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Stability for holomorphic spheres and Morse theory

Ralph L. Cohen·John D. S. Jones·Graeme B. Segal
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Abstract

In this paper we study the question of when does a closed, simply connected, integral symplectic manifold (W,omega) have the stability property for its spaces of based holomorphic spheres? This property states that in a stable limit under certain gluing operators, the space of based holomorphic maps from a sphere to X, becomes homotopy equivalent to the space of all continuous maps, lim_{->} Hol_{x_0}(P^1,X) = Omega^2 X. This limit will be viewed as a kind of stabilization of Hol_{x_0}(P^1,X). We conjecture that this stability holds if and only if an evaluation map E: lim_{->} Hol_{x_0}(P^1,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 (defined in [4]) of the symplectic action functional on the Z-cover of the loop space, L~X, defined by the symplectic form, has a classifying space that realizes the homotopy type of L~X. 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 [7].

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Ralph L. Cohen, John D. S. Jones, Graeme B. Segal. 1999-04-28. Stability for holomorphic spheres and Morse theory. https://arxiv.org/abs/math/9904185

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