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International Journal of Mathematics and Mathematical Sciences
VolumeΒ 2012Β (2012), Article IDΒ 867247, 8 pages
http://dx.doi.org/10.1155/2012/867247
Research Article

Examples of Rational Toral Rank Complex

Faculty of Education, Kochi University, 2-5-1 Akebono-Cho, Kochi 780-8520, Japan

Received 21 December 2011; Accepted 6 March 2012

Academic Editor: FrankΒ Werner

Copyright Β© 2012 Toshihiro Yamaguchi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

There is a CW complex 𝒯(𝑋), which gives a rational homotopical classification of almost free toral actions on spaces in the rational homotopy type of X associated with rational toral ranks and also presents certain relations in them. We call it the rational toral rank complex of X. It represents a variety of toral actions. In this note, we will give effective 2-dimensional examples of it when X is a finite product of odd spheres. This is a combinatorial approach in rational homotopy theory.

1. Introduction

Let 𝑋 be a simply connected CW complex with dimπ»βˆ—(𝑋;β„š)<∞ and π‘Ÿ0(𝑋) be the rational toral rank of 𝑋, which is the largest integer π‘Ÿ such that an π‘Ÿ-torus π‘‡π‘Ÿ=𝑆1×⋅⋅⋅×𝑆1 (π‘Ÿ-factors) can act continuously on a CW-complex π‘Œ in the rational homotopy type of 𝑋 with all its isotropy subgroups finite (such an action is called almost free) [1]. It is a very interesting rational invariant. For example, the inequalityπ‘Ÿ0(𝑋)=π‘Ÿ0(𝑋)+π‘Ÿ0𝑆2𝑛<π‘Ÿ0𝑋×𝑆2𝑛(βˆ—) can hold for a formal space 𝑋 and an integer 𝑛>1 [2]. It must appear as one phenomenon in a variety of almost free toral actions. The example (*) is given due to Halperin by using Sullivan minimal model [3].

Put the Sullivan minimal model 𝑀(𝑋)=(Λ𝑉,𝑑) of 𝑋. If an π‘Ÿ-torus π‘‡π‘Ÿ acts on 𝑋 by πœ‡βˆΆπ‘‡π‘ŸΓ—π‘‹β†’π‘‹, there is a minimal KS extension with |𝑑𝑖|=2 for 𝑖=1,…,π‘Ÿξ€·β„šξ€Ίπ‘‘1,…,π‘‘π‘Ÿξ€»ξ€ΈβŸΆξ€·β„šξ€Ίπ‘‘,01,…,π‘‘π‘Ÿξ€»ξ€ΈβŸΆβŠ—βˆ§π‘‰,𝐷(βˆ§π‘‰,𝑑)(1.1) with 𝐷𝑑𝑖=0 and 𝐷𝑣≑𝑑𝑣 modulo the ideal (𝑑1,…,π‘‘π‘Ÿ) for π‘£βˆˆπ‘‰ which is induced from the Borel fibration [4]π‘‹βŸΆπΈπ‘‡π‘ŸΓ—πœ‡π‘‡π‘Ÿπ‘‹βŸΆπ΅π‘‡π‘Ÿ.(1.2) According to [1, Proposition 4.2], π‘Ÿ0(𝑋)β‰₯π‘Ÿ if and only if there is a KS extension of above satisfying dimπ»βˆ—(β„š[𝑑1,…,π‘‘π‘Ÿ]βŠ—βˆ§π‘‰,𝐷)<∞. Moreover, then π‘‡π‘Ÿ acts freely on a finite complex that has the same rational homotopy type as 𝑋. So we will discuss this note by Sullivan models.

