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 $mod~2$-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 $mod~2$-cohomology of various twisted, connective covers of real K-theory.