In a well known work, Graeme Segal proved that the space of holomorphic maps from
a Riemann surface to a complex projective space is homology equivalent to the
corresponding space of continuous maps through a range of dimensions increasing
with degree. In this paper, we address if a similar result holds when other (not
necessarily integrable) almost complex structures are put on projective space. We
take almost complex structures that are compatible with the underlying symplectic
structure. We obtain the following result: the inclusion of the space of based
degree–
–holomorphic
maps from
to
into the double loop
space of
is a homology
surjection for dimensions
.
The proof involves constructing a gluing map analytically in a way similar to McDuff
and Salamon, and Sikorav, and then comparing it to a combinatorial gluing map
studied by Cohen, Cohen, Mann, and Milgram.
Keywords
almost complex structure, little disks operad, gluing
