Twisted K-theory and its equivariant version have seen numerous applications in recent years, e.g. to model certain types of charges and fields in Quantum Field Theories and to implement T-duality isomorphisms. For manifolds, they typically come in play when spin structures which give rise to K-theory orientations are not present. In conjuction with of index theory of Dirac operators with twists they have been used to obtain obstructions to positive scalar curvature for certain non-spin manifolds. The talk will give a survey on models for the construction of twisted K-theory groups test cases for their use, and for computation tools. In the spirit of the conference, we will discuss in some detail index theoretic aspects.

Let $(M,\mathcal{F})$ be a compact foliated manifold and let $G$ be its monodromy groupoid. We can then define a long exact sequence of K-theory groups of $C^\ast$-algebras, called the adiabatic exact sequence, which is the receptacle of index theoretic invariants: the primary ones in the K-theory of the groupoid $C^\ast$-algebra of $G$ and the secondary invariants in the K-theory group associated to its adiabatic deformation, the so-called $\rho$-classes. Let us fix a complete transversal $T$ of $(M,\mathcal{F})$ and the étale subgroupoid $\Gamma$ of $G$ obtained by restriction to $T$; then for any dense and holomorphically closed subalgebra $\mathcal{A}(\Gamma)$ in $C_r^\ast(\Gamma)$, we are going to define a Chern type mapping from the adiabatic exact sequence to a long exact sequence of periodic homology groups, which in particular sends the $\rho$-classes to the delocalized periodic homology group of $\mathcal{A}(\Gamma)$. Under suitable hypotheses about the foliation, through pairing with delocalized periodic cocycles, it is possible to extract numeric $\rho$-invariants, associated to Dirac type operators, in order to obtain geometric and topological applications about $(M,\mathcal{F})$.

The existence of spin structures on moduli spaces depends on the calculation of the 2nd Stiefel-Whitney class of the KO-index of a family of twisted Dirac operators. This question is particularly important for Calabi-Yau 4-folds, where spin structures on moduli spaces enter the construction of K-theoretic enumerative invariants. I will explain how to solve this problem using the 10-dimensional Diaconescu-Moore-Witten $E_8$-index, together with Adams spectral sequence calculations of spin bordism groups of mapping spaces.

We will discuss the curvature terms appearing in the Bochner-Lichnerowicz-Weitzebock formulae for the Dirac operator twisted with tensor bundles. We give explicit conditions on the eigenvalues of the curvature operator which imply that the curvature terms are positive definite, and discuss topological implications for spin manifolds. In the case of the tangent bundle and the bundle of symmetric two tensors we show that, up to connected sums, if the the Witten genus of a 5-connected manifold of dimension at least 13 vanishes then the manifold admits a metric for which the curvaure terms are positive definite. If the dimension is less than 72 that is a necessary and sufficient condition. This is joint work with Renato Bettiol.