The space of metrics of positive scalar curvature
Publications Mathématiques de l'IHÉS, Tome 120 (2014), pp. 335-367.

We study the topology of the space of positive scalar curvature metrics on high dimensional spheres and other spin manifolds. Our main result provides elements in higher homotopy and homology groups of these spaces, which, in contrast to previous approaches, are of infinite order and survive in the (observer) moduli space of such metrics.

Along the way we construct smooth fiber bundles over spheres whose total spaces have non-vanishing A ^-genera, thus establishing the non-multiplicativity of the A ^-genus in fiber bundles with simply connected base.

DOI : https://doi.org/10.1007/s10240-014-0062-9
MANUSCRIPT : 62
PUBLISHER-ID : s10240-014-0062-9
Mots clés : Modulus Space, Normal Bundle, Homotopy Group, Spin Manifold, Positive Scalar Curvature
@article{PMIHES_2014__120__335_0,
     author = {Hanke, Bernhard and Schick, Thomas and Steimle, Wolfgang},
     title = {The space of metrics of positive scalar curvature},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     pages = {335--367},
     publisher = {Springer Berlin Heidelberg},
     address = {Berlin/Heidelberg},
     volume = {120},
     year = {2014},
     doi = {10.1007/s10240-014-0062-9},
     zbl = {1321.58008},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1007/s10240-014-0062-9/}
}
TY  - JOUR
AU  - Hanke, Bernhard
AU  - Schick, Thomas
AU  - Steimle, Wolfgang
TI  - The space of metrics of positive scalar curvature
JO  - Publications Mathématiques de l'IHÉS
PY  - 2014
DA  - 2014///
SP  - 335
EP  - 367
VL  - 120
PB  - Springer Berlin Heidelberg
PP  - Berlin/Heidelberg
UR  - http://archive.numdam.org/articles/10.1007/s10240-014-0062-9/
UR  - https://zbmath.org/?q=an%3A1321.58008
UR  - https://doi.org/10.1007/s10240-014-0062-9
DO  - 10.1007/s10240-014-0062-9
LA  - en
ID  - PMIHES_2014__120__335_0
ER  - 
Hanke, Bernhard; Schick, Thomas; Steimle, Wolfgang. The space of metrics of positive scalar curvature. Publications Mathématiques de l'IHÉS, Tome 120 (2014), pp. 335-367. doi : 10.1007/s10240-014-0062-9. http://archive.numdam.org/articles/10.1007/s10240-014-0062-9/

[1.] Akutagawa, K.; Botvinnik, B. The relative Yamabe invariant, Commun. Anal. Geom., Volume 10 (2002), pp. 925-954 | Zbl 1034.58007

[2.] Besse, A. Einstein Manifolds (1987) | Zbl 0613.53001

[3.] Botvinnik, B.; Hanke, B.; Schick, T.; Walsh, M. Homotopy groups of the moduli space of metrics of positive scalar curvature, Geom. Topol., Volume 14 (2010), pp. 2047-2076 | Article | Zbl 1201.58006

[4.] Casson, A. J. Fibrations over spheres, Topology, Volume 6 (1967), pp. 489-499 | Article | Zbl 0163.45302

[5.] Chern, S.-S.; Hirzebruch, F.; Serre, J.-P. On the index of a fibered manifold, Proc. Am. Math. Soc., Volume 8 (1957), pp. 587-596 | Article | Zbl 0083.17801

[6.] Crowley, D.; Schick, T. The Gromoll filtration, KO-characteristic classes and metrics of positive scalar curvature, Geom. Topol., Volume 17 (2013), pp. 1773-1790 | Article | Zbl 1285.57015

[7.] Davis, J. F. Manifold aspects of the Novikov conjecture, Surveys of Surgery Theory (2000), pp. 195-224 | Zbl 0948.57001

[8.] Farrell, T. The obstruction to fibering a manifold over a circle, Indiana Univ. Math. J., Volume 21 (1971/1972), pp. 315-346 | Article | Zbl 0242.57016

[9.] Farrell, T.; Jones, L. A topological analogue of Mostow’s rigidity theorem, J. Am. Math. Soc., Volume 2 (1989), pp. 257-370 | Zbl 0696.57018

[10.] Gromov, M.; Lawson, H. B. Jr. The classification of simply connected manifolds of positive scalar curvature, Ann. Math. (2), Volume 111 (1980), pp. 423-434 | Article | Zbl 0463.53025

[11.] Gromov, M.; Lawson, H. B. Jr. Positive scalar curvature and the Dirac operator on complete Riemannian manifolds, Publ. Math. IHES, Volume 58 (1983), pp. 295-408 | Article | Numdam | Zbl 0538.53047

[12.] Hatcher, A. Concordance spaces, higher simple homotopy theory, and applications, Algebraic and Geometric Topology, Stanford 1976 (1978), pp. 3-21 | Zbl 0406.57031

[13.] Hitchin, N. Harmonic spinors, Adv. Math., Volume 14 (1974), pp. 1-55 | Article | Zbl 0284.58016

[14.] Igusa, K. The space of framed functions, Trans. Am. Math. Soc., Volume 301 (1987), pp. 431-477 | Article | Zbl 0624.57026

[15.] Igusa, K. The stability theorem for smooth pseudoisotopies, K-Theory, Volume 2 (1988), pp. 1-355 | Article | Zbl 0691.57011

[16.] Kreck, M.; Stolz, S. Nonconnected moduli spaces of positive sectional curvature metrics, J. Am. Math. Soc., Volume 6 (1993), pp. 825-850 | Article | Zbl 0793.53041

[17.] Lawson, H. B. Jr.; Michelsohn, M.-L. Spin Geometry (1989) | Zbl 0688.57001

[18.] Melrose, R. The Atiyah-Patodi-Singer Index Theorem (1993) | Zbl 0796.58050

[19.] Milnor, J.; Stasheff, J. Characteristic Classes (1974) | Zbl 0298.57008

[20.] Schick, T. A counterexample to the (unstable) Gromov-Lawson-Rosenberg conjecture, Topology, Volume 37 (1998), pp. 1165-1168 | Article | Zbl 0976.53052

[21.] Stolz, S. Simply connected manifolds of positive scalar curvature, Ann. Math., Volume 136 (1992), pp. 511-540 | Article | Zbl 0784.53029

[22.] Stolz, S. Positive scalar curvature metrics—existence and classification questions, Proc. Int. Cong. Math (1994), pp. 625-636 | Zbl 0848.57021

[23.] Walsh, M. Metrics of positive scalar curvature and generalized Morse functions, part II, Trans. Am. Math. Soc., Volume 366 (2014), pp. 1-50 | Article | Zbl 1294.53040

[24.] Weinberger, S. On smooth surgery, Commun. Pure Appl. Math., Volume 43 (1990), pp. 695-696 | Article | Zbl 0715.57011

Cité par Sources :