** 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.