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 -genera, thus establishing the non-multiplicativity of the -genus in fiber bundles with simply connected base.
@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 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/ DO - 10.1007/s10240-014-0062-9 LA - en ID - PMIHES_2014__120__335_0 ER -
%0 Journal Article %A Hanke, Bernhard %A Schick, Thomas %A Steimle, Wolfgang %T The space of metrics of positive scalar curvature %J Publications Mathématiques de l'IHÉS %D 2014 %P 335-367 %V 120 %I Springer Berlin Heidelberg %C Berlin/Heidelberg %U http://archive.numdam.org/articles/10.1007/s10240-014-0062-9/ %R 10.1007/s10240-014-0062-9 %G en %F PMIHES_2014__120__335_0
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.] The relative Yamabe invariant, Commun. Anal. Geom., Volume 10 (2002), pp. 925-954 | Zbl
[2.] Einstein Manifolds (1987) | Zbl
[3.] Homotopy groups of the moduli space of metrics of positive scalar curvature, Geom. Topol., Volume 14 (2010), pp. 2047-2076 | DOI | Zbl
[4.] Fibrations over spheres, Topology, Volume 6 (1967), pp. 489-499 | DOI | Zbl
[5.] On the index of a fibered manifold, Proc. Am. Math. Soc., Volume 8 (1957), pp. 587-596 | DOI | Zbl
[6.] The Gromoll filtration, KO-characteristic classes and metrics of positive scalar curvature, Geom. Topol., Volume 17 (2013), pp. 1773-1790 | DOI | Zbl
[7.] Manifold aspects of the Novikov conjecture, Surveys of Surgery Theory (2000), pp. 195-224 | Zbl
[8.] The obstruction to fibering a manifold over a circle, Indiana Univ. Math. J., Volume 21 (1971/1972), pp. 315-346 | DOI | Zbl
[9.] A topological analogue of Mostow’s rigidity theorem, J. Am. Math. Soc., Volume 2 (1989), pp. 257-370 | Zbl
[10.] The classification of simply connected manifolds of positive scalar curvature, Ann. Math. (2), Volume 111 (1980), pp. 423-434 | DOI | Zbl
[11.] Positive scalar curvature and the Dirac operator on complete Riemannian manifolds, Publ. Math. IHES, Volume 58 (1983), pp. 295-408 | DOI | Numdam | Zbl
[12.] Concordance spaces, higher simple homotopy theory, and applications, Algebraic and Geometric Topology, Stanford 1976 (1978), pp. 3-21 | Zbl
[13.] Harmonic spinors, Adv. Math., Volume 14 (1974), pp. 1-55 | DOI | Zbl
[14.] The space of framed functions, Trans. Am. Math. Soc., Volume 301 (1987), pp. 431-477 | DOI | Zbl
[15.] The stability theorem for smooth pseudoisotopies, K-Theory, Volume 2 (1988), pp. 1-355 | DOI | Zbl
[16.] Nonconnected moduli spaces of positive sectional curvature metrics, J. Am. Math. Soc., Volume 6 (1993), pp. 825-850 | DOI | Zbl
[17.] Spin Geometry (1989) | Zbl
[18.] The Atiyah-Patodi-Singer Index Theorem (1993) | Zbl
[19.] Characteristic Classes (1974) | Zbl
[20.] A counterexample to the (unstable) Gromov-Lawson-Rosenberg conjecture, Topology, Volume 37 (1998), pp. 1165-1168 | DOI | Zbl
[21.] Simply connected manifolds of positive scalar curvature, Ann. Math., Volume 136 (1992), pp. 511-540 | DOI | Zbl
[22.] Positive scalar curvature metrics—existence and classification questions, Proc. Int. Cong. Math (1994), pp. 625-636 | Zbl
[23.] Metrics of positive scalar curvature and generalized Morse functions, part II, Trans. Am. Math. Soc., Volume 366 (2014), pp. 1-50 | DOI | Zbl
[24.] On smooth surgery, Commun. Pure Appl. Math., Volume 43 (1990), pp. 695-696 | DOI | Zbl
Cité par Sources :