We want to give a classification of rationally almost free toral actions on 𝑋 associated with rational toral ranks and also present certain relations in them. Recall a finite-based CW complex 𝒯(𝑋) in [5, Section 5]. Put π’³π‘Ÿ={(β„š[𝑑1,…,π‘‘π‘Ÿ]βŠ—βˆ§π‘‰,𝐷)} the set of isomorphism classes of KS extensions of 𝑀(𝑋)=(Λ𝑉,𝑑) such that dimπ»βˆ—(β„š[𝑑1,…,π‘‘π‘Ÿ]βŠ—βˆ§π‘‰,𝐷)<∞. First, the set of 0-cells 𝒯0(𝑋) is the finite sets {(𝑠,π‘Ÿ)βˆˆβ„€β‰₯0Γ—β„€β‰₯0} where the point 𝑃𝑠,π‘Ÿ of the coordinate (𝑠,π‘Ÿ) exists if there is a model (Ξ›π‘Š,π‘‘π‘Š)βˆˆπ’³π‘Ÿ and π‘Ÿ0(Ξ›π‘Š,π‘‘π‘Š)=π‘Ÿ0(𝑋)βˆ’π‘ βˆ’π‘Ÿ. Of course, the model may not be uniquely determined. Note that the base point 𝑃0,0=(0,0) always exists by 𝑋 itself.

Next, 1-skeltons (vertexes) of the 1-skelton 𝒯1(𝑋) are represented by a KS-extension (β„š[𝑑],0)β†’(β„š[𝑑]βŠ—Ξ›π‘Š,𝐷)β†’(Ξ›π‘Š,π‘‘π‘Š) with dimπ»βˆ—(β„š[𝑑]βŠ—βˆ§π‘Š,𝐷)<∞ for (Ξ›π‘Š,π‘‘π‘Š)βˆˆπ’³π‘Ÿ, where π‘Š=β„š(𝑑1,…,π‘‘π‘Ÿ)βŠ•π‘‰ and π‘‘π‘Š|𝑉=𝑑. It is given as

867247.fig.001
where 𝑃 exists by (Ξ›π‘Š,π‘‘π‘Š), and 𝑄 exists by (β„š[𝑑]βŠ—Ξ›π‘Š,𝐷). The 2 cell is given if there is a (homotopy) commutative diagram of restrictions

867247.fig.002
which represents (a horizontal deformation of)

867247.fig.003
Here π‘ƒπ‘Ž exists by (Ξ›π‘Š,π‘‘π‘Š), 𝑃𝑏(or 𝑃𝑑) by (β„š[π‘‘π‘Ÿ+1]βŠ—Ξ›π‘Š,π·π‘Ÿ+1), 𝑃𝑐 by (β„š[π‘‘π‘Ÿ+1,π‘‘π‘Ÿ+2]βŠ—Ξ›π‘Š,𝐷), and 𝑃𝑑(or 𝑃𝑏) by (β„š[π‘‘π‘Ÿ+2]βŠ—Ξ›π‘Š,π·π‘Ÿ+2). Then we say that a 2 cell attaches to (the tetragon) π‘ƒπ‘Žπ‘ƒπ‘π‘ƒπ‘π‘ƒπ‘‘. Thus, we can construct the 2-skelton 𝒯2(𝑋).

Generally, an 𝑛-cell is given by an 𝑛-cube where a vertex of (β„š[π‘‘π‘Ÿ+1,…,π‘‘π‘Ÿ+𝑛]βŠ—Ξ›π‘Š,𝐷) of height π‘Ÿ+𝑛, 𝑛-vertexes {(β„š[π‘‘π‘Ÿ+1,…,βˆ¨π‘‘π‘Ÿ+𝑖,…,π‘‘π‘Ÿ+𝑛]βŠ—Ξ›π‘Š,𝐷(𝑖))}1≀𝑖≀𝑛 of height π‘Ÿ+π‘›βˆ’1, …, a vertex (Ξ›π‘Š,π‘‘π‘Š) of height π‘Ÿ. Here ∨ is the symbol which removes the below element, and the differential 𝐷(𝑖) is the restriction of 𝐷.

