Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 16 (2020), 127, 15 pages      arXiv:2008.13754
Contribution to the Special Issue on Scalar and Ricci Curvature in honor of Misha Gromov on his 75th Birthday

Width, Largeness and Index Theory

Rudolf Zeidler
Mathematical Institute, University of Münster, Einsteinstr. 62, 48149 Münster, Germany

Received September 01, 2020, in final form November 26, 2020; Published online December 02, 2020

In this note, we review some recent developments related to metric aspects of scalar curvature from the point of view of index theory for Dirac operators. In particular, we revisit index-theoretic approaches to a conjecture of Gromov on the width of Riemannian bands $M \times [-1,1]$, and on a conjecture of Rosenberg and Stolz on the non-existence of complete positive scalar curvature metrics on $M \times {\mathbb R}$. We show that there is a more general geometric statement underlying both of them implying a quantitative negative upper bound on the infimum of the scalar curvature of a complete metric on $M \times {\mathbb R}$ if the scalar curvature is positive in some neighborhood. We study ($\hat{A}$-)iso-enlargeable spin manifolds and related notions of width for Riemannian manifolds from an index-theoretic point of view. Finally, we list some open problems arising in the interplay between index theory, largeness properties and width.

Key words: scalar curvature; comparison geometry; index theory; Dirac operator; Callias-type operator; enlargeability; largeness properties.

