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


SIGMA 17 (2021), 062, 39 pages      arXiv:2005.02744      https://doi.org/10.3842/SIGMA.2021.062
Contribution to the Special Issue on Scalar and Ricci Curvature in honor of Misha Gromov on his 75th Birthday

Positive Scalar Curvature on Spin Pseudomanifolds: the Fundamental Group and Secondary Invariants

Boris Botvinnik a, Paolo Piazza b and Jonathan Rosenberg c
a) Department of Mathematics, University of Oregon, Eugene OR 97403-1222, USA
b) Dipartimento di Matematica ''Guido Castelnuovo'', Sapienza Università di Roma, Piazzale Aldo Moro 5, 00185 Roma, Italy
c) Department of Mathematics, University of Maryland, College Park, MD 20742-4015, USA

Received May 26, 2020, in final form June 08, 2021; Published online June 24, 2021

Abstract
In this paper we continue the study of positive scalar curvature (psc) metrics on a depth-1 Thom-Mather stratified space $M_\Sigma$ with singular stratum $\beta M$ (a closed manifold of positive codimension) and associated link equal to $L$, a smooth compact manifold. We briefly call such spaces manifolds with $L$-fibered singularities. Under suitable spin assumptions we give necessary index-theoretic conditions for the existence of wedge metrics of positive scalar curvature. Assuming in addition that $L$ is a simply connected homogeneous space of positive scalar curvature, $L=G/H$, with the semisimple compact Lie group $G$ acting transitively on $L$ by isometries, we investigate when these necessary conditions are also sufficient. Our main result is that our conditions are indeed sufficient for large classes of examples, even when $M_\Sigma$ and $\beta M$ are not simply connected. We also investigate the space of such psc metrics and show that it often splits into many cobordism classes.

Key words: positive scalar curvature; pseudomanifold; singularity; bordism; transfer; $K$-theory; index; rho-invariant.

