Positive Scalar Curvature due to the Cokernel of the Classifying Map

Thomas Schick ; Vito Felice Zenobi

Symmetry, integrability and geometry: methods and applications, Tome 16 (2020)

Résumé

This paper contributes to the classification of positive scalar curvature metrics up to bordism and up to concordance. Let $M$ be a closed spin manifold of dimension $\ge 5$ which admits a metric with positive scalar curvature. We give lower bounds on the rank of the group of psc metrics over $M$ up to bordism in terms of the corank of the canonical map $KO_*(M)\to KO_*(B\pi_1(M))$, provided the rational analytic Novikov conjecture is true for $\pi_1(M)$.

Détail

Keywords: positive scalar curvature, Stolz exact sequence, analytic surgery exact sequence, secondary index theory, higher index theory, $K$-theory.
Mots-clés : bordism, concordance

Bibliographie

[1] Baum P., Connes A., Higson N., “Classifying space for proper actions and $K$-theory of group $C^\ast$-algebras”, $C^\ast$-Algebras: 1943–1993 (San Antonio, TX, 1993), Contemp. Math., 167, Amer. Math. Soc., Providence, RI, 1994, 240–291 | DOI | MR

[2] Botvinnik B., Gilkey P. B., “The eta invariant and metrics of positive scalar curvature”, Math. Ann., 302 (1995), 507–517 | DOI | MR | Zbl

[3] Buggisch L., The spectral flow theorem for families of twisted Dirac operators, Ph.D. Thesis, Westfälische Wilhelms-Universität Münster, 2019

[4] Carr R., “Construction of manifolds of positive scalar curvature”, Trans. Amer. Math. Soc., 307 (1988), 63–74 | DOI | MR | Zbl

[5] Cecchini S., Seyedhosseini M., Zenobi V. F., Relative torsion and bordism classes of positive scalar metrics on manifolds with boundary, arXiv: 2009.13718

[6] Ebert J., “The two definitions of the index difference”, Trans. Amer. Math. Soc., 369 (2017), 7469–7507, arXiv: 1308.4998 | DOI | MR | Zbl

[7] Ebert J., Frenck G., The Gromov–Lawson–Chernysh surgery theorem, arXiv: 1807.06311

[8] Ebert J., Randal-Williams O., “Infinite loop spaces and positive scalar curvature in the presence of a fundamental group”, Geom. Topol., 23 (2019), 1549–1610, arXiv: 1711.11363 | DOI | MR | Zbl

[9] Frenck G., H-Space structures on spaces of metrics of positive scalar curvature, arXiv: 2004.01033

[10] 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

[11] Hebestreit F., Joachim M., “Twisted spin cobordism and positive scalar curvature”, J. Topol., 13 (2020), 1–58, arXiv: 1311.3164 | DOI | MR | Zbl

[12] Kazaras D., Ruberman D., Saveliev N., “On positive scalar curvature cobordisms and the conformal Laplacian on end-periodic manifolds”, Comm. Anal. Geom. (to appear) , arXiv: 1902.00443

[13] Leichtnam E., Piazza P., “On higher eta-invariants and metrics of positive scalar curvature”, $K$-Theory, 24 (2001), 341–359 | DOI | MR | Zbl

[14] Piazza P., Schick T., “Groups with torsion, bordism and rho invariants”, Pacific J. Math., 232 (2007), 355–378, arXiv: math.GN/0604319 | DOI | MR | Zbl

[15] Piazza P., Schick T., “Rho-classes, index theory and Stolz' positive scalar curvature sequence”, J. Topol., 7 (2014), 965–1004, arXiv: 1210.6892 | DOI | MR | Zbl

[16] Piazza P., Schick T., Zenobi V. F., Higher rho numbers and the mapping of analytic surgery to homology, arXiv: 1905.11861

[17] Piazza P., Schick T., Zenobi V. F., “On positive scalar curvature bordism”, Comm. Anal. Geom. (to appear) , arXiv: 1912.09168

[18] Rosenberg J., “$C^\ast$-algebras, positive scalar curvature and the Novikov conjecture. II”, Geometric Methods in Operator Algebras (Kyoto, 1983), Pitman Res. Notes Math. Ser., 123, Longman Sci. Tech., Harlow, 1986, 341–374 | MR

[19] Schick T., “Real versus complex $K$-theory using Kasparov's bivariant $KK$-theory”, Algebr. Geom. Topol., 4 (2004), 333–346, arXiv: math.KT/0311295 | DOI | MR | Zbl

[20] Stolz S., Concordance classes of positive scalar curvature metrics, Preprint, 1998 http://www3.nd.edu/s̃tolz/concordance.ps

[21] Wall C. T. C., “Finiteness conditions for ${\rm CW}$-complexes”, Ann. of Math., 81 (1965), 56–69 | DOI | MR | Zbl

[22] Wall C. T. C., “Geometrical connectivity. I”, J. London Math. Soc., 3 (1971), 597–604 | DOI | MR | Zbl

[23] Weinberger S., Yu G., “Finite part of operator $K$-theory for groups finitely embeddable into Hilbert space and the degree of nonrigidity of manifolds”, Geom. Topol., 19 (2015), 2767–2799, arXiv: 1308.4744 | DOI | MR | Zbl

[24] Xie Z., Yu G., “Positive scalar curvature, higher rho invariants and localization algebras”, Adv. Math., 262 (2014), 823–866, arXiv: 1302.4418 | DOI | MR | Zbl

[25] Xie Z., Yu G., Zeidler R., On the range of the relative higher index and the higher rho-invariant for positive scalar curvature, arXiv: 1712.03722

[26] Zeidler R., “Positive scalar curvature and product formulas for secondary index invariants”, J. Topol., 9 (2016), 687–724, arXiv: 1412.0685 | DOI | MR | Zbl

[27] Zenobi V. F., “Adiabatic groupoid and secondary invariants in K-theory”, Adv. Math., 347 (2019), 940–1001, arXiv: 1609.08015 | DOI | MR | Zbl