds with positive scalar curvature”, Manuscripta Math., 28 (1979), 159–183 | DOI | MR | Zbl

[35] Schoen R., Yau S.-T., “Positive scalar curvature and minimal hypersurface singularities”, Surveys in Differential Geometry, Surv. Differ. Geom., 24, Int. Press, Boston, MA, 2022, 441–480, arXiv: 1704.05490 | DOI | MR

[36] Smale N., “Generic regularity of homologically area minimizing hypersurfaces in eight-dimensional manifolds”, Comm. Anal. Geom., 1 (1993), 217–228 | DOI | MR | Zbl

[37] Stolz S., “Simply connected manifolds of positive scalar curvature”, Ann. of Math., 136 (1992), 511–540 | DOI | MR | Zbl

[38] Stolz S., “Splitting certain ${\rm MSpin}$-module spectra”, Topology, 33 (1994), 159–180 | DOI | MR | Zbl

[39] Su G., Wang X., Zhang W., “Nonnegative scalar curvature and area decreasing maps on complete foliated manifolds”, J. Reine Angew. Math., 790 (2022), 85–113, arXiv: 2104.03472 | DOI | MR | Zbl

[40] Wall C.T.C., “Determination of the cobordism ring”, Ann. of Math., 72 (1960), 292–311 | DOI | MR | Zbl

[41] Zeidler R., “Width, largeness and index theory”, SIGMA, 16 (2020), 127, 15 pp., arXiv: 2008.13754 | DOI | MR | Zbl

[42] Zeidler R., “Band width estimates via the Dirac operator”, J. Differential Geom., 122 (2022), 155–183, arXiv: 1905.08520 | DOI | MR | Zbl