×

The action of the mapping class group on metrics of positive scalar curvature. (English) Zbl 1490.58007

The diffeomorphism group of a manifold acts on the space of positive scalar curvature (psc) metrics and, thus, induces an action of its mapping class group, \(\pi_0(Diff(M))\), by homotopy classes of homotopy self-equivalences of \(\mathcal{R}^+(M)\), giving a group homomorphism: \[ \pi_0(\mathsf{Diff(M)}) \longrightarrow \pi_0(h\mathsf{Aut}(\mathcal{R}^+(M)) ). \] The work under review displays sufficient conditions under which the previous map factors through a certain cobordism group \(\Omega_d\) of closed \(n\)-dimensional manifolds. For this, it is first required that the dimension of the manifold be \(n\geq 6\) and, secondly, that both the manifolds and cobordisms involved be spin or, else, enriched by a tangential structure.
The key to the aforementioned factorization and, perhaps, the most important result of the paper (see Theorem G and its generalization Theorem 3.6), is the construction of a functor from the cobordism category (spin or with a tangential structure) to the homotopy category of spaces. This functor is completely determined by the following: on objects it sends a (cobordism class of a) manifold to \(\mathcal{R}^+(M) \in h\mathsf{Top}\), while a morphism, given by a cobordism \(W\) from \(M_0\) to \(M_1\), is sent to an automorphism \(\mathcal{S}_W\) of surgery-theoretic nature. The definition of \(\mathcal{S}_W\) relies on handle decompositions, specially on Hatcher-Igusa’s 2-index theorem, and on the geometric deformation/construction of psc-metrics due to V. Chernysh and M. Walsh, which themselves can be seen as a refinement of the Surgery Theorem of Gromov-Lawson and Schoen-Yau.
In this way, a diffeomorphism \(f \in \mathsf{Diff(M)}\) gives rise to its mapping torus \(T_f \in \Omega_d(M)\) which then gives the automorphism \(\mathcal{S}_{T_f}\). This factorization is presented as a rigidity result once the cobordism group is -first of all- abelian and, also, rather small and known in certain cases (see Theorem B). Furthermore, the functor \(\mathcal{S}_W\) is not just a diffeomorfism invariant of the cobordism \(W\) but an invariant of the cobordism (with corners) class of it.
The paper has a detailed exposition of the categories involved while often relying on the technical details of M. Walsh [Metrics of positive scalar curvature and generalised Morse functions. I. Providence, RI: American Mathematical Society (AMS) (2011; Zbl 1251.53001); Proc. Am. Math. Soc. 141, No. 7, 2475–2484 (2013; Zbl 1285.57016); Trans. Am. Math. Soc. 366, No. 1, 1–50 (2014; Zbl 1294.53040)], which it subtly refines. However, it is not without its geometric merit. In particular, it is shown that if a spin diffeomorphism \(f\) of the \(n\)-sphere, \(n\geq 6\), sends the round metric \(g_0\) to an homotopic one, \(f^*g_0\), then the automorphism \(f^*\) is necessarily trivial, i.e., homotopic to the identity in \(\mathsf{Aut}(\mathcal{R}^+(\mathcal{S}^n))\). This last result generalizes to simply-connected spin manifolds of dimension at least \(7\), (see Corollary C and Proposition D).

MSC:

