## Annals of K-Theory

### Positive scalar curvature and low-degree group homology

#### Abstract

Let $Γ$ be a discrete group. Assuming rational injectivity of the Baum–Connes assembly map, we provide new lower bounds on the rank of the positive scalar curvature bordism group and the relative group in Stolz’ positive scalar curvature sequence for $B Γ$. The lower bounds are formulated in terms of the part of degree up to $2$ in the group homology of $Γ$ with coefficients in the $ℂ Γ$-module generated by finite order elements. Our results use and extend work of Botvinnik and Gilkey which treated the case of finite groups. Further crucial ingredients are a real counterpart to the delocalized equivariant Chern character and Matthey’s work on explicitly inverting this Chern character in low homological degrees.

#### Article information

Source
Ann. K-Theory, Volume 3, Number 3 (2018), 565-579.

Dates
Revised: 21 December 2017
Accepted: 6 January 2018
First available in Project Euclid: 24 July 2018

https://projecteuclid.org/euclid.akt/1532397768

Digital Object Identifier
doi:10.2140/akt.2018.3.565

Mathematical Reviews number (MathSciNet)
MR3830202

Zentralblatt MATH identifier
06911677

#### Citation

Bárcenas, Noé; Zeidler, Rudolf. Positive scalar curvature and low-degree group homology. Ann. K-Theory 3 (2018), no. 3, 565--579. doi:10.2140/akt.2018.3.565. https://projecteuclid.org/euclid.akt/1532397768

#### References

• P. Baum and A. Connes, “Chern character for discrete groups”, pp. 163–232 in A fête of topology, edited by Y. Matsumoto et al., Academic Press, Boston, 1988.
• P. Baum and A. Connes, “Geometric $K$-theory for Lie groups and foliations”, Enseign. Math. $(2)$ 46:1-2 (2000), 3–42.
• B. Botvinnik and P. B. Gilkey, “The eta invariant and metrics of positive scalar curvature”, Math. Ann. 302:3 (1995), 507–517.
• R. R. Bruner and J. P. C. Greenlees, Connective real $K$-theory of finite groups, Mathematical Surveys and Monographs 169, American Mathematical Society, Providence, RI, 2010.
• N. Higson and G. Kasparov, “$E$-theory and $K\!K$-theory for groups which act properly and isometrically on Hilbert space”, Invent. Math. 144:1 (2001), 23–74.
• N. Higson and J. Roe, “$K$-homology, assembly and rigidity theorems for relative eta invariants”, Pure Appl. Math. Q. 6:2 (2010), 555–601.
• M. Matthey, “The Baum–Connes assembly map, delocalization and the Chern character”, Adv. Math. 183:2 (2004), 316–379.
• P. Piazza and T. Schick, “Groups with torsion, bordism and rho invariants”, Pacific J. Math. 232:2 (2007), 355–378.
• P. Piazza and T. Schick, “Rho-classes, index theory and Stolz' positive scalar curvature sequence”, J. Topol. 7:4 (2014), 965–1004.
• J. Rosenberg and S. Stolz, “Metrics of positive scalar curvature and connections with surgery”, pp. 353–386 in Surveys on surgery theory, vol. 2, edited by S. Cappell et al., Ann. of Math. Stud. 149, Princeton University Press, 2001.
• S. Weinberger and G. Yu, “Finite part of operator $K$-theory for groups finitely embeddable into Hilbert space and the degree of nonrigidity of manifolds”, Geom. Topol. 19:5 (2015), 2767–2799.
• Z. Xie and G. Yu, “Positive scalar curvature, higher rho invariants and localization algebras”, Adv. Math. 262 (2014), 823–866.
• Z. Xie and G. Yu, “Higher rho invariants and the moduli space of positive scalar curvature metrics”, Adv. Math. 307 (2017), 1046–1069.
• R. Zeidler, “Positive scalar curvature and product formulas for secondary index invariants”, J. Topol. 9:3 (2016), 687–724.
• R. Zeidler, Secondary large-scale index theory and positive scalar curvature, Ph.D. thesis, Georg-August-Universität Göttingen, 2016, https://tinyurl.com/Zeidler-thesis.
• V. F. Zenobi, “Mapping the surgery exact sequence for topological manifolds to analysis”, J. Topol. Anal. 9:2 (2017), 329–361.