We will call this connected regular complex 𝒯(𝑋)=βˆͺ𝑛β‰₯0𝒯𝑛(𝑋) the rational toral rank complex (r.t.r.c.) of 𝑋. Since π‘Ÿ0(𝑋)<∞ in our case, it is a finite complex. For example, when 𝑋=𝑆3×𝑆3 and π‘Œ=𝑆5, we have𝒯(𝑋)βˆ¨π’―(π‘Œ)=𝒯1(𝑋)βˆ¨π’―1(π‘Œ)=𝒯1(π‘‹Γ—π‘Œ)=𝒯(π‘‹Γ—π‘Œ),(1.3) which is an unusual case. Then, of course, π‘Ÿ0(𝑋)+π‘Ÿ0(π‘Œ)=π‘Ÿ0(π‘‹Γ—π‘Œ). Recall that π‘Ÿ0(𝑆3×𝑆3)+π‘Ÿ0(𝑆7)=π‘Ÿ0(𝑆3×𝑆3×𝑆7) but 𝒯1(𝑆3×𝑆3)βˆ¨π’―1(𝑆7)βŠŠπ’―1(𝑆3×𝑆3×𝑆7) [5, Example 3.5]. In Section 2, we see that r.t.r.c. is not complicated as a CW complex but delicate. We see in Theorems 2.2 and 2.3 that the differences between 𝑋=𝑍×𝑆7 and π‘Œ=𝑍×𝑆9 for some products 𝑍 of odd spheres make certain different homotopy types of r.t.r.c., respectively. Remark that the above inequality (*) is a property on 𝒯0(𝑋) or 𝒯1(𝑋) as the example of Theorem 2.4(1). We see in Theorem 2.4(2) an example that 𝒯1(𝑋)=𝒯1(𝑋×ℂ𝑃𝑛) but 𝒯2(𝑋)βŠŠπ’―2(𝑋×ℂ𝑃𝑛), which is a higher-dimensional phenomenon of (*).

2. Examples

In this section, the symbol π‘ƒπ‘–π‘ƒπ‘—π‘ƒπ‘˜π‘ƒπ‘™ means the tetragon, which is the cycle with vertexes 𝑃𝑖, 𝑃𝑗, π‘ƒπ‘˜, 𝑃𝑙, and edges 𝑃𝑖𝑃𝑗, π‘ƒπ‘—π‘ƒπ‘˜, π‘ƒπ‘˜π‘ƒπ‘™, 𝑃𝑙𝑃𝑖.

In general, it is difficult to show that a point of 𝒯0(𝑋) does not exist on a certain coordinate. So the following lemma is useful for our purpose.

Lemma 2.1. If 𝑋 has the rational homotopy type of the product of finite odd spheres and finite complex projective spaces, then (1,π‘Ÿ)βˆ‰π’―0(𝑋) for any π‘Ÿ.

Proof. Suppose that 𝑋 has the rational homotopy type of the product of 𝑛 odd spheres and π‘š complex projective spaces. Put a minimal model 𝐴=(β„š[𝑑1,…,π‘‘π‘›βˆ’1,π‘₯1,…,π‘₯π‘š]βŠ—Ξ›(𝑣1,…,𝑣𝑛,𝑦1,…,π‘¦π‘š),𝐷) with |𝑑1|=β‹―=|π‘‘π‘›βˆ’1|=|π‘₯1|=β‹―=|π‘₯π‘š|=2 and |𝑣𝑖|,|𝑦𝑖| odd. If dimπ»βˆ—(𝐴)<∞, then 𝐴 is pure; that is, 𝐷𝑣𝑖,π·π‘¦π‘–βˆˆβ„š[𝑑1,…,π‘‘π‘›βˆ’1,π‘₯1,…,π‘₯π‘š] for all 𝑖. Therefore, from [2, Lemma 2.12], π‘Ÿ0(𝐴)=1. Thus, we have (1,π‘Ÿ0(𝑋)βˆ’1)=(1,π‘›βˆ’1)βˆ‰π’―0(𝑋).

Theorem 2.2. Put 𝑋=𝑆3×𝑆3×𝑆3×𝑆7×𝑆7 and π‘Œ=𝑆3×𝑆3×𝑆3×𝑆7×𝑆9. Then 𝒯1(𝑋)=𝒯1(π‘Œ). But 𝒯(𝑋) is contractible and 𝒯(π‘Œ)≃𝑆2.

