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The space of positive scalar curvature metrics on a manifold with boundary. (English) Zbl 1453.58005

Author’s abstract: We study the space of Riemannian metrics with positive scalar curvature on a compact manifold with boundary. These metrics extend a fixed boundary metric and take a product structure on a collar neighbourhood of the boundary. We show that the weak homotopy type of this space is preserved by certain surgeries on the boundary in co-dimension at least three. Thus, under reasonable circumstances there is a weak homotopy equivalence between the space of such metrics on a compact spin manifold \(W\), of dimension \(n\ge 6\) and whose boundary inclusion is 2-connected, and the corresponding space of metrics of positive scalar curvature on the standard disk \(D^n\). Indeed, for certain boundary metrics, this space is weakly homotopy equivalent to the space of all metrics of positive scalar curvature on the standard sphere \(S^n\). Finally, we prove analogous results for the more general space where the boundary metric is left unfixed.

MSC:

58D17 Manifolds of metrics (especially Riemannian)
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
55P10 Homotopy equivalences in algebraic topology
58B05 Homotopy and topological questions for infinite-dimensional manifolds
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[1] Besse, Arthur L.Einstein manifolds. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 10.Springer-Verlag, Berlin,(1987). xii+510 pp. ISBN 978-3-540-74120-6MR0867684,Zbl 0613.53001, doi:10.1007/978-3-540-74311-8. · Zbl 0613.53001
[2] Botvinnik, Boris; Ebert, Johannes; Randal-Williams, Oscar.Infinite loop spaces and positive scalar curvature.Invent. Math.209(2017), no. 3, 749-835. MR3681394,Zbl 1377.53067,arXiv:1411.7408, doi:10.1007/s00222-017-0719-3.854,857, 858 · Zbl 1377.53067
[3] Botvinnik, Boris; Hanke, Bernhard; Schick, Thomas; Walsh, Mark.Homotopy groups of the moduli space of metrics of positive scalar curvature.Geom. Topol.14(2010), no. 4, 2047-2076.MR2680210,Zbl 1201.58006,arXiv:0907.5188, doi:10.2140/gt.2010.14.2047.856 · Zbl 1201.58006
[4] Botvinnik, Boris; Gilkey, Peter B.The eta invariant and metrics of positive scalar curvature.Math. Ann.302(1995), no. 3, 507-517.MR1339924,Zbl 0835.58034, doi:10.1007/BF01444505.856 · Zbl 0835.58034
[5] Chernysh, Vladislav.On the homotopy type of the spaceR+(M). Preprint, 2004. arXiv:math.GT/0405235.857,868,874,875,876,879
[6] Chernysh, Vladislav.A quasifibration of spaces of positive scalar curvature metrics. Proc. Amer. Math. Soc.134(2006), no. 9, 2771-2777.MR2213758,Zbl 1092.58005, arXiv:0405436, doi:10.1090/S0002-9939-06-08539-X.857,859,911,922,923 · Zbl 1092.58005
[7] Carr Rodney.Construction of manifolds of positive scalar curvature.Trans. Amer. Math. Soc.307(1988), no. 1, 63-74.MR0936805,Zbl 0654.53049, doi:10.2307/2000751. 856,857,875,876 · Zbl 0654.53049
[8] Crowley, Diamuld; Schick, Thomas.The Gromoll filtration,KO-characteristic classes and metrics of positive scalar curvature.Geom. Topol.17(2013), no. 3, 1773- 1789.MR3073935,Zbl 1285.57015,arXiv:1204.6474, doi:10.2140/gt.2013.17.1773.854, 856 · Zbl 1285.57015
[9] Ebert, Johannes.; Frenck, Georg.The Gromov-Lawson-Chernysh surgery theorem. Preprint, 2018.arXiv:1807.06311v1.854,857,859,868,871,874,875,876,877, 879,881,882,911,923,924
[10] Ebert, Johannes; Randal-Williams, Oscar.Infinite loop spaces and positive scalar curvature in the presence of a fundamental group.Geom. Topol.23(2019), no. 3, 1549-1610.MR3956897,Zbl 07079063,arXiv:1711.11363, doi:10.2140/gt.2019.23.1549. 854 · Zbl 1515.53052
[11] Ebert, Johannes; Randal-Williams, Oscar.The positive scalar curvature cobordism category. Preprint, 2019.arXiv:1904.12951. · Zbl 1515.53052
[12] Gajer, Pawel.Riemannian metrics of positive scalar curvature on compact manifolds with boundary.Ann. Global Anal. Geom.5(1987), no. 3, 179-191.MR962295,Zbl 0665.53040, doi:10.1007/BF00128019.855,861,874,875,876 · Zbl 0665.53040
