The Lefschetz fixed point theorem and its converse have many generalizations. One of these generalizations is to endomorphisms of a space relative to a fixed subspace. In this paper we define relative Lefschetz numbers and Reidemeister traces using traces in bicategories with shadows. We use the functoriality of this trace to identify different forms of these invariants and to prove a relative Lefschetz fixed point theorem and its converse.
"Relative fixed point theory." Algebr. Geom. Topol. 11 (2) 839 - 886, 2011. https://doi.org/10.2140/agt.2011.11.839