Proof. Let 𝑀(𝑋)=(Λ𝑉,0)=(Ξ›(𝑣1,𝑣2,𝑣3,𝑣4,𝑣5),0) with |𝑣1|=|𝑣2|=|𝑣3|=3 and |𝑣4|=|𝑣5|=7. Then 𝒯0𝑃(𝑋)=0,0,𝑃0,1,𝑃0,2,𝑃0,3,𝑃0,4,𝑃0,5,𝑃2,1,𝑃2,2,𝑃2,3,𝑃3,1,𝑃3,2ξ€Ύ.(2.1) For example, they are given as follows.(0) 𝑃0,0 is given by (Λ𝑉,0).(1) 𝑃0,1 is given by (β„š[𝑑1]βŠ—Ξ›π‘‰,𝐷) with 𝐷𝑣1=𝑑21 and 𝐷𝑣2=𝐷𝑣3=𝐷𝑣4=𝐷𝑣5=0.(2) 𝑃0,2 is given by (β„š[𝑑1,𝑑2]βŠ—Ξ›π‘‰,𝐷) with 𝐷𝑣1=𝑑21, 𝐷𝑣2=𝑑22, and 𝐷𝑣3=𝐷𝑣4=𝐷𝑣5=0.(3) 𝑃0,3 is given by (β„š[𝑑1,𝑑2,𝑑3]βŠ—Ξ›π‘‰,𝐷) with 𝐷𝑣1=𝑑21, 𝐷𝑣2=𝑑22, 𝐷𝑣3=𝑑23, and 𝐷𝑣4=𝐷𝑣5=0.(4) 𝑃0,4 is given by (β„š[𝑑1,𝑑2,𝑑3,𝑑4]βŠ—Ξ›π‘‰,𝐷) with 𝐷𝑣1=𝑑21, 𝐷𝑣2=𝑑22, 𝐷𝑣3=𝑑23, 𝐷𝑣4=𝑑44, and 𝐷𝑣5=0.(5) 𝑃0,5 is given by (β„š[𝑑1,𝑑2,𝑑3,𝑑4,𝑑5]βŠ—Ξ›π‘‰,𝐷) with 𝐷𝑣1=𝑑21, 𝐷𝑣2=𝑑22, 𝐷𝑣3=𝑑23, 𝐷𝑣4=𝑑44, and 𝐷𝑣5=𝑑45.(6) 𝑃2,1 is given by (β„š[𝑑1]βŠ—Ξ›π‘‰,𝐷) with 𝐷𝑣1=𝐷𝑣2=𝐷𝑣3=𝐷𝑣5=0 and 𝐷𝑣4=𝑣1𝑣2𝑑1+𝑑41(7) 𝑃2,2 is given by (β„š[𝑑1,𝑑2]βŠ—Ξ›π‘‰,𝐷) with 𝐷𝑣1=𝐷𝑣2=0, 𝐷𝑣3=𝑑22, 𝐷𝑣4=𝑣1𝑣2𝑑1+𝑑21, and 𝐷𝑣5=0.(8) 𝑃2,3 is given by (β„š[𝑑1,𝑑2,𝑑3]βŠ—Ξ›π‘‰,𝐷) with 𝐷𝑣1=𝐷𝑣2=0, 𝐷𝑣3=𝑑22, 𝐷𝑣4=𝑑21+𝑣1𝑣2𝑑1, and 𝐷𝑣5=𝑑43.(9) 𝑃3,1 is given by (β„š[𝑑1]βŠ—Ξ›π‘‰,𝐷) with 𝐷𝑣1=𝐷𝑣2=𝐷𝑣3=0, 𝐷𝑣4=𝑣1𝑣2𝑑1+𝑑41, and 𝐷𝑣5=𝑣1𝑣3𝑑1.(10) 𝑃3,2 is given by (β„š[𝑑1,𝑑2]βŠ—Ξ›π‘‰,𝐷) with 𝐷𝑣4=𝑣1𝑣2𝑑1+𝑑41 and 𝐷𝑣5=𝑣1𝑣3𝑑1+𝑑42.(11) 𝑃4,1, that is, a point of the coordinate (4,1) does not exist. Indeed, if it exists, it must be given by a model (β„š[𝑑1]βŠ—Ξ›π‘‰,𝐷) whose differential is 𝐷𝑣1=𝐷𝑣2=𝐷𝑣3=0 and 𝐷𝑣4,𝐷𝑣5βˆˆβ„š[𝑑1]βŠ—Ξ›(𝑣1,𝑣2,𝑣3) by degree reason. But, for any 𝐷 satisfying such conditions, we have dimπ»βˆ—(β„š[𝑑1,𝑑2]βŠ—Ξ›π‘‰,𝐷)<∞ for a KS extension ξ€·β„šξ€Ίπ‘‘2ξ€»ξ€ΈβŸΆξ‚€β„šξ€Ίπ‘‘,01,𝑑2ξ€»ξ‚π·ξ‚βŸΆξ€·β„šξ€Ίπ‘‘βŠ—Ξ›π‘‰,1ξ€»ξ€ΈβŠ—Ξ›π‘‰,𝐷,(2.2) that is, π‘Ÿ0(β„š[𝑑1]βŠ—Ξ›π‘‰,𝐷)>0. It contradicts the definition of 𝑃4,1.
𝒯1(𝑋) is given as

