Twisted Spin cobordism and positive scalar curvature

Abstract

We show how a suitably twisted Spin-cobordism spectrum connects to the question of existence of metrics of positive scalar curvature on closed, smooth manifolds by building on fundamental work of Gromov, Lawson, Rosenberg, Stolz and others. We then investigate this parametrised spectrum, compute its $mod2$-cohomology and generalise the Anderson-Brown-Peterson splitting of the regular Spin-cobordism spectrum to the twisted case. Along the way we also describe the $mod2$-cohomology of various twisted, connective covers of real K-theory.

61 pages. Reworked exposition following a referee's report, to appear in Journal of Topology

References in corpus (5)

• An $\infty$-categorical approach to $R$-line bundles, $R$-module Thom spectra, and twisted $R$-homology
• Units of ring spectra and Thom spectra
• Unit spectra of K-theory from strongly self-absorbing C*-algebras
• Multiplicative parametrized homotopy theory via symmetric spectra in retractive spaces
• Homotopical and operator algebraic twisted K-theory

Cited by in corpus (12)

• The Gromov-Lawson-Chernysh surgery theorem
• The positive scalar curvature cobordism category
• The action of the mapping class group on metrics of positive scalar curvature
• Bordism for the 2-group symmetries of the heterotic and CHL strings
• Homotopical and operator algebraic twisted K-theory
• On the homotopy type of the space of metrics of positive scalar curvature
• Positive Scalar Curvature due to the Cokernel of the Classifying Map
• The Smith Fiber Sequence and Invertible Field Theories
• Bosonization and Anomaly Indicators of (2+1)-D Fermionic Topological Orders
• Twisted differential KO-theory
• The Disc-structure space
• Line Bundle Twists for Unitary Bordism are Ghosts