In this thesis the close relationship between the topological $K$-homology group of the spacetime manifold $X$ of string theory and D-branes in string theory is examined. An element of the $K$-homology group is given by an equivalence class of $K$-cycles $[M,E,\phi]$, where $M$ is a closed spin$^c$ manifold, $E$ is a complex vector bundle over $M$, and $\phi: M\rightarrow X$ is a continuous map. It is proposed that a $K$-cycle $[M,E,\phi]$ represents a D-brane configuration wrapping the subspace $\phi(M)$. As a consequence, the $K$-homology element defined by $[M,E,\phi]$ represents a class of D-brane configurations that have the same physical charge. Furthermore, the $K$-cycle representation of D-branes resembles the modern way of characterizing fundamental strings, in which the strings are represented as two-dimensional surfaces with maps into the spacetime manifold. This classification of D-branes also suggests the possibility of physically interpreting D-branes wrapping singular subspaces of spacetime, enlarging the known types of singularities that string theory can cope with. / Master of Science

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We construct geometric models for K-homology with coefficients based on the theory of Z/k-manifolds. To do so, we generalize the operations and relations Baum and Douglas put on spinc-manifolds to spinc Z/kZ-manifolds. We then de fine a model for K-homology with coefficients in Z/k using cycles of the form ((Q,P), (E,F), f) where (Q, P) is a spinc Z/k-manifold, (E, F) is a Z/k-vector bundle over (Q, P) and f is a continuous map from (Q, P) into the space whose K-homology we are modelling. Using results of Rosenberg and Schochet, we then construct an analytic model for K-homology with coefficients in Z/k and a natural map from our geometric model to this analytic model. We show that this map is an isomorphism in the case of finite CW-complexes. Finally, using direct limits, we produced geometric models for K-homology with coefficients in any countable abelian group.

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