UC Santa Cruz

Given a topological space $X$ and two self-maps $f,g:X \to X$, coincidence theory asks the question: which points $x \in X$ satisfy the property $f(x)=g(x)$? For $g$ (or $f$) equal to the identity map, this question reduces to asking how many fixed points $g$ (respectively $f$) has. In this manner, coincidence theory is a straight-forward generalization of fixed point theory. The classical approach to fixed point and coincidence theory examines the induced maps on the homology groups of the space in question. This thesis presents the classical Lefschetz number and the Lefschetz coincidence number as a foundation with the goal of generalizing these ideas to the stable homotopy category. Instead of passing directly from the category of ``nice" topological spaces to graded $\Q$-vector spaces using rational homology, we generate spectra from these topological spaces first and then apply rational homology. This formulation of fixed point theory and coincidence theory in the stable homotopy category is carried out by using Atiyah duality with a nod towards Spanier-Whitehead duality, and traces in symmetric monoidal categories.