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This thesis analyzes topological aspects of infinite rank bundles with gauge groups and invertible pseudodifferential operators as transition maps. These bundles appear naturally in infinite dimensional geometry, for instance, when studying the geometry of mapping spaces between closed manifolds. In the first part, using an analogue of Chern-Weil theory, we construct characteristic classes for these bundles (called exotic characteristic classes) and use them to find topologically nontrivial examples. In particular, we construct nonzero elements in the cohomology of a mapping space. In the second part, we define a K-theory ring for psendodifferential bundles and use exotic classes to find nonzero elements in this ring.
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