Diffeomorphic souls and disconnected moduli spaces of nonnegatively curved metrics
Annales de l'Institut Fourier, Volume 72 (2022) no. 1, pp. 109-122.

We give examples of open manifolds that carry infinitely many complete metrics of nonnegative sectional curvature such that they all have the same soul, and their isometry classes lie in different connected components of the moduli space. All previously known examples of this kind have souls of codimension one. In our examples the souls have codimensions three and two.

Nous donnons des exemples de variétés ouvertes qui admettent une infinité de métriques complètes à courbure sectionnelle non négative telles que leurs âmes soient identiques et que leurs classes d’équivalence se trouvent dans des composantes connexes différentes de l’espace de modules. Tous les exemples de ce genre, connus auparavant, ont une âme de codimension un. Dans les exemples que nous présentons, les âmes sont de codimension trois et deux.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/aif.3471
Classification: 53C20
Keywords: Nonnegative curvature, soul, moduli space, positive scalar curvature.
Belegradek, Igor 1; González-Álvaro, David 2

1 School of Mathematics Georgia Institute of Technology Atlanta, GA, USA 30332
2 ETSI de Caminos, Canales y Puertos Universidad Politécnica de Madrid 28040, Spain
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
@article{AIF_2022__72_1_109_0,
     author = {Belegradek, Igor and Gonz\'alez-\'Alvaro, David},
     title = {Diffeomorphic souls and disconnected moduli spaces of nonnegatively curved metrics},
     journal = {Annales de l'Institut Fourier},
     pages = {109--122},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {72},
     number = {1},
     year = {2022},
     doi = {10.5802/aif.3471},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3471/}
}
TY  - JOUR
TI  - Diffeomorphic souls and disconnected moduli spaces of nonnegatively curved metrics
JO  - Annales de l'Institut Fourier
PY  - 2022
DA  - 2022///
SP  - 109
EP  - 122
VL  - 72
IS  - 1
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3471/
UR  - https://doi.org/10.5802/aif.3471
DO  - 10.5802/aif.3471
LA  - en
ID  - AIF_2022__72_1_109_0
ER  - 
%0 Journal Article
%T Diffeomorphic souls and disconnected moduli spaces of nonnegatively curved metrics
%J Annales de l'Institut Fourier
%D 2022
%P 109-122
%V 72
%N 1
%I Association des Annales de l’institut Fourier
%U https://doi.org/10.5802/aif.3471
%R 10.5802/aif.3471
%G en
%F AIF_2022__72_1_109_0
Belegradek, Igor; González-Álvaro, David. Diffeomorphic souls and disconnected moduli spaces of nonnegatively curved metrics. Annales de l'Institut Fourier, Volume 72 (2022) no. 1, pp. 109-122. doi : 10.5802/aif.3471. https://aif.centre-mersenne.org/articles/10.5802/aif.3471/

[1] Alexandrino, Marcos M.; Bettiol, Renato G. Lie groups and geometric aspects of isometric actions, Springer, Cham, 2015, x+213 pages | DOI | MR | Zbl

[2] Bahuaud, Eric; Guenther, Christine; Isenberg, James Convergence stability for Ricci flow, J. Geom. Anal., Volume 30 (2020) no. 1, pp. 310-336 | DOI | MR | Zbl

[3] Belegradek, Igor Pinching, Pontrjagin classes, and negatively curved vector bundles, Invent. Math., Volume 144 (2001) no. 2, pp. 353-379 | DOI | MR | Zbl

[4] Belegradek, Igor Vector bundles with infinitely many souls, Proc. Amer. Math. Soc., Volume 131 (2003) no. 7, pp. 2217-2221 | DOI | MR | Zbl

[5] Belegradek, Igor; Farrell, F. Thomas; Kapovitch, Vitali Space of nonnegatively curved metrics and pseudoisotopies, J. Differential Geom., Volume 105 (2017) no. 3, pp. 345-374 | DOI | MR | Zbl

[6] Belegradek, Igor; Kwasik, Slawomir; Schultz, Reinhard Moduli spaces of nonnegative sectional curvature and non-unique souls, J. Differential Geom., Volume 89 (2011) no. 1, pp. 49-85 | DOI | MR | Zbl

[7] Belegradek, Igor; Kwasik, Slawomir; Schultz, Reinhard Codimension two souls and cancellation phenomena, Adv. Math., Volume 275 (2015), pp. 1-46 | DOI | MR | Zbl

[8] Botvinnik, Boris; Gilkey, Peter B. The eta invariant and metrics of positive scalar curvature, Math. Ann., Volume 302 (1995) no. 3, pp. 507-517 | DOI | MR | Zbl

[9] Brendle, Simon Ricci flow and the sphere theorem, Graduate Studies in Mathematics, 111, American Mathematical Society, Providence, RI, 2010, viii+176 pages | DOI | MR | Zbl

[10] Cheeger, Jeff; Gromoll, Detlef On the structure of complete manifolds of nonnegative curvature, Ann. of Math. (2), Volume 96 (1972), pp. 413-443 | DOI | MR | Zbl

[11] Crowley, Diarmuid; Escher, Christine M. A classification of S 3 -bundles over S 4 , Differential Geom. Appl., Volume 18 (2003) no. 3, pp. 363-380 | DOI | MR | Zbl

[12] Dessai, Anand On the moduli space of nonnegatively curved metrics on Milnor spheres (2017) (https://arxiv.org/abs/1712.08821)

[13] Dessai, Anand Moduli space of nonnegatively curved metrics on manifolds of dimension 4k+1 (2020) (https://arxiv.org/abs/2005.04741)

