The present Ph.D. thesis consists of three parts.
The first part calculates the effect of changing the spin-structure on a manifold on the Rosenberg index \(\alpha([M])\in K_\ast(C^* \pi_1 M)\). Concretely, if \([M_1]\) and \([M_2]\) denote the fundamental classes corresponding to the respective spin-structures and if we denote by \(x\in H^1(M;\mathbb{Z}/2\mathbb{Z})\) the cohomology class relating the two spin-structures, then \(\alpha([M_2]) = (\Phi_x)_\ast\circ\alpha([M_1])\) for an automorphism \(\Phi_x: C^*\pi_1 M \to C^*\pi_1 M\) that depends only on \(x\).
The second part of the Ph.D.thesis constructs a natural transformation of bordism groups \(\Omega_*^{\mathrm{spin}^c}(-) \to \Omega_{*+1}^{\mathrm{spin}}(-\times B(\mathbb{Z}/n\mathbb{Z}))\) for any odd number \(n \geq 3\) by a circle bundle construction: if a manifold \(M\) is \(\mathrm{spin}^c\), then its second Stiefel-Whitney class has an integral lift. This integral lift corresponds to a circle bundle over \(M\), whose total space is a \(\mathrm{spin}\)-manifold. The author goes on to show that if \(M\) is a totally non-spin, but \(\mathrm{spin}^c\)-manifold, then the Rosenberg index of the total space of the circle bundle vanishes. On the other hand, using the minimal hypersurface method of Schoen and Yau the author shows that there are examples of this situation where the total space does not admit a positice scalar curvature metric.
In the last part of the Ph.D.thesis the author constructs wrong way maps on generalized homology theories of classifying spaces, solving the corresponding problem raised by Hanke and Schick. Concretely, if \(M\) is a closed and connected manifold and \(N\) a closed and connected submanifold with trivialized normal bundle and of codimension two, and if the induced map \(\pi_1(N) \to \pi_1(M)\) is injective and \(\pi_2(N) \to \pi_2(M)\) is surjective, then there is a map \(E_\ast(B\pi_1 M) \to E_{\ast-2}(B\pi_1 N)\) for every generalized homology theory (satisfying some assumptions) and this map is compatible with the corresponding Pontryagin-Thom map for the pair \((M,N)\).