The topology of spaces of coprime polynomials

Guest, Martin A.; Kozłowski, Andrzej; Yamaguchi, Kohhei

doi:10.1007/bf02571953

Cited by 26 publications

(30 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The problem of finding such models is also considered in several papers (e.g. [3], [6], [7], [9], [12]) and it is originally appeared in the areas of geometry and mathematical physics. Second, the problem of finding a Morse theoretic interpretation of Segal's theorem also arises in our case.…”

Section: Which Is a Homology Equivalence Up To Dimension D(d N) Whementioning

confidence: 99%

Conﬁguration Space Models for Spaces of Maps from a Riemann Surface to Complex Projective Space

Yamaguchi¹

2003

Publ. Res. Inst. Math. Sci.

View full text Add to dashboard Cite

show abstract

Section: Which Is a Homology Equivalence Up To Dimension D(d N) Whementioning

confidence: 99%

Conﬁguration Space Models for Spaces of Maps from a Riemann Surface to Complex Projective Space

Yamaguchi¹

2003

Publ. Res. Inst. Math. Sci.

View full text Add to dashboard Cite

show abstract

“…Because this was now well explained in several papers (cf. [5], [6], [7], [8], [10], [11]), we only sketch the rough idea.…”

Section: §2 Stabilized Spacesmentioning

confidence: 99%

“…d+1 (C) denote the stabilization map given by adding a point from the edge in a usual way (see [5], [6], [8], [11]), and let lim …”

Section: §2 Stabilized Spacesmentioning

confidence: 99%

See 1 more Smart Citation

The Topology of the Space of Holomorphic Maps with Bounded Multiplicity

Yamaguchi¹

2006

Publ. Res. Inst. Math. Sci.

View full text Add to dashboard Cite

show abstract

“…Suppose that r min (Σ) ≥ 2. By using the string representation of elements of π 1 (Hol * D (S 2 , X Σ )) as in [11,§Appendix], one can show that π 1 (Hol * D (S 2 , X Σ )) is an abelian group and the case r min (Σ) = 2 was obtained. Now assume that r min (Σ) ≥ 3.…”

mentioning

confidence: 99%

The homotopy type of spaces of rational curves on a toric variety

Kozłowski¹,

Yamaguchi²

2018

Topology and its Applications

Self Cite

View full text Add to dashboard Cite

Spaces of holomorphic maps from the Riemann sphere to various complex manifolds (holomorphic curves ) have played an important role in several area of mathematics. In a seminal paper G. Segal investigated the homotopy type of holomorphic curves on complex projective spaces and M. Guest on compact smooth toric varieties.. Recently Mostovoy and Villanueva, obtained a far reaching generalisation of these results, and in particular (for holomorphic curves) improved the stability dimension obtained by Guest. In this paper, we generalize their result to holomorphic curves, on certain non-compact smooth toric varieties.In [17], the present authors studied this problem of the case m = 1 for a certain family of non-compact smooth toric subvarieties X I of CP n , and showed that the result of Mostovoy-Villanueva [21] can be extended to this case (with the same stability dimension).In this paper, we shall prove that this result can be further extended to non-compact smooth toric varieties X which satisfy certain two conditions (see the conditions (1.12.1) and (1.12.2)). These conditions are satisfied for a wide range of smooth toric varieties (including all compact ones).In fact, we will do better and show that under the certain condition "homology equivalence"can be replaced by "homotopy equivalence"(up to the same dimension).The broad outline of our argument is analogous to Segal's seminal paper [23] (a brief sketch of such an argument is given in [10, §5]). Namely, for a smooth toric variety X, we first prove that there is a homotopy equivalence between certain limits of spaces Hol * D (S 2 , X) of holomorphic maps, stabilized with respect to a suitably defined degree D, and the double loop space Map * (S 2 , X) = Ω 2 X. We can refer to this as the stable result. The method used to prove it is a generalization of the scanning map technique used by Segal in [23]. In particular, we describe a generalization to the case of toric varieties of a fibration sequence that plays the key role in Segal's argument (see Proposition 3.4).Note that in [21] a quite different stabilization is used, which is based on the Stone-Weierstrass theorem. This stabilization has the advantage that it can used in the case of holomorphic maps from CP m to a compact toric variety X for any m ≥ 1. However, the usefulness of the Stone-Weierstrass theorem is based on the fact that two holomorphic maps that are 'uniformly close', with respect to some metric, are actually homotopic. This is true when the metric on X is complete (e.g. when X is compact), but not for general X. We are, therefore, unsure if our results can be extended to the case Hol * (CP m , X), for m > 1. Even if this is possible, we believe that our generalization of Segal's argument to the case of toric varieties is of some independent interest.The second part of the paper is concerned with establishing "homology stability dimensions" for the inclusion map from Hol * D (S 2 , X) to the double loop space Ω 2 D X of maps of degree D. These stability dimensions depend both on the degree D ...

show abstract

The topology of spaces of coprime polynomials

Cited by 26 publications

References 6 publications

Conﬁguration Space Models for Spaces of Maps from a Riemann Surface to Complex Projective Space

Conﬁguration Space Models for Spaces of Maps from a Riemann Surface to Complex Projective Space

The Topology of the Space of Holomorphic Maps with Bounded Multiplicity

The homotopy type of spaces of rational curves on a toric variety

Contact Info

Product

Resources

About