[14] Dessai, Anand; González-Álvaro, David Moduli space of metrics of nonnegative sectional or positive Ricci curvature on homotopy real projective spaces, Trans. Amer. Math. Soc., Volume 374 (2021) no. 1, pp. 1-33 | DOI | MR | Zbl

[15] Dessai, Anand; Klaus, Stephan; Tuschmann, Wilderich Nonconnected moduli spaces of nonnegative sectional curvature metrics on simply connected manifolds, Bull. Lond. Math. Soc., Volume 50 (2018) no. 1, pp. 96-107 | DOI | MR | Zbl

[16] Ebin, David Gregory On the space of Riemannian metrics, ProQuest LLC, Ann Arbor, MI, 1968 Thesis (Ph.D.)–Massachusetts Institute of Technology | MR

[17] Ebin, David Gregory The manifold of Riemannian metrics, Global Analysis (Proc. Sympos. Pure Math., Vol. XV, Berkeley, Calif., 1968) (1970), pp. 11-40 | MR | Zbl

[18] Goette, S.; Kerin, M.; Shankar, K. Highly connected 7-manifolds and non-negative sectional curvature, Ann. of Math. (2), Volume 191 (2020) no. 3, pp. 829-892 | DOI | MR | Zbl

[19] González-Álvaro, David; Zibrowius, Marcus Open manifolds with non-homeomorphic positively curved souls, Math. Proc. Cambridge Philos. Soc., Volume 169 (2020) no. 2, pp. 357-376 | DOI | MR | Zbl

[20] Goodman, McFeely Jackson Moduli spaces of Ricci positive metrics in dimension five (2020) (https://arxiv.org/abs/2002.00333)

[21] Goodman, McFeely Jackson On the moduli spaces of metrics with nonnegative sectional curvature, Ann. Global Anal. Geom., Volume 57 (2020) no. 2, pp. 305-320 | DOI | MR | Zbl

[22] Gromov, Mikhael; Lawson, H. Blaine Jr. Positive scalar curvature and the Dirac operator on complete Riemannian manifolds, Inst. Hautes Études Sci. Publ. Math. (1983) no. 58, p. 83-196 (1984) | DOI | Numdam | MR | Zbl

[23] Grove, Karsten; Ziller, Wolfgang Curvature and symmetry of Milnor spheres, Ann. of Math. (2), Volume 152 (2000) no. 1, pp. 331-367 | DOI | MR | Zbl

[24] Himmelberg, C. J. Pseudo-metrizability of quotient spaces, Fund. Math., Volume 63 (1968), pp. 1-6 | DOI | MR | Zbl

[25] Kapovitch, Vitali; Petrunin, Anton; Tuschmann, Wilderich Non-negative pinching, moduli spaces and bundles with infinitely many souls, J. Differential Geom., Volume 71 (2005) no. 3, pp. 365-383 | DOI | MR | Zbl

[26] Kreck, Matthias Differential algebraic topology. From stratifolds to exotic spheres, Graduate Studies in Mathematics, 110, American Mathematical Society, Providence, RI, 2010, xii+218 pages | DOI | MR | Zbl

[27] Kreck, Matthias; Stolz, Stephan Nonconnected moduli spaces of positive sectional curvature metrics, J. Amer. Math. Soc., Volume 6 (1993) no. 4, pp. 825-850 | DOI | MR | Zbl

[28] May, P.; Nardin, D. A fibration of classifying spaces (mathoverflow.net/questions/182618)

[29] Milnor, John On manifolds homeomorphic to the 7-sphere, Ann. of Math. (2), Volume 64 (1956), pp. 399-405 | DOI | MR | Zbl

[30] Mosher, Robert E.; Tangora, Martin C. Cohomology operations and applications in homotopy theory, Harper & Row, Publishers, New York-London, 1968, x+214 pages | MR | Zbl

[31] Munkres, James R. Topology, Prentice Hall, Inc., Upper Saddle River, NJ, 2000, xvi+537 pages (Second edition of [MR0464128]) | MR | Zbl

[32] Ottenburger, Sadeeb A classification of 5-dimensional manifolds, souls of codimension two and non-diffeomorphic pairs (2011) (https://arxiv.org/abs/1103.0099)

[33] Porteous, Ian R. Clifford algebras and the classical groups, Cambridge Studies in Advanced Mathematics, 50, Cambridge University Press, Cambridge, 1995, x+295 pages | DOI | MR | Zbl

[34] Šarafutdinov, Vladimir A. Convex sets in a manifold of nonnegative curvature, Mat. Zametki, Volume 26 (1979) no. 1, p. 129-136, 159 | MR

[35] Tuschmann, Wilderich; Wiemeler, Michael On the topology of moduli spaces of non-negatively curved Riemannian metrics (2017) (https://arxiv.org/abs/1712.07052)

[36] Tuschmann, Wilderich; Wraith, David J. Moduli spaces of Riemannian metrics, Oberwolfach Seminars, 46, Birkhäuser Verlag, Basel, 2015, x+123 pages (Second corrected printing) | DOI | MR | Zbl

[37] Wang, McKenzie Y.; Ziller, Wolfgang Einstein metrics on principal torus bundles, J. Differential Geom., Volume 31 (1990) no. 1, pp. 215-248 | DOI | MR | Zbl

[38] Wraith, David J. On the moduli space of positive Ricci curvature metrics on homotopy spheres, Geom. Topol., Volume 15 (2011) no. 4, pp. 1983-2015 | DOI | MR | Zbl

[39] Yim, Jin-Whan Space of souls in a complete open manifold of nonnegative curvature, J. Differential Geom., Volume 32 (1990) no. 2, pp. 429-455 | DOI | MR | Zbl

[40] Ziller, Wolfgang On M. Mueter’s Ph.D. Thesis on Cheeger deformations (https://arxiv.org/abs/0909.0161)

Cited by Sources: