The strong Novikov conjecture for low degree cohomology

We show that for each discrete group Γ, the rational assembly map is injective on classes dual to , where Λ^* is the subring generated by cohomology classes of degree at most 2 (and where the pairing uses the Chern character). Our result implies homotopy invariance of higher signatures associated to classes in Λ^*. This consequence was first established by Connes–Gromov–Moscovici (Geom. Funct. Anal. 3(1): 1–78, 1993) and Mathai (Geom. Dedicata 99: 1–15, 2003). Note, however that the above injectivity statement does not follow from their methods. Our approach is based on the construction of flat twisting bundles out of sequences of almost flat bundles as first described in our work (Hanke and Schick, J. Differential Geom. 74: 293–320, 2006). In contrast to the argument in Connes-Gromov-Moscovici (Geom. Funct.Anal. 3(1): 1–78, 1993), our approach is independent of (and indeed gives a new proof of) the result of Hilsum–Skandalis (J. Reine Angew. Math. 423: 73–99, 1999) on the homotopy invariance of the index of the signature operator twisted with bundles of small curvature.