867247.fig.004

For example, the edges (1 simplexes) 𝑃0,0𝑃0,1,𝑃0,1𝑃0,2,𝑃0,2𝑃0,3,𝑃0,3𝑃0,4,…,𝑃0,0𝑃3,1,𝑃3,1𝑃3,2ξ€Ύ(2.3) are given as follows.(1)𝑃0,1𝑃3,2 is given by the projection (β„š[𝑑1,𝑑2]βŠ—Ξ›π‘‰,𝐷)β†’(β„š[𝑑1]βŠ—Ξ›π‘‰,𝐷1) where 𝐷𝑣1=𝐷𝑣2=𝐷𝑣3=0, 𝐷𝑣4=𝑣1𝑣2𝑑2+𝑑41, 𝐷𝑣5=𝑣1𝑣3𝑑2+𝑑42, and 𝐷1𝑣1=𝐷1𝑣2=𝐷1𝑣3=𝐷1𝑣5=0 and 𝐷1𝑣4=𝑑41.(2)𝑃2,1𝑃3,2 is given by 𝐷𝑣1=𝐷𝑣2=𝐷𝑣3=0, 𝐷𝑣4=𝑣1𝑣2𝑑1+𝑑41, and 𝐷𝑣5=𝑣1𝑣3𝑑2+𝑑42.(3)𝑃3,1𝑃3,2 is given by 𝐷𝑣1=𝐷𝑣2=𝐷𝑣3=0, 𝐷𝑣4=𝑣1𝑣2𝑑1+𝑑41, and 𝐷𝑣5=𝑣1𝑣3𝑑1+𝑑42.
𝒯2(𝑋) is given as follows.
(1)𝑃0,0𝑃2,1𝑃3,2𝑃3,1 is attached by a 2 cell from 𝐷𝑣1=𝐷𝑣2=𝐷𝑣3=0, 𝐷𝑣4=𝑣1𝑣2(𝑑1+𝑑2)+𝑑41 and 𝐷𝑣5=𝑣1𝑣3𝑑2+𝑑42. (Then 𝑃2,1 is given by 𝐷1𝑣4=𝑣1𝑣2𝑑1+𝑑41, 𝐷1𝑣5=0, and 𝑃3,1 is given by 𝐷2𝑣4=𝑣1𝑣2𝑑2, 𝐷2𝑣5=𝑣1𝑣3𝑑2+𝑑42.)(2)𝑃0,0𝑃0,1𝑃3,2𝑃3,1 is attached by a 2 cell from 𝐷𝑣1=𝐷𝑣2=𝐷𝑣3=0, 𝐷𝑣4=𝑣1𝑣2𝑑2+𝑑41, and 𝐷𝑣5=𝑣1𝑣3𝑑2+𝑑42.(3)𝑃0,0𝑃0,1𝑃2,2𝑃2,1 is attached by a 2 cell from 𝐷𝑣1=𝐷𝑣2=𝐷𝑣3=0, 𝐷𝑣4=𝑣1𝑣2𝑑2+𝑑42, and 𝐷𝑣5=𝑑41.(4)𝑃0,1𝑃0,2𝑃2,3𝑃2,2 is attached by a 2 cell from 𝐷𝑣1=𝐷𝑣2=0, 𝐷𝑣3=𝑑23, 𝐷𝑣4=𝑣1𝑣2𝑑2+𝑑42, and 𝐷𝑣5=𝑑41.(5)𝑃0,0𝑃0,1𝑃3,2𝑃2,1 is not attached by a 2 cell. Indeed, assume that a 2 cell attaches on it. Notice that 𝑃3,2 is given by (β„š[𝑑1,𝑑2]βŠ—Ξ›π‘‰,𝐷) with 𝐷𝑣1=𝐷𝑣2=𝐷𝑣3=0 and 𝐷𝑣4𝑣=𝛼1,𝑣2,𝑣3ξ€Έ+𝑓,𝐷𝑣5𝑣=𝛽1,𝑣2,𝑣3ξ€Έ+𝑔,(2.4) where 𝛼,π›½βˆˆ(𝑣1,𝑣2,𝑣3) and {𝑓,𝑔} is a regular sequence in β„š[𝑑1,𝑑2]. Since 𝑃0,1𝑃3,2βˆˆπ’―1(𝑋), both 𝛼 and 𝛽 must