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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

For positive integers $d,m,n\geq 1$ with $(m,n)\not= (1,1)$ and $\Bbb K=\Bbb R$ or $\Bbb C$, let $Q^{d,m}_{n}(\Bbb K)$ denote the space of $m$-tuples $(f_1(z),\cdots ,f_m(z))\in \Bbb K [z]^m$ of $\Bbb K$-coefficients monic polynomials of the same degree $d$ such that polynomials $\{f_k(z)\}_{k=1}^m$ have no common {\it real} root of multiplicity $\geq n$ (but may have complex common root of any multiplicity). %% These spaces can be regarded as one of generalizations of the spaces defined and studied by Arnold and Vassiliev, and they may be also considered as the real analogues of the spaces studied by B. Farb and J. Wolfson. In this paper, we shall determine their homotopy types explicitly and generalize some previously obtained results., Comment: arXiv admin note: text overlap with arXiv:1612.06793

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.

We show that the space A ˜ d ( m , n ) consisting of all real projective classes of ( n + 1 ) -tuples of real coefficients homogeneous polynomials of degree d in ( m + 1 ) variables, without common real roots except zero, has the same homology as the space Map ( R P m , R P n ) of continuous maps from the m-dimensional real projective space R P m into the n real dimensional projective space R P n up to dimension ( n − m ) ( d + 1 ) − 1 . This considerably improves the main result of Adamaszek et al. (2011) [1] .

The problem of approximating the infinite dimensional space of all continuous maps from an algebraic variety $X$ to an algebraic variety $Y$ by finite dimensional spaces of algebraic maps arises in several areas of geometry and mathematical physics. An often considered formulation of the problem (sometimes called the Atiyah-Jones problem after \cite{AJ}) is to determine a (preferably optimal) integer $n_D$ such that the inclusion from this finite dimensional algebraic space into the corresponding infinite dimensional one induces isomorphisms of homology (or homotopy) groups through dimension $n_D$, where $D$ denotes a tuple of integers called the "degree" of the algebraic maps and $n_D\to\infty$ as $D\to\infty$. In this paper we investigate this problem in the case when $X$ is a real projective space and $Y$ is a smooth compact toric variety.

For a complex (quasi-) projective variety X CP N with π2(X )= Z and an integer d ≥ 0, let Hol ∗(S 2 ,X ) denote the space consisting of all basepoint preserving holomorphic maps f from S 2 to X with degree d. We study the topology of certain subspaces of Hol ∗(S 2 ,X ) defined using the concept of multiplicity of roots, and we show that the Atiyah-Jones-Segal type theorem ([1], [11]) holds for these subspaces if X is belong to a certain family of complex quasi-projective varieties.

We study the homotopy types of the spaces of free algebraic loops on real projective spaces. In particular, we show that the Morse theoretic principle holds for these spaces in the stable homotopy category.

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 describe an alternative approach to some results of Vassiliev on spaces of polynomials, by using the scanning method which was used by Segal in his investigation of spaces of rational functions. We explain how these two approaches are related by the Smale-Hirsch Principle or the h-Principle of Gromov. We obtain several generalizations, which may be of interest in their own right.

We consider the inclusion of the space of algebraic (regular) maps between real algebraic varieties in the space of all continuous maps. For a certain class of real algebraic varieties, which include real projective spaces, it is well known that the space of real algebraic maps is a dense subset of the space of all continuous maps. Our first result shows that, for this class of varieties, the inclusion is also a homotopy equivalence. After proving this, we restrict the class of varieties to real projective spaces. In this case, the space of algebraic maps has a 'minimum degree' filtration by finite-dimensional subspaces and it is natural to expect that the homotopy types of the terms of the filtration approximate closer and closer the homotopy type of the space of continuous mappings as the degree increases. We prove this and compute the lower bounds of this approximation of these spaces. This result can be seen as a generalization of the results of Mostovoy, Vassiliev and others on the topology of the space of real rational maps and the space of real polynomials without n-fold roots. It can also be viewed as a real analogue of Mostovoy's work on the topology of the space of holomorphic maps between complex projective spaces, which generalizes Segal's work on the space of complex rational maps.

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.

We study the existence problem of a 2 n dimensional Poincare complex whose homology is isomorphic to that of n dimensional complex projective space when n = 4 .

For integers $d \geq 0$ and $1 \neq k \neq n$, let $\mathrm{Hol}_{d}(\mathbb{C}P^{k},\mathbb{C}P^{n})$ denote the space consisting of all holomorphic maps $f : \mathbb{C}P^{k}\to \mathbb{C}P^{n}$ of degree $d$. We shall compute the fundamental group of $\mathrm{Hol}_{d}(\mathbb{C}P^{k},\mathbb{C}P^{n})$ and study $\mathrm{PGL}_{n+1}(\mathbb{C})$-action on it.

In this paper we study the homotopy types of mod $p$ truncated configuration spaces. In particular, we investigate a finite dimensional configuration space model for the based loop space of mod $p$ lens space as an application.

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 $n = 3$.

We describe an alternative approach to some results of Vassiliev ([Va1]) on spaces of polynomials, by using the “scanning method” which was used by Segal ([Se2]) in his investigation of spaces of rational functions. We explain how these two approaches are related by the Smale-Hirsch Principle or the h-Principle of Gromov. We obtain several generalizations, which may be of interest in their own right.

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

For an integer d⩾0, let Hol d ( C P k , C P n ) denote the space consisting of all holomorphic maps f : C P k → C P n of degree d. We study the homogeneous space structure of Hol d ( C P k , C P n ) for the case d=1. In particular we explicitly determine its homotopy type.