On positive scalar curvature bordism. (English) Zbl 07732156
The Stolz positive scalar curvature exact sequence, originally introduced in an unpublished preprint of Stolz [https://www3.nd.edu/~stolz/concordance.ps] and re-popularized by P. Piazza and T. Schick [J. Topol. 7, No. 4, 965–1004 (2014; Zbl 1320.58012)],
\[
\cdots \to \mathrm{Pos}^{\mathrm{spin}}_n(\mathrm{B}\Gamma) \to \Omega_n^{\mathrm{spin}}(\mathrm{B}\Gamma) \to \mathrm{R}^{\mathrm{spin}}_n(\mathrm{B}\Gamma) \to \mathrm{Pos}_{n-1}^{\mathrm{spin}}(\mathrm{B}\Gamma) \to \cdots,
\]
organizes the “classification problem” for positive scalar curvature (psc) metrics on spin manifolds with fundamental group \(\Gamma\) in a cobordism-theoretic language.
The present paper considers the case that the group is of the form \(\Gamma = G \times \mathbb{Z}\). The main result (Theorem 1.3) states that if \(4 \leq n \equiv 0 \mod 4\) and \(G\) is finitely presented with non-trivial torsion, then the psc bordism group \(\mathrm{Pos}^{\mathrm{spin}}_n(\mathrm{B}\Gamma)\) is infinite. A slightly weaker result is established in the case \(4 \leq n \equiv 2 \mod 4\), where the same conclusion is derived if \(G\) is finite and contains an element that is not conjugate to its inverse.
The proof proceeds by combining standard methods from higher index theory: A well-known delocalized index map to extract the torsion information from \(G\), and a kind of Künneth theorem applied to parts of the Stolz sequence to pass from \(G\) to \(\Gamma = G \times \mathbb{Z}\). The paper thereby both simplifies and vastly generalizes earlier work of D. Kazaras et al. [Commun. Anal. Geom. 30, No. 4, 869–890 (2022; Zbl 1514.57034)].
As pointed out in Remark 2.8 of the paper, finer results on \(\mathrm{Pos}^{\mathrm{spin}}_n(\mathrm{B}\Gamma)\) can be obtained if one is willing to assume that \(G\) satisfies additional hypotheses (such as the Strong Novikov Conjecture or being finitely embeddable into a Hilbert space), for instance by applying results from [Z. Xie et al., Adv. Math. 390, Article ID 107897, 24 p. (2021; Zbl 1475.58018); S. Weinberger and G. Yu, Geom. Topol. 19, No. 5, 2767–2799 (2015; Zbl 1328.19011)].
The present paper considers the case that the group is of the form \(\Gamma = G \times \mathbb{Z}\). The main result (Theorem 1.3) states that if \(4 \leq n \equiv 0 \mod 4\) and \(G\) is finitely presented with non-trivial torsion, then the psc bordism group \(\mathrm{Pos}^{\mathrm{spin}}_n(\mathrm{B}\Gamma)\) is infinite. A slightly weaker result is established in the case \(4 \leq n \equiv 2 \mod 4\), where the same conclusion is derived if \(G\) is finite and contains an element that is not conjugate to its inverse.
The proof proceeds by combining standard methods from higher index theory: A well-known delocalized index map to extract the torsion information from \(G\), and a kind of Künneth theorem applied to parts of the Stolz sequence to pass from \(G\) to \(\Gamma = G \times \mathbb{Z}\). The paper thereby both simplifies and vastly generalizes earlier work of D. Kazaras et al. [Commun. Anal. Geom. 30, No. 4, 869–890 (2022; Zbl 1514.57034)].
As pointed out in Remark 2.8 of the paper, finer results on \(\mathrm{Pos}^{\mathrm{spin}}_n(\mathrm{B}\Gamma)\) can be obtained if one is willing to assume that \(G\) satisfies additional hypotheses (such as the Strong Novikov Conjecture or being finitely embeddable into a Hilbert space), for instance by applying results from [Z. Xie et al., Adv. Math. 390, Article ID 107897, 24 p. (2021; Zbl 1475.58018); S. Weinberger and G. Yu, Geom. Topol. 19, No. 5, 2767–2799 (2015; Zbl 1328.19011)].
Reviewer: Rudolf Zeidler (Münster)
MSC:
| 19K56 | Index theory |
| 57R90 | Other types of cobordism |
| 58J20 | Index theory and related fixed-point theorems on manifolds |
| 58J22 | Exotic index theories on manifolds |
