Let Ratk(CPn)
denote the space of based holomorphic maps of degree k from the
Riemannian sphere S2 to the complex projective space
CPn. The basepoint condition
we assume is that f(∞)=[1, …, 1].
Such holomorphic maps are given by rational functions:
Ratk(CPn)
={(p0(z), …,
pn(z))[ratio ]each
pi(z) is a monic, degree-k
polynomial
and such that there are no roots common to all
pi(z)}. (1.1)
The study of the topology of Ratk(CPn)
originated in [10]. Later, the stable
homotopy type of Ratk(CPn)
was described in [3] in terms of configuration spaces
and
Artin's braid groups. Let W(S2n)
denote the homotopy theoretic fibre of the
Freudenthal suspension E[ratio ]S2n→
ΩS2n+1. Then we have the following
sequence of
fibrations: Ω2S2n+1→
W(S2n)→S2n→
ΩS2n+1. A theorem in [10]
tells us that the inclusion Ratk(CPn)→
Ω2kCPn≃
Ω2S2n+1
is a homotopy equivalence up to dimension
k(2n−1). Thus if we form the direct limit
Rat∞(CPn)=
limk→∞
Ratk(CPn), we have,
in particular, that Rat∞(CPn)
is homotopy equivalent to Ω2S2n+1.
If we take the results of [3] and [10]
into account, we naturally encounter the
following problem: how to construct spaces
Xk(CPn),
which are natural generalizations of
Ratk(CPn), so that
X∞(CPn) approximates
W(S2n). Moreover, we study the stable
homotopy type of
Xk(CPn).
The purpose of this paper is to give an answer to this problem. The
results are
stated after the following definition.