Is there a reference giving a method to compare the homology of $F(X,Y)$, the space of continuous maps from $X$ to $Y$, with that of $F((X,x_0),(Y,y_0))$, the space of basepoint-preserving maps?
I'm assuming that all basepoints are non-degenerate, i.e. the inclusions $x_0 \to X$ and $y_0\to Y$ are cofibrations. More specifically, I'm interested in the case where $X$ is a compact manifold and $Y$ is the classifying space of a compact Lie group.
I've done some preliminary work, writing the inclusion map $F((X,x_0),(Y,y_0))\to F(X,Y)$ as a fibration with fiber a loop space, but I feel I'm reinventing the wheel and someone has already done this much more efficiently that I am likely to do.