Stability for holomorphic spheres and Morse theory
by
Ralph L. Cohen, John D.S. Jones, and Graeme B. Segal
AMS Classification numbers: 57R19, 58fF09, 32H02
Addresses of Authors:
Cohen: Dept. of Mathematics, Stanford University,
Stanford, Ca. 94305
Jones: Dept. of Mathematics, University of Warwick,
Coventry, England
Segal: Dept. of Pure Math. and Math. Statistics, Cambridge University,
Cambridge, England
Email addresses of Authors:
Cohen: ralph@math.stanford.edu
Jones: jdsj@maths.warwick.ac.uk
Segal: G.B.Segal@dpmms.cam.ac.uk
In this paper we study the question of when does a closed, simply
connected, integral
symplectic manifold $(X, \omega)$ have the "stability property" for
its spaces of based
holomorphic spheres? This property states that in a stable limit
under certain gluing operations,
the space of based holomorphic maps from a sphere to $X$, becomes
homotopy equivalent to the space of all
continuous maps,
lim_k Hol_k(S^2, X) = \Omega^2 X
Here "=" means homotopy equivalent. The "degree k" is the evaluation of
the integral cohomology
class represented by the symplectic form on the map S^2 --> X. We
describe this limit as a kind of group completion of
Hol(S^2, X). We conjecture that this stability property holds if and
only if an evaluation map $E:
lim_k Hol_k(S^2, X) ---> X is a quasifibration. In this paper we
will prove that in the presence of
this quasifibration condition, then the stability property holds if and
only if the Morse theoretic flow
category of the symplectic action functional on the universal cover of
the loop space, LX, has a
classifying space that realizes the homotopy type of LX. We conjecture
that in the presence of this
quasifibration condition, this Morse theoretic condition always holds.
We will prove this in the case of
X a homogeneous space, thereby giving an alternate proof of the
stability theorem for holomorphic
spheres for a projective homogeneous variety originally due to Gravesen.