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  1. Baum P., Karoubi M., On the Baum-Connes conjecture in the real case, Q. J. Math. 55 (2004), 231-235, arXiv:math.OA/0509495.
  2. Brunnbauer M., Hanke B., Large and small group homology, J. Topol. 3 (2010), 463-486, arXiv:0902.0869.
  3. Bunke U., A $K$-theoretic relative index theorem and Callias-type Dirac operators, Math. Ann. 303 (1995), 241-279.
  4. Cecchini S., Callias-type operators in $C^*$-algebras and positive scalar curvature on noncompact manifolds, J. Topol. Anal. 12 (2020), 897-939, arXiv:1611.01800.
  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.
  6. Cecchini S., Schick T., Enlargeable metrics on nonspin manifolds, Proc. Amer. Math. Soc., to appear, arXiv:1810.02116.
  7. Chang S., Coarse obstructions to positive scalar curvature in noncompact arithmetic manifolds, J. Differential Geom. 57 (2001), 1-21, arXiv:math.DG/0005115.
  8. Chang S., Weinberger S., Yu G., Taming 3-manifolds using scalar curvature, Geom. Dedicata 148 (2010), 3-14.
  9. Ebert J., Elliptic regularity for Dirac operators on families of noncompact manifolds, arXiv:1608.01699.
  10. Engel A., Wrong way maps in uniformly finite homology and homology of groups, J. Homotopy Relat. Struct. 13 (2018), 423-441, arXiv:1602.03374.
  11. Engel A., Rough index theory on spaces of polynomial growth and contractibility, J. Noncommut. Geom. 13 (2019), 617-666, arXiv:1505.03988.
  12. Gromov M., Metric inequalities with scalar curvature, Geom. Funct. Anal. 28 (2018), 645-726, arXiv:1710.04655.
  13. Gromov M., Four lectures on scalar curvature, arXiv:1908.10612.
  14. Gromov M., Lawson Jr. H.B., Spin and scalar curvature in the presence of a fundamental group. I, Ann. of Math. 111 (1980), 209-230.
  15. Gromov M., Lawson Jr. H.B., Positive scalar curvature and the Dirac operator on complete Riemannian manifolds, Inst. Hautes Études Sci. Publ. Math. 58 (1983), 83-196.
  16. Hanke B., Kotschick D., Roe J., Schick T., Coarse topology, enlargeability, and essentialness, Ann. Sci. Éc. Norm. Supér. (4) 41 (2008), 471-493, arXiv:0707.1999.
  17. Hanke B., Pape D., Schick T., Codimension two index obstructions to positive scalar curvature, Ann. Inst. Fourier (Grenoble) 65 (2015), 2681-2710, arXiv:1402.4094.
  18. Hanke B., Schick T., Enlargeability and index theory, J. Differential Geom. 74 (2006), 293-320, arXiv:math.GT/0403257.
  19. Hanke B., Schick T., Enlargeability and index theory: infinite covers, $K$-Theory 38 (2007), 23-33, arXiv:math.GT/0604540.
  20. Higson N., A note on the cobordism invariance of the index, Topology 30 (1991), 439-443.
  21. Hitchin N., Harmonic spinors, Adv. Math. 14 (1974), 1-55.
  22. Joyce D.D., Compact $8$-manifolds with holonomy ${\rm Spin}(7)$, Invent. Math. 123 (1996), 507-552.
  23. Kubota Y., The relative Mishchenko-Fomenko higher index and almost flat bundles II: Almost flat index pairing, arXiv:1908.10733.
  24. Kubota Y., Schick T., The Gromov-Lawson codimension 2 obstruction to positive scalar curvature and the $C^*$-index, Geom. Topol., to appear, arXiv:1909.09584.
  25. Lawson Jr. H.B., Michelsohn M.L., Spin geometry, Princeton Mathematical Series, Vol. 38, Princeton University Press, Princeton, NJ, 1989.
  26. Lichnerowicz A., Spineurs harmoniques, C. R. Acad. Sci. Paris 257 (1963), 7-9.
  27. Mišvcenko A.S., Fomenko A.T., The index of elliptic operators over $C^{\ast} $-algebras, Math. USSR Izv. 15 (1980), 87-112.
  28. Nitsche M., Schick T., Zeidler R., Transfer maps in generalized group homology via submanifolds, arXiv:1906.01190.
  29. Roe J., Index theory, coarse geometry, and topology of manifolds, CBMS Regional Conference Series in Mathematics, Vol. 90, Amer. Math. Soc., Providence, RI, 1996.
  30. Rosenberg J., $C^{\ast} $-algebras, positive scalar curvature, and the Novikov conjecture, Inst. Hautes Études Sci. Publ. Math. (1983), 197-212.
  31. Rosenberg J., $C^\ast$-algebras, positive scalar curvature and the Novikov conjecture. II, in Geometric Methods in Operator Algebras (Kyoto, 1983), Pitman Res. Notes Math. Ser., Vol. 123, Longman Sci. Tech., Harlow, 1986, 341-374.
  32. Rosenberg J., $C^\ast$-algebras, positive scalar curvature, and the Novikov conjecture. III, Topology 25 (1986), 319-336.
  33. Rosenberg J., Stolz S., Manifolds of positive scalar curvature, in Algebraic Topology and its Applications, Math. Sci. Res. Inst. Publ., Vol. 27, Springer, New York, 1994, 241-267.
  34. Rosenberg J., Stolz S., A ''stable'' version of the Gromov-Lawson conjecture, in The Čech Centennial (Boston, MA, 1993), Contemp. Math., Vol. 181, Amer. Math. Soc., Providence, RI, 1995, 405-418, arXiv:dg-ga/9407002.
  35. Rosenberg J., Stolz S., Metrics of positive scalar curvature and connections with surgery, in Surveys on Surgery Theory, Vol. 2, Ann. of Math. Stud., Vol. 149, Princeton University Press, Princeton, NJ, 2001, 353-386.
  36. Schick T., A counterexample to the (unstable) Gromov-Lawson-Rosenberg conjecture, Topology 37 (1998), 1165-1168, arXiv:math.GT/0403063.
  37. Schick T., Real versus complex $K$-theory using Kasparov's bivariant $KK$-theory, Algebr. Geom. Topol. 4 (2004), 333-346, arXiv:math.KT/0311295.
  38. Schick T., The topology of positive scalar curvature, in Proceedings of the International Congress of Mathematicians - Seoul 2014, Vol. II, Kyung Moon Sa, Seoul, 2014, 1285-1307, arXiv:1405.4220.
  39. Schick T., Zadeh M.E., Large scale index of multi-partitioned manifolds, J. Noncommut. Geom. 12 (2018), 439-456, arXiv:1308.0742.
  40. Schoen R., Yau S.T., On the structure of manifolds with positive scalar curvature, Manuscripta Math. 28 (1979), 159-183.
  41. Stolz S., Concordance classes of positive scalar curvature metrics, Preprint, 1998, available at
  42. Stolz S., Simply connected manifolds of positive scalar curvature, Ann. of Math. 136 (1992), 511-540.
  43. Stolz S., Manifolds of positive scalar curvature, in Topology of High-Dimensional Manifolds (Trieste, 2001), ICTP Lect. Notes, Vol. 9, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2002, 661-709.
  44. Taubes C.H., The Seiberg-Witten invariants and symplectic forms, Math. Res. Lett. 1 (1994), 809-822.
  45. Zeidler R., An index obstruction to positive scalar curvature on fiber bundles over aspherical manifolds, Algebr. Geom. Topol. 17 (2017), 3081-3094, arXiv:1512.06781.
  46. Zeidler R., Band width estimates via the Dirac operator, J. Differential Geom., to appear, arXiv:1905.08520.

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