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

Hallam, Michael; Mathai, Varghese

Ann. \(K\)-Theory 5, No. 3, 639-676 (2020).

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.

Reviewer: Christopher Wulff (Göttingen)

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

Keywords:

positive scalar curvature metrics; maximal Baum-Connes conjecture; end-periodic manifolds; end-periodic \(K\)-homology; end-periodic eta invariant; vanishing theorems; end-periodic bordism

Citations:

Zbl 1348.58012; Zbl 1227.19006; Zbl 0835.58034; Zbl 0507.55004

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