[13] Galatius, S.; Randal-Williams, Oscar.Stable moduli spaces of high-dimensional manifolds.Acta Math.212(2014), no. 2, 257-377.MRMR3207759,Zbl 1377.55012, arXiv:1201.3527, doi:10.1007/s11511-014-0112-7.857 · Zbl 1377.55012
[14] Gromov, Mikhael; Lawson, H. Blaine, Jr.The classification of simply connected manifolds of positive scalar curvature.Ann. of Math.111(1980), no. 3, 423-434. MR0577131,Zbl 0463.53025, doi:10.2307/1971103.855,856,861,868,870,871,922 · Zbl 0463.53025
[15] Hanke, Bernhard; Schick, Thomas; Steimle, WolfgangThe space of metrics of positive scalar curvature.Publ. Math. Inst. Hautes ´Etudes Sci.120(2014) 335-367. MR3270591,Zbl 1321.58008,arXiv:1212.0068v3, doi:10.1007/s10240-014-0062-9.854, 856 · Zbl 1321.58008
[16] Hatcher, Allen.Algebraic topology.Cambridge University Press, Cambridge, (2002). xii+544 pp. ISBN: 0-521-79160-X; 0-521-79540-0.MR1867354,Zbl 1044.55001. 876 · Zbl 1044.55001
[17] Hirsch, Morris W.Differential topology. Graduate Texts in Mathematics, No. 33.Springer-Verlag New York-Heidelberg, (1976) x+222 pp. ISBN: 978-0-387-90148-0 MR0448362,Zbl 0356.57001, doi:10.1007/978-1-4684-9449-5.870,871 · Zbl 0356.57001
[18] Hitchin, Nigel.Harmonic spinors.Advances in Math.14(1974), 1-55.MR0358873, Zbl 0284.58016, doi:10.1016/0001-8708(74)90021-8.856,857 · Zbl 0284.58016
[19] Hoelzel, Sebastian.Surgery stable curvature conditions.Math. Ann.365(2016), no. 1-2, 13-47.MR3498902,Zbl 1348.53043,arXiv:1303.6531, doi:10.1007/s00208-0151265-1. · Zbl 1348.53043
[20] Kreck, Matthias; Stolz, Stephan.Nonconnected moduli spaces of positive sectional curvature metrics.J. Amer. Math. Soc.6(1993), no. 4, 825-850.MR1205446,Zbl 0793.53041, doi:10.2307/2152742.856 · Zbl 0793.53041
[21] Marques, Fernando Coda.Deforming three-manifolds with positive scalar curvature.Ann. of Math.176(2012), no. 2, 815-863.MR2950765,Zbl 06093942, arXiv:0907.2444, doi:10.4007/annals.2012.176.2.3.854 · Zbl 1319.53028
[22] Palais, Richard S.Homotopy theory of infinite dimensional manifolds.Topology5 (1966), 1-16.MR0189028,Zbl 0138.18302, doi:10.1016/0040-9383(66)90002-4.876 · Zbl 0138.18302
[23] Peterson, Peter.Riemannian Geometry. Second edition. Graduate Texts in Mathematics, 171.Springer, New York, (2006). xvi+ 405 pp. ISBN 978-1-4419-2123-9. MR2243772,Zbl 1220.53002, doi:10.1007/978-0-387-29403-2.864 · Zbl 1220.53002
[24] Rosenberg, Jonathan; Stolz, Stephan.Metrics of positive scalar curvature and connections with surgery.Surveys on Surgery Theory,Vol. 2, 353-386. Ann. of Math. Studies 149,Princeton Univ. Press, Princeton, NJ, (2001).MR1818778,Zbl 0971.57003. 856,857,868,871 · Zbl 0971.57003
[25] Schoen, Richard M.; Yau, Shing-Tung.On the structure of manifolds with positive scalar curvature.Manuscripta Math.28(1979), no. 1-3, 159-183.MR0535700,Zbl 0423.53032, doi:10.1007/BF01647970. · Zbl 0423.53032
[26] Stolz, Stephan.Simply connected manifolds of positive scalar curvature.Ann. of Math.136, (1992), no. 3, 511-540.MR1189863,Zbl 0784.53029, doi:10.2307/2946598. 856 · Zbl 0784.53029
[27] Tuschmann, Wilderich; Wraith, David J.Moduli spaces of Riemannian metrics. Second corrected printing. Oberwolfach Seminars, 46.Birkh¨auser Verlag, Basel, (2015 x+123 pp. ISBN: 978-3-0348-0948-1.MR3445334,Zbl 1336.53002, doi:10.1007/978-30348-0948-1.858 · Zbl 1336.53002
[28] Walsh, Mark G.Metrics of positive scalar curvature and generalised Morse functions, part I.Mem. Amer. Math. Soc.209(2011), no. 983, xviii+80 pp. ISBN: 978-08218-5304-7.MR2789750,Zbl 1251.53001,arXiv:0811.1245, doi:10.1090/S0065-9266-1000622-8.855,857,864,865,866,868,871,872,874,875,876,903,922 · Zbl 1251.53001
[29] Walsh, Mark G.Cobordism invariance of the homotopy type of the space of positive scalar curvature metrics.Proc. Amer. Math. Soc.141(2013), no. 7, 2475-2484. MR3043028,Zbl 1285.57016,arXiv:1109.6878, doi:10.1090/S0002-9939-2013-11647-3. 857 · Zbl 1285.57016
[30] Walsh, Mark G.
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