be contained in the ideal (𝑑𝑖) for some 𝑖. Also they are not in (𝑑1𝑑2) by degree reason. Furthermore, since 𝑃2,1𝑃3,2βˆˆπ’―1(𝑋), we can put that both 𝛼 and 𝛽 are contained in the monogenetic ideal (𝑣𝑖𝑣𝑗) for some 1≀𝑖<𝑗≀3 without losing generality. Then, dimπ»βˆ—(β„š[𝑑1,𝑑2,𝑑3]βŠ—Ξ›π‘‰,𝐷)<∞ for a KS extension ξ€·β„šξ€Ίπ‘‘3ξ€»ξ€ΈβŸΆξ‚€β„šξ€Ίπ‘‘,01,𝑑2,𝑑3ξ€»ξ‚π·ξ‚βŸΆξ€·β„šξ€Ίπ‘‘βŠ—Ξ›π‘‰,1,𝑑2ξ€»ξ€ΈβŠ—Ξ›π‘‰,𝐷,(2.5) by putting ξ‚π·π‘£π‘˜=𝑑23 for π‘˜βˆˆ{1,2,3} with π‘˜β‰ π‘–,𝑗 and 𝐷𝑣𝑛=𝐷𝑣𝑛 for π‘›β‰ π‘˜. Thus, we have π‘Ÿ0(β„š[𝑑1,𝑑2]βŠ—Ξ›π‘‰,𝐷)>0. It contradicts to the definition of 𝑃3,2.
Notice there is no 3 cell since it must attach to a 3 cube (in graphs) in general. Thus, we see that 𝒯(𝑋)=𝒯2(𝑋) is contractible.
On the other hand, let 𝑀(π‘Œ)=(Ξ›π‘Š,0)=(Ξ›(𝑀1,𝑀2,𝑀3,𝑀4,𝑀5),0) with |𝑀1|=|𝑀2|=|𝑀3|=3, |𝑀4|=7 and |𝑀5|=9. Then we see that 𝒯1(𝑋)=𝒯1(π‘Œ) from same arguments. But, in 𝒯2(π‘Œ), 𝑃0,0𝑃0,1𝑃3,2𝑃2,1 is attached by a 2 cell since we can put 𝐷𝑀1=𝐷𝑀2=𝐷𝑀3=0 and 𝐷𝑀4=𝑀1𝑀2𝑑2+𝑑42,𝐷𝑀5=𝑀1𝑀3𝑑1𝑑2+𝑑51,(2.6) by degree reason. Here 𝑃0,1 is given by 𝐷1𝑀4=0, 𝐷1𝑀5=𝑑51, and 𝑃2,1 is given by 𝐷2𝑀4=𝑀1𝑀2𝑑2+𝑑42, 𝐷2𝑀5=0. Others are same as 𝒯2(𝑋). Then three 2 cells on 𝑃0,0𝑃0,1𝑃3,2𝑃2,1, 𝑃0,0𝑃2,1𝑃3,2𝑃3,1, and 𝑃0,0𝑃0,1𝑃3,2𝑃3,1 in 𝒯2(π‘Œ) make the following:
867247.fig.005
to be homeomorphic to 𝑆2. Thus 𝒯(π‘Œ)=𝒯2(π‘Œ)≃𝑆2.