58D17 Manifolds of metrics (especially Riemannian)
57R65 Surgery and handlebodies
53C27 Spin and Spin\({}^c\) geometry
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Anderson, DW; Brown, EH Jr; Peterson, FP, Spin cobordism, Bull. Am. Math. Soc., 72, 256-260 (1966) · Zbl 0136.44103 · doi:10.1090/S0002-9904-1966-11486-6
[2] Adachi, M., Embeddings and Immersions. Translations of mathematical monographs (1993), Providence: American Mathematical Society, Providence · Zbl 0810.57001
[3] Botvinnik, B.; Ebert, J.; Randal-Williams, O., Infinite loop spaces and positive scalar curvature, Invent. Math., 209, 3, 749-835 (2017) · Zbl 1377.53067 · doi:10.1007/s00222-017-0719-3
[4] Chernysh, V.: On the homotopy type of the space \({\cal{R}}^+(m), 2004\), arXiv:math/0405235
[5] Dieck, T., Algebraic Topology (2008), Zurich: EMS textbooks in mathematics, European Mathematical Society, Zurich · Zbl 1156.55001 · doi:10.4171/048
[6] J. Ebert. Characteristic classes of spin surface bundles: applications of the Madsen-Weiss theory, volume 381 of Bonner Mathematische Schriften [Bonn Mathematical Publications]. Universität Bonn, Mathematisches Institut, Bonn: Dissertation, p. 2006. Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn (2006) · Zbl 1121.57301
[7] Ebert, J., Frenck, G.: The Gromov-Lawson-Chernysh surgery theorem. Bol. Soc. Mat. Mex. (3) (2020). doi:10.1007/s40590-021-00310-w · Zbl 1467.53044
[8] Ebert, J.; Randal-Williams, O., Infinite loop spaces and positive scalar curvature in the presence of a fundamental group, Geom. Topol., 23, 3, 1549-1610 (2019) · Zbl 1515.53052 · doi:10.2140/gt.2019.23.1549
[9] Ebert, J., Randal-Williams, O.: The positive scalar curvature cobordism category. ArXiv e-prints, 2019, arxiv:1904.12951
[10] Frenck, G.: The Action of the mapping class group on spaces of metrics of positive scalar curvature. PhD thesis, WWU Münster, Available through the author’s website, 2019 · Zbl 1418.57001
[11] Frenck, G.: H-space structures on spaces of metrics of positive scalar curvature, 2020, arXiv:2004.01033
[12] Gromov, M.; Lawson, HB Jr, The classification of simply connected manifolds of positive scalar curvature, Ann. Math. (2), 111, 3, 423-434 (1980) · Zbl 0463.53025 · doi:10.2307/1971103
[13] Galatius, S.; Randal-Williams, O., Stable moduli spaces of high-dimensional manifolds, Acta Math., 212, 2, 257-377 (2014) · Zbl 1377.55012 · doi:10.1007/s11511-014-0112-7
[14] Galatius, S.; Randal-Williams, O., Abelian quotients of mapping class groups of highly connected manifolds, Math. Ann., 365, 1-2, 857-879 (2016) · Zbl 1343.55005 · doi:10.1007/s00208-015-1300-2
[15] Gay, D., Wehrheim, K., Woodward, C.: Connected cerf theory. in preparation, 2012. https://math.berkeley.edu/ katrin/papers/cerf.pdf
[16] Hatcher, AE, Higher simple homotopy theory, Ann. Math. (2), 102, 1, 101-137 (1975) · Zbl 0305.57009 · doi:10.2307/1970977
[17] Hirsch, M., Differential Topology. Graduate Texts in Mathematics (1976), New York: Springer, New York · Zbl 0356.57001
[18] Hitchin, N., Harmonic spinors, Adv. Math., 14, 1-55 (1974) · Zbl 0284.58016 · doi:10.1016/0001-8708(74)90021-8
[19] Hebestreit, F.; Joachim, M., Twisted spin cobordism and positive scalar curvature, J. Topol., 13, 1, 1-58 (2020) · Zbl 1523.55006 · doi:10.1112/topo.12122
[20] Igusa, K., The stability theorem for smooth pseudoisotopies, K-Theory, 2, 1-2, vi+355 (1988) · Zbl 0691.57011 · doi:10.1007/BF00533643
[21] Kervaire, MA, Le théorème de Barden-Mazur-Stallings, Comment. Math. Helv., 40, 31-42 (1965) · Zbl 0135.41503 · doi:10.1007/BF02564363
[22] Kreck, M., Cobordism of odd-dimensional diffeomorphisms, Topology, 15, 4, 353-361 (1976) · Zbl 0335.57021 · doi:10.1016/0040-9383(76)90029-X
[23] Kreck, M., Surgery and duality, Ann. Math. (2), 149, 3, 707-754 (1999) · Zbl 0935.57039 · doi:10.2307/121071
[24] MacLane, S., Categories for the working mathematician (1971), New York-Berlin: Springer, New York-Berlin · Zbl 0705.18001
[25] Palais, RS, Homotopy theory of infinite dimensional manifolds, Topology, 5, 1-16 (1966) · Zbl 0138.18302 · doi:10.1016/0040-9383(66)90002-4
[26] Perlmutter, N.: Cobordism categories and parametrized morse theory, 2017, arXiv:1703.01047
[27] Smale, S., On the structure of manifolds, Am. J. Math., 84, 387-399 (1962) · Zbl 0109.41103 · doi:10.2307/2372978
[28] Schoen, R.; Yau, ST, On the structure of manifolds with positive scalar curvature, Manuscripta Math., 28, 1-3, 159-183 (1979) · Zbl 0423.53032 · doi:10.1007/BF01647970
[29] Thom, R., Quelques propriétés globales des variétés différentiables, Comment. Math. Helv., 28, 17-86 (1954) · Zbl 0057.15502 · doi:10.1007/BF02566923
[30] Wall, CTC, Geometrical connectivity, I. J. Lond. Math. Soc., 2, 3, 597-604 (1971) · Zbl 0214.22304 · doi:10.1112/jlms/s2-3.4.597
[31] Walsh, M., Metrics of positive scalar curvature and generalised Morse functions, Part I, Mem. Am. Math. Soc., 209, 983, xviii+80 (2011) · Zbl 1251.53001 · doi:10.1090/S0065-9266-10-00622-8
[32] Walsh, M., Cobordism invariance of the homotopy type of the space of positive scalar curvature metrics, Proc. Am. Math. Soc., 141, 7, 2475-2484 (2013) · Zbl 1285.57016 · doi:10.1090/S0002-9939-2013-11647-3
[33] Walsh, M., Metrics of positive scalar curvature and generalised Morse functions. Part II, Trans. Am. Math. Soc., 366, 1, 1-50 (2014) · Zbl 1294.53040 · doi:10.1090/S0002-9947-2013-05715-7
[34] Wall, CTC, Differential Topology (2016), Cambridge: Cambridge University Press, Cambridge · Zbl 1358.57001 · doi:10.1017/CBO9781316597835
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.