Positive scalar curvature metrics via end-periodic manifolds.(English)Zbl 1454.58020

The paper under review uses the index theory for end-periodic manifolds developed by T. Mrowka et al. [Compos. Math. 152, No. 2, 399–444 (2016; Zbl 1348.58012)] to investigate the moduli space of Riemannian metrics of positiv scalar curvature on closed even-dimensional spin-manifolds. The index invariant employed is the end-periodic $$\rho$$-invariant. One main result of the paper is that it can be used to different extents as an obstruction to positive scalar curvature. This is an even dimensional analogue of results by N. Higson and J. Roe [Pure Appl. Math. Q. 6, No. 2, 555–601 (2010; Zbl 1227.19006)] involving the usual $$\rho$$-invariant on odd dimensional manifolds. Furthermore, the authors also give a more conceptual proof for a theorem by Mrowka et al. about the number of components of the moduli space of Riemannian metrics of positive scalar curvature.
The technical innovation of the paper is the definition of abelian groups $$K^{\mathrm{ep}}_1(B\pi)$$, $$\Omega^{\mathrm{ep,spin}}_m(B\pi)$$, $$\Omega^{\mathrm{ep,spin},+}_m(B\pi)$$ and $$S_1^{\mathrm{ep}}(\sigma_1,\sigma_2)$$, which are end-periodic versions of the topological realization of $$K$$-homology by P. Baum and R. G. Douglas [Contemp. Math. 10, 1–31 (1982; Zbl 0507.55004)], the spin-bordism and psc spin-bordism groups and the structure group of N. Higson and J. Roe [Pure Appl. Math. Q. 6, No. 2, 555–601 (2010; Zbl 1227.19006)], respectively. They are isomorphic to their non-end-periodic counterparts, but their cycles are better adapted to the problem under investigation. The end-periodic $$\rho$$-invariant defines homomorphisms on them with values in $$\mathbb{R}/\mathbb{Z}$$ for the first two and in $$\mathbb{R}$$ for the last two.
End-periodic manifolds are merely a tool in this paper, but not objects of interest themselves. They provide a suitable notion of bordism between the closed oriented even-dimensional manifolds which their ends are modeled on, and thereby constitute part of the relations in the definitions of the above-mentioned groups.

MSC:

 58J28 Eta-invariants, Chern-Simons invariants 19K33 Ext and $$K$$-homology 19K56 Index theory 53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions 53C27 Spin and Spin$${}^c$$ geometry 58D27 Moduli problems for differential geometric structures
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References:

 [1] ; Atiyah, Colloque “Analyse et Topologie” en l’Honneur de Henri Cartan. Astérisque, 32-33, 43 (1976) [2] 10.1017/S0305004100049410 · Zbl 0297.58008 [3] 10.1017/S0305004100051872 · Zbl 0314.58016 [4] 10.1017/S0305004100052105 · Zbl 0325.58015 [5] ; Baum, Operator algebras and applications I. Proc. Sympos. Pure Math., 38, 117 (1982) [6] 10.1016/j.geomphys.2013.03.010 · Zbl 1282.58018 [7] 10.1017/S030500411400005X · Zbl 1290.58015 [8] 10.4171/JNCG/209 · Zbl 1342.58014 [9] 10.1007/BF01444505 · Zbl 0835.58034 [10] 10.1007/s00208-016-1364-7 · Zbl 1370.19001 [11] 10.1007/BF01388468 · Zbl 0547.58032 [12] ; Gilkey, Invariance theory, the heat equation, and the Atiyah-Singer index theorem. Mathematics Lecture Series, 11 (1984) · Zbl 0565.58035 [13] 10.2307/1971103 · Zbl 0463.53025 [14] 10.2307/1971198 · Zbl 0445.53025 [15] 10.1007/BF02953774 · Zbl 0538.53047 [16] 10.1007/s10240-005-0030-5 · Zbl 1073.19003 [17] 10.4310/PAMQ.2010.v6.n2.a11 · Zbl 1227.19006 [18] 10.1016/0001-8708(74)90021-8 · Zbl 0284.58016 [19] ; Keswani, New York J. Math., 5, 53 (1999) [20] 10.1016/S0040-9383(99)00045-2 · Zbl 0983.19002 [21] 10.1007/BF00130915 · Zbl 0815.53048 [22] 10.1016/0022-1236(92)90022-B · Zbl 0783.57015 [23] 10.1016/0377-0257(93)80040-i [24] ; Miyazaki, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 30, 549 (1984) [25] 10.4310/jdg/1320067650 · Zbl 1238.57028 [26] 10.1112/S0010437X15007502 · Zbl 1348.58012 [27] 10.4171/JNCG/2 · Zbl 1158.58012 [28] 10.2140/pjm.2007.232.355 · Zbl 1152.58020 [29] ; Rosenberg, Geometric methods in operator algebras. Pitman Res. Notes Math. Ser., 123, 341 (1986) [30] 10.1007/BF01647970 · Zbl 0423.53032 [31] 10.4310/jdg/1214440981 · Zbl 0615.57009 [32] 10.1073/pnas.85.15.5362 · Zbl 0659.57016
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