Theorem 2.3. Put 𝑋=𝑆3×𝑆3×𝑆3×𝑆3×𝑆7×𝑆7 and π‘Œ=𝑆3×𝑆3×𝑆3×𝑆3×𝑆7×𝑆9. Then 𝒯1(𝑋)=𝒯1(π‘Œ). But 𝒯(𝑋)≃𝑆2 and 𝒯(π‘Œ)β‰ƒβˆ¨6𝑖=1𝑆2𝑖.

Proof. We see as the proof of Theorem 2.2 that 𝒯0𝑃(𝑋)=0,0,𝑃0,1,𝑃0,2,𝑃0,3,𝑃0,4,𝑃0,5,𝑃0,6,𝑃2,1,𝑃2,2,𝑃2,3,𝑃2,4,𝑃3,1,𝑃3,2,𝑃3,3,𝑃4,1,𝑃4,2ξ€Ύ(2.7) and both 𝒯1(𝑋) and 𝒯1(π‘Œ) are given as

867247.fig.006

For all tetragons in 𝒯1(𝑋) except the following 4 tetragons: (1)𝑃0,0𝑃0,1𝑃3,2𝑃2,1, (2) 𝑃0,1𝑃0,2𝑃3,3𝑃2,2, (3)𝑃0,0𝑃0,1𝑃4,2𝑃2,1, and (4) 𝑃0,0𝑃0,1𝑃4,2𝑃3,1, 2 cells attach in 𝒯2(𝑋). The proof is similar to it of Theorem 2.2. Thus we see that 𝒯2(𝑋) is homotopy equivalent to
867247.fig.007
which is homeomorphic to 𝑆2. For example, when 𝑀(𝑋)=(Λ𝑉,0)=(Ξ›(𝑣1,𝑣2,𝑣3,𝑣4,𝑣5,𝑣6),0) with |𝑣1|=|𝑣2|=|𝑣3|=|𝑣4|=3 and |𝑣5|=|𝑣6|=7, 2 cells attach 𝑃0,0𝑃2,1𝑃4,2𝑃3,1, 𝑃0,0𝑃3,1𝑃4,2𝑃4,1 and 𝑃0,0𝑃2,1𝑃4,2𝑃4,1 from 𝐷𝑣1=β‹―=𝐷𝑣4=0, 𝐷𝑣5=𝑣1𝑣2𝑑1+𝑑41,𝐷𝑣6=𝑣1𝑣3𝑑1+𝑣2𝑣4𝑑2+𝑑42,𝐷𝑣5=𝑣1𝑣2𝑑1+𝑑41,𝐷𝑣6=𝑣1𝑣3𝑑1+𝑑2ξ€Έ+𝑣2𝑣4𝑑2+𝑑42,𝐷𝑣5=𝑣1𝑣2𝑑1+𝑑41,𝐷𝑣6=𝑣1𝑣3𝑑2+𝑣2𝑣4𝑑2+𝑑42,(2.8) respectively.
In 𝒯2(π‘Œ), 2 cells attach all tetragons in 𝒯1(π‘Œ) by degree reason. For example, when 𝑀(π‘Œ)=(Ξ›π‘Š,0)=(Ξ›(𝑀1,𝑀2,𝑀3,𝑀4,𝑀5,𝑀6),0) with |𝑀1|=|𝑀2|=|𝑀3|=|𝑀4|=3, |𝑀5|=7 and |𝑀6|=9, put 𝐷𝑀1=𝐷𝑀2=𝐷𝑀3=0 and(1)𝐷𝑀4=0,𝐷𝑀5=𝑀1𝑀3𝑑2+𝑑42,𝐷𝑀6=𝑀2𝑀3𝑑1𝑑2+𝑑51, (2)𝐷𝑀4=𝑑23,𝐷𝑀5=𝑀1𝑀3𝑑2+𝑑42,𝐷𝑀6=𝑀2𝑀3𝑑1𝑑2+𝑑51, (3)𝐷𝑀4=0,𝐷𝑀5=𝑀1𝑀2𝑑2+𝑑42,𝐷𝑀6=𝑀3𝑀4𝑑1𝑑2+𝑑51, (4)𝐷𝑀4=0,𝐷𝑀5=𝑀1𝑀3𝑑2+𝑑42,𝐷𝑀6=𝑀1𝑀4𝑑22+𝑀2𝑀3𝑑1𝑑2+𝑑51, for (1)~(4) of above. Then we can check that 𝒯(π‘Œ)β‰ƒβˆ¨6𝑖=1𝑆2𝑖 (𝒯(π‘Œ) cannot be embedded in ℝ3).

