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We investigate the homotopy type of the space of tuples of polynomials inducing base-point preserving algebraic maps from the circle S1 to a toric variety X{\Sigma}. In particular, we prove a homotopy stability result for this space by combining the Vassiliev spectral sequence and the scanning map., Comment: arXiv admin note: text overlap with arXiv:1707.02603

Spaces of holomorphic maps from the Riemann sphere to various complex manifolds have played an important role in several areas of mathematics (e.g. linear control theory and mathematical physics ( [2] , [3] )). G. Segal [22] investigated the homotopy type of spaces of holomorphic maps on complex projective spaces and M. Guest [10] generalized Segal's result for compact smooth toric varieties. Recently Mostovoy–Villanueva [20] improved the homology stability dimension obtained by Guest. In this paper we generalize their result [20] for certain non-compact smooth toric varieties by the careful analysis of toric varieties with the scanning maps.

Let Hold be the space consisting of all holomorphic maps f : S 2 → S 2 of degree d. The group Hol1 = PSL2( ) acts on Hold freely by the post-composition and we shall study the orbit space Xd = Hol1\Hold. As an application we shall determine the homotopy types of the universal covering spaces of Hold and Xd explicitly.

The purpose of this paper is to study the topology of certain toric varieties X I , arising as quotients of the action of C ⁎ on complements of arrangements of coordinate subspaces in C n , and to improve the homotopy stability dimension for the inclusion map i d : Hol d ⁎ ( S 2 , X I ) → Map d ⁎ ( S 2 , X I ) given in [11] by making use of the Vassiliev spectral sequence. We also improve the homotopy stability dimension of this inclusion given by G. Segal [18] for X I = C P n − 1 and n ≥ 3 .

We construct the universal coverings of spaces of self-holomorphic maps on the complex projective space $\mathbb{C}P_{n}$ by using the resultants, and we study their homotopy types.

We study the homotopy groups of spaces of continuous maps between real projective spaces and we generalize the results given in [5], [8] and [12]. In particular, we determine the rational homotopy types of these spaces and compute their fundamental groups explicitly.

Let $P_{n}^{d}(\mathbb{R})$ denote the space consisiting of all monic polynomials $f(z) \in \mathbb{R}[z]$ of degree $d$ which have no real roots of multplicity $\geq n$. In this paper we study the homotopy types of the spaces $P_{n}^{d}(\mathbb{R})$ for the case $n = 3$.

In this paper, we classify the homotopy types of spaces of monic polynomials which have no $n$ -fold real roots or spaces of $n$ -tuples of monic polynomials which have no common real roots, by using the "scanning method"([9]) and Vassiliev's spectral sequence ([15], [16]). In particular, we show that such spaces are finite dimensional models for the infinite dimensional loop space of spheres.

For each positive integer d, let Hold denote the space of all holomorphic (equivalently, algebraic) maps of degree d from the Riemann sphere S = C ∪ ∞ to itself. This space is of interest both from a classical and a modern point of view (see [1], [5]). Let Hold be the subspace of Hold consisting of maps which preserve a basepoint of S. It is well known that Hol1 is the group of fractional linear transformations PSL2(C) and that Hol1 may be identified with the affine transformation group of C. It is an elementary fact that Hold and Hold are connected spaces. The fundamental groups of these spaces are Z/2d,Z respectively; these computations are due to Epshtein ([6]) and Jones (see [8]). The following more general result was obtained by Segal