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References

  1. Albin P., Gell-Redman J., The index formula for families of Dirac type operators on pseudomanifolds, arXiv:1712.08513.
  2. Albin P., Gell-Redman J., The index of Dirac operators on incomplete edge spaces, SIGMA 12 (2016), 089, 45 pages, arXiv:1312.4241.
  3. Albin P., Gell-Redman J., Piazza P., Higher index theory for Dirac operators on wedge pseudomanifolds, in preparation.
  4. Albin P., Leichtnam E., Mazzeo R., Piazza P., The signature package on Witt spaces, Ann. Sci. Éc. Norm. Supér. (4) 45 (2012), 241-310, arXiv:1112.0989.
  5. Albin P., Piazza P., Stratified surgery and $K$-theory invariants of the signature operator, Ann. Sci. Éc. Norm. Supér. (4), to appear, arXiv:1710.00934.
  6. Azzali S., Wahl C., Two-cocycle twists and Atiyah-Patodi-Singer index theory, Math. Proc. Cambridge Philos. Soc. 167 (2019), 437-487, arXiv:1312.6373.
  7. Baum P., Douglas R.G., Taylor M.E., Cycles and relative cycles in analytic $K$-homology, J. Differential Geom. 30 (1989), 761-804.
  8. Benameur M.-T., Roy I., The Higson-Roe exact sequence and $\ell^2$ eta invariants, J. Funct. Anal. 268 (2015), 974-1031, arXiv:1409.2717.
  9. Bismut J.-M., Cheeger J., $\eta$-invariants and their adiabatic limits, J. Amer. Math. Soc. 2 (1989), 33-70.
  10. Botvinnik B., Manifolds with singularities accepting a metric of positive scalar curvature, Geom. Topol. 5 (2001), 683-718, arXiv:math.DG/9910177.
  11. Botvinnik B., Ebert J., Randal-Williams O., Infinite loop spaces and positive scalar curvature, Invent. Math. 209 (2017), 749-835, arXiv:1411.7408.
  12. Botvinnik B., Gilkey P., The eta invariant and metrics of positive scalar curvature, Math. Ann. 302 (1995), 507-517.
  13. Botvinnik B., Gilkey P., Stolz S., The Gromov-Lawson-Rosenberg conjecture for groups with periodic cohomology, J. Differential Geom. 46 (1997), 374-405.
  14. Botvinnik B., Piazza P., Rosenberg J., Positive scalar curvature on simply connected spin pseudomanifolds, J. Topol. Anal., to appear, arXiv:1908.04420.
  15. Botvinnik B., Rosenberg J., Positive scalar curvature on manifolds with fibered singularities, arXiv:1808.06007.
  16. Botvinnik B., Walsh M.G., Homotopy invariance of the space of metrics with positive scalar curvature on manifolds with singularities, SIGMA 17 (2021), 034, 27 pages, arXiv:2005.03073.
  17. Buggisch L., The spectral flow theorems for families of twisted Dirac operators, Ph.D. Thesis, University of Münster, 2018, available at https: d-nb.info/1190724960/34.
  18. Bunke U., A $K$-theoretic relative index theorem and Callias-type Dirac operators, Math. Ann. 303 (1995), 241-279.
  19. Debord C., Lescure J.-M., Rochon F., Pseudodifferential operators on manifolds with fibred corners, Ann. Inst. Fourier (Grenoble) 65 (2015), 1799-1880, arXiv:1112.4575.
  20. Dwyer W., Schick T., Stolz S., Remarks on a conjecture of Gromov and Lawson, in High-Dimensional Manifold Topology, World Sci. Publ., River Edge, NJ, 2003, 159-176, arXiv:math.GT/0208011.
  21. Ebert J., The two definitions of the index difference, Trans. Amer. Math. Soc. 369 (2017), 7469-7507, arXiv:1308.4998.
  22. Ebert J., Index theory in spaces of manifolds, Math. Ann. 374 (2019), 931-962, arXiv:1608.01701.
  23. Ebert J., Frenck G., The Gromov-Lawson-Chernysh surgery theorem, Bol. Soc. Mat. Mex. 27 (2021), 37, 43 pages, arXiv:1807.06311.
  24. 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.
  25. Gromov M., Lawson Jr. H.B., The classification of simply connected manifolds of positive scalar curvature, Ann. of Math. 111 (1980), 423-434.
  26. 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.
  27. Joyce D., A new construction of compact 8-manifolds with holonomy ${\rm Spin}(7)$, J. Differential Geom. 53 (1999), 89-130, arXiv:math.DG/9910002.
  28. Kasparov G.G., Equivariant $KK$-theory and the Novikov conjecture, Invent. Math. 91 (1988), 147-201.
  29. Kasparov G.G., $K$-theory, group $C^*$-algebras, and higher signatures (conspectus), in Novikov Conjectures, Index Theorems and Rigidity, Vol. 1 (Oberwolfach, 1993), London Math. Soc. Lecture Note Ser., Vol. 226, Cambridge University Press, Cambridge, 1995, 101-146.
  30. Kreck M., Stolz S., Nonconnected moduli spaces of positive sectional curvature metrics, J. Amer. Math. Soc. 6 (1993), 825-850.
  31. Lawson Jr. H.B., Michelsohn M.-L., Spin geometry, Princeton Mathematical Series, Vol. 38, Princeton University Press, Princeton, NJ, 1989.
  32. Leichtnam E., Piazza P., Spectral sections and higher Atiyah-Patodi-Singer index theory on Galois coverings, Geom. Funct. Anal. 8 (1998), 17-58.
  33. Leichtnam E., Piazza P., On higher eta-invariants and metrics of positive scalar curvature, $K$-Theory 24 (2001), 341-359.
  34. Leichtnam E., Piazza P., Dirac index classes and the noncommutative spectral flow, J. Funct. Anal. 200 (2003), 348-400.
  35. Mazzeo R., Elliptic theory of differential edge operators. I, Comm. Partial Differential Equations 16 (1991), 1615-1664.
  36. Piazza P., Schick T., Bordism, rho-invariants and the Baum-Connes conjecture, J. Noncommut. Geom. 1 (2007), 27-111, arXiv:math.KT/0407388.
  37. Piazza P., Schick T., Groups with torsion, bordism and rho invariants, Pacific J. Math. 232 (2007), 355-378, arXiv:math.GN/0604319.
  38. Piazza P., Schick T., Rho-classes, index theory and Stolz' positive scalar curvature sequence, J. Topol. 7 (2014), 965-1004, arXiv:1210.6892.
  39. Piazza P., Vertman B., Eta and rho invariants on manifolds with edges, Ann. Inst. Fourier (Grenoble) 69 (2019), 1955-2035, arXiv:1604.07420.
  40. Piazza P., Zenobi V.F., Singular spaces, groupoids and metrics of positive scalar curvature, J. Geom. Phys. 137 (2019), 87-123, arXiv:1803.02697.
  41. 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.
  42. Rosenberg J., $C^\ast$-algebras, positive scalar curvature, and the Novikov conjecture. III, Topology 25 (1986), 319-336.
  43. Schick T., A counterexample to the (unstable) Gromov-Lawson-Rosenberg conjecture, Topology 37 (1998), 1165-1168, arXiv:math.GT/0403063.
  44. Stolz S., Concordance classes of positive scalar curvature metrics, available at https://www3.nd.edu/~stolz/preprint.html.
  45. Stolz S., Positive scalar curvature metrics - existence and classification questions, in Proceedings of the International Congress of Mathematicians, Vols. 1, 2 (Zürich, 1994), Birkhäuser, Basel, 1995, 625-636.
  46. Stolz S., Manifolds of positive scalar curvature, in Topology of High-Dimensional Manifolds, No. 1, 2 (Trieste, 2001), ICTP Lect. Notes, Vol. 9, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2002, 661-709.
  47. Wahl C., The Atiyah-Patodi-Singer index theorem for Dirac operators over $C^\ast$-algebras, Asian J. Math. 17 (2013), 265-319, arXiv:0901.0381.
  48. Xie Z., Yu G., A relative higher index theorem, diffeomorphisms and positive scalar curvature, Adv. Math. 250 (2014), 35-73, arXiv:1204.3664.
  49. 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.

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