Theorem 2.4. Even when π‘Ÿ0(𝑋)=π‘Ÿ0(𝑋×ℂ𝑃𝑛) for the 𝑛-dimensional complex projective space ℂ𝑃𝑛, it does not fold that 𝒯(𝑋)=𝒯(𝑋×ℂ𝑃𝑛) in general. For example,(1)When 𝑋=𝑆3×𝑆3×𝑆3×𝑆3×𝑆7 and 𝑛=4, then 𝒯1(𝑋)βŠŠπ’―1(𝑋×ℂ𝑃4).(2)When 𝑋=𝑆3×𝑆3×𝑆3×𝑆7×𝑆7 and 𝑛=4, then 𝒯1(𝑋)=𝒯1(𝑋×ℂ𝑃4) but 𝒯2(𝑋)βŠŠπ’―2(𝑋×ℂ𝑃4).

Proof. Put 𝑀(ℂ𝑃𝑛)=(Ξ›(π‘₯,𝑦),𝑑) with 𝑑π‘₯=0 and 𝑑𝑦=π‘₯𝑛+1 for |π‘₯|=2 and |𝑦|=2𝑛+1. Put (β„š[𝑑1,…,π‘‘π‘Ÿ]βŠ—Ξ›π‘‰βŠ—Ξ›(π‘₯,𝑦),𝐷) the model of a Borel space πΈπ‘‡π‘ŸΓ—π‘‡π‘Ÿ(𝑋×ℂ𝑃𝑛) of 𝑋×ℂ𝑃𝑛.
(1) 𝒯1(𝑋) and 𝒯1(𝑋×ℂ𝑃4) are given as

867247.fig.008
respectively. For 𝑀(𝑋)=(Λ𝑉,0)=(Ξ›(𝑣1,𝑣2,𝑣3,𝑣4,𝑣5),0) with |𝑣1|=|𝑣2|=|𝑣3|=|𝑣4|=3 and |𝑣5|=7. Here 𝑃4,1 is given by 𝐷𝑣𝑖=0 for 𝑖=1,2,3,4 and 𝐷𝑣5=𝑣1𝑣2𝑑1+𝑣3𝑣4𝑑1+𝑑41. It is contained in both 𝒯0(𝑋) and 𝒯0(𝑋×ℂ𝑃4). On the other hand, 𝑃3,2 is given by 𝐷𝑣𝑖=0 for 𝑖=1,2,3, 𝐷𝑣4=𝑑22, 𝐷𝑣5=𝑣1𝑣2𝑑1+𝑑41, 𝐷π‘₯=0, and 𝐷𝑦=π‘₯5+𝑣1𝑣3𝑑21. Then 𝑃3,1 is given by 𝐷𝑣𝑖=0 for 𝑖=1,2,3,4, 𝐷𝑣5=𝑣1𝑣2𝑑1+𝑑41, 𝐷π‘₯=0, and 𝐷𝑦=π‘₯5+𝑣1𝑣3𝑑21. They are contained only in 𝒯0(𝑋×ℂ𝑃4).
(2) Both 𝒯1(𝑋) and 𝒯1(𝑋×ℂ𝑃4) are same as one in Theorem 2.2. Notice that 𝑃0,0𝑃0,1𝑃3,2𝑃2,1 is attached by a 2 cell in 𝒯2(𝑋×ℂ𝑃4) from 𝐷𝑣𝑖=0 for 𝑖=1,2,3, 𝐷𝑣4=𝑣1𝑣2𝑑1+𝑑41, 𝐷𝑣5=𝑑42, 𝐷π‘₯=0, and 𝐷𝑦=π‘₯5+𝑣1𝑣3𝑑1𝑑2. So 𝒯(𝑋×ℂ𝑃4)=𝒯(π‘Œ) for π‘Œ=𝑆3×𝑆3×𝑆3×𝑆7×𝑆9.

Remark 2.5. The author must mention about the spaces 𝑋1 and 𝑋2 in [5, Examples 3.8 and 3.9] such that 𝒯1(𝑋1)=𝒯1(𝑋2). We can check that 2 cells attach on both 𝑃0𝑃5𝑃9𝑃8 of them (compare [5, page 506]).

Remark 2.6. In [5, Question 1.6], a rigidity problem is proposed. It says that does 𝒯0(𝑋) with coordinates determine 𝒯1(𝑋)? For 𝒯(𝑋), it is false as we see in above examples. But it seems that there are certain restrictions. For example, is 𝒯2(𝑋) simply connected?

Acknowledgment

This paper is dedicated to Yves FΓ©lix on his 60th birthday.

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