Examples of Rational Toral Rank Complex

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International Journal of Mathematics and Mathematical Sciences Volume 2012, Article ID 867247,8pages

doi:10.1155/2012/867247

Research Article

Examples of Rational Toral Rank Complex

Toshihiro Yamaguchi

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

Correspondence should be addressed to Toshihiro Yamaguchi,tyamag@kochi-u.ac.jp

Received 21 December 2011; Accepted 6 March 2012

Academic Editor: Frank Werner

Copyrightq2012 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.

There is a CW complexTX, 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

LetX be a simply connected CW complex with dimH∗X;Q< ∞andr0Xbe the rational

toral rank ofX, which is the largest integerrsuch that anr-torusTr _S1_{× · · · ×}_S1_r_-factors

can act continuously on a CW-complexY 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 inequality

r0X r0X r0

S2n< r0

X×S2n ∗

can hold for a formal spaceXand an integern >12. 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 model3.

Put the Sullivan minimal modelMX ΛV, dofX. If anr-torusTr _{acts on}_X _by

μ:Tr_×_X _→ _X_{, there is a minimal KS extension with}_|t

i|2 fori1, . . . , r

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withDti 0 andDv ≡dvmodulo the idealt1, . . . , trforv∈V which is induced from the

Borel fibration4

X−→ETr ×μ_Tr X −→BTr. 1.2

According to1, Proposition 4.2,r0X ≥ r if and only if there is a KS extension of above

satisfying dimH∗Qt1, . . . , tr⊗∧V, D<∞. Moreover, thenTracts 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 onXassociated

with rational toral ranks and also present certain relations in them. Recall a finite-based CW complex TXin 5, Section 5. PutXr {Qt1, . . . , tr⊗ ∧V, D} the set of isomorphism

classes of KS extensions ofMX ΛV, dsuch that dimH∗Qt1, . . . , tr⊗∧V, D<∞. First,

the set of 0-cells T0X is the finite sets {s, r ∈ Z≥0 × Z≥0} where the point Ps,r of the

coordinates, rexists if there is a modelΛW, dW∈ Xr andr0ΛW, dW r0X−s−r. Of

course, the model may not be uniquely determined. Note that the base pointP0,0 0,0

always exists byXitself.

Next, 1-skeltonsvertexesof the 1-skeltonT1Xare represented by a KS-extension

Qt,0 → Qt⊗ΛW, D → ΛW, dWwith dimH∗Qt⊗∧W, D<∞forΛW, dW∈ Xr,

whereWQt1, . . . , tr⊕VanddW|V d. It is given as

or or

Q Q Q or· · ·,

P P

P

whereP exists byΛW, dW, andQexists byQt⊗ΛW, D. The 2 cell is given if there is a

homotopycommutative diagram of restrictions

( W, dW)

(Q[tr+1]⊗ W, Dr+1)

Λ

(Q[tr+2]⊗ΛW,Dr+2)

(Q[tr+1, tr+2]⊗ΛW,D)

which representsa horizontal deformation of

.

Pc

Pb

Pa

Pd

HerePaexists byΛW, dW,PborPdbyQtr1⊗ΛW, Dr1,PcbyQtr1, tr2⊗ΛW, D,

andPdorPb byQtr2⊗ΛW, Dr2. Then we say that a 2 cell attaches tothe tetragon

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Generally, ann-cell is given by ann-cube where a vertex ofQtr1, . . . , trn⊗ΛW, Dof

heightrn,n-vertexes{Qtr1, . . . ,

∨

tri, . . . , trn⊗ΛW, Di}1≤i