Title: Gluing maps and the homology of spaces of rational J-holomorphic curves in CP^2

Abstract: Graeme Segal proved that the space of holomorphic maps from a Riemann surface to a complex projective space is homology equivalent to the corresponding continuous mapping space through a range of dimensions increasing with degree. We investigate if a similar result holds when other (not necessarily integrable) almost complex structures are put on projective space. We obtain the following result: the inclusion of the space of based degree $k$ $J$-holomorphic maps from CP^1 to CP^2 into the double loop space of $\p$ is a homology surjection for dimensions <3k-2. Using automatic transversality, Gromov proved that the topology of the space of rational degree one curves in CP^2 is independent of choice of almost complex structure. Again using automatic transversality, we construct a gluing map which allows us to leverage this fact about degree one J holomorphic mapping spaces to better understand the topology of higher degree mapping spaces.