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Positive scalar curvature of totally nonspin manifolds


Author: Stanley Chang
Journal: Proc. Amer. Math. Soc. 138 (2010), 1621-1632
MSC (2010): Primary 32Q10
DOI: https://doi.org/10.1090/S0002-9939-09-09483-0
Published electronically: December 16, 2009
MathSciNet review: 2587446
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Abstract: In this paper we address the issue of positive scalar curvature on oriented nonspin compact manifolds whose universal cover is also nonspin. We provide a conjecture for an obstruction to such curvature in this venue that takes into account all the data known to date. The conjecture is proved for a wide class of closed manifolds based on their fundamental group structure.


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  • 1. P. Baum, A. Connes and N. Higson, Classifying space for proper actions and $ K$-theory of group $ C\sp *$-algebras, $ C\sp *$-algebras: 1943-1993 (San Antonio, TX, 1993), 240-291, Contemp. Math., 167, Amer. Math. Soc., Providence, RI, 1994. MR 1292018 (96c:46070)
  • 2. H. Bechtell, Elementary groups, Trans. Amer. Math. Soc. 114 (1965), 355-362. MR 0175967 (31:243)
  • 3. B. Botvinnik and J. Rosenberg, The Yamabe invariant for non-simply connected manifolds, J. Differential Geom. 62 (2002), no. 2, 175-208. MR 1988502 (2004j:53045)
  • 4. U. Christ and J. Lohkamp, Singular minimal hypersurfaces and scalar curvature, preprint, arXiv:math/0609338.
  • 5. J. Davis and K. Pearson, The Gromov-Lawson-Rosenberg conjecture for cocompact Fuchsian groups, Proc. Amer. Math. Soc. 131 (2003), no. 11, 3571-3578 (electronic). MR 1991770 (2004f:53039)
  • 6. M. Dunwoody, Accessibility and groups of cohomological dimension one, Proc. London Math. Soc. (3) 38 (1979), no. 2, 193-215. MR 531159 (80i:20024)
  • 7. B. Eckmann, Cohomology of groups and transfer, Ann. of Math. (2) 58 (1953), 481-493. MR 0058600 (15:397a)
  • 8. M. Gromov and B. Lawson, The classification of simply connected manifolds of positive scalar curvature, Annals of Mathematics 111 (1980), 423-434. MR 577131 (81h:53036)
  • 9. M. Joachim, Toral classes and the Gromov-Lawson-Rosenberg conjecture for elementary abelian $ 2$-groups, Arch. Math. (Basel) 83 (2004), no. 5, 461-466. MR 2102644 (2005g:53050)
  • 10. M. Joachim and T. Schick, Positive and negative results concerning the Gromov-Lawson-Rosenberg conjecture, Geometry and topology: Aarhus (1998), 213-226, Contemp. Math., 258, Amer. Math. Soc., Providence, RI, 2000. MR 1778107 (2002g:53079)
  • 11. R. Jung and S. Stolz, private communication.
  • 12. L. Kappe and J. Kirtland, Finite groups with trivial Frattini subgroup, Arch. Math. (Basel) 80 (2003), no. 3, 225-234. MR 1981175 (2004f:20046)
  • 13. S. Kwasik and R. Schultz, Positive scalar curvature and periodic fundamental groups, Comment. Math. Helv. 65 (1990), no. 2, 271-286. MR 1057244 (91k:57027)
  • 14. I. Leary and B. Nucinkis, Every CW-complex is a classifying space for proper bundles, Topology 40 (2001), no. 3, 539-550. MR 1838994 (2002c:55022)
  • 15. J. Lohkamp, Positive scalar curvature in $ \hbox{dim}\ge8$ , C. R. Math. Acad. Sci. Paris 343 (2006), no. 9, 585-588. MR 2269869 (2008d:53079)
  • 16. W. Lück, Survey on classifying spaces for families of subgroups, Infinite groups: Geometric, combinatorial and dynamical aspects, 269-322, Progr. Math., 248, Birkhäuser, Basel, 2005. MR 2195456 (2006m:55036)
  • 17. W. Lück, The type of the classifying space for a family of subgroups, J. Pure Appl. Algebra 149 (2000), no. 2, 177-203. MR 1757730 (2001i:55018)
  • 18. W. Lück and R. Stamm, Computations of $ K$- and $ L$-theory of cocompact planar groups, $ K$-Theory 21 (2000), no. 3, 249-292. MR 1803230 (2001k:19004)
  • 19. D. Robinson, A course in the theory of groups, second edition, Graduate Texts in Mathematics, 80, Springer-Verlag, New York, 1996. MR 1357169 (96f:20001)
  • 20. J. Rosenberg, $ C^\ast$-algebras, positive scalar curvature, and the Novikov conjecture, Inst. Hautes Études Sci. Publ. Math., no. 58 (1983), 197-212. MR 720934 (85g:58083)
  • 21. J. Rosenberg, $ C^\ast$-algebras, positive scalar curvature and the Novikov conjecture. II, Geometric methods in operator algebras (Kyoto, 1983), 341-374, Pitman Res. Notes Math. Ser., 123, Longman Sci. Tech., Harlow, 1986. MR 866507 (88f:58140)
  • 22. J. Rosenberg, $ C^\ast$-algebras, positive scalar curvature, and the Novikov conjecture. III, Topology 25 (1986), no. 3, 319-336. MR 842428 (88f:58141)
  • 23. J. Rosenberg and S. Stolz, A ``stable'' version of the Gromov-Lawson conjecture, The Čech centennial (Boston, MA, 1993), 405-418, Contemp. Math., 181, Amer. Math. Soc., Providence, RI, 1995. MR 1321004 (96m:53042)
  • 24. T. Schick, A counterexample to the (unstable) Gromov-Lawson-Rosenberg conjecture, Topology 37 (1998), no. 6, 1165-1168. MR 1632971 (99j:53049)
  • 25. R. Schoen and S.-T. Yau, On the structure of manifolds with positive scalar curvature, Manuscripta Math. 28 (1979), no. 1-3, 159-183. MR 535700 (80k:53064)
  • 26. R. Schultz, Positive scalar curvature and odd order abelian fundamental groups, Proc. Amer. Math. Soc. 125 (1997), no. 3, 907-915. MR 1363184 (97j:53041)

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Additional Information

Stanley Chang
Affiliation: Department of Mathematics, Wellesley College, Wellesley, Massachusetts 02481
Email: schang@wellesley.edu

DOI: https://doi.org/10.1090/S0002-9939-09-09483-0
Keywords: Nonspin manifolds, proper classifying spaces, positive scalar curvature
Received by editor(s): April 9, 2007
Received by editor(s) in revised form: January 8, 2008
Published electronically: December 16, 2009
Additional Notes: This research was partially supported by NSF Grant DMS-9971657
Communicated by: Alexander N. Dranishnikov
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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