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 usual 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. In an Appendix we provide a comparison of our geometric models of twisted spin cobordism and twisted K‐theory with others arising from abstract homotopy theory.

Positive Scalar Curvature due to the Cokernel of the Classifying Map

“…In the following, we discuss the details how Gromov-Lawson admissible bordism W can be obtained, focusing on the not quite so obvious question why finitely many surgery steps suffice. The result appears also, e.g., as [18,Theorem 2.2] where the finiteness questions are not discussed or in a much more general setup in [11,Appendix 2].…”

The Gromov–Lawson–Chernysh surgery theorem

“…W are both 2-connected. However, the assumptions of the proposition only imply that B is of type ðF 2 Þ, and an additional argument is required (which goes back to [20][Theorem 2.2] and is explained also in [10]). By h-surgeries in the interior of W, we can replace W by a cobordism W 0 so that W 0 !…”

Homotopical and operator algebraic twisted K-theory

“…Their projectivizations will form the basis for our discussion of twisted K -theory and twisted Spin-cobordism. They were originally devised by Joachim and the second author in [14] for this purpose.…”

“…As an additional input, we use the Atiyah-Bott-Shapiro orientations α : MSpin −→ KO and α c : MSpin c −→ KU from [18,Section 6]. Recall first that the maps α and α c are obtained from the unit maps S n → KO n or S n → KU n (see [14,Construction 3.4.1]), by extending them over MSpin n = (P O n ) + ∧ O(n) S n and its complex analogue in the unique P O n -or P U n -equivariant fashion. Also recall that Aut I (K) = PO I acts on KO (and that P S 1 U I acts on KU) by the constructions of the previous section.…”

“…of commutative parametrized ring spectra, geometric incarnations of the twisted Atiyah-Bott-Shapiro orientations. These maps are refinements of the analogous map considered in [14] based on the framework for parametrized homotopy theory of May and Sigurdsson [19], which did not allow for multiplicative considerations. By our results above, the left one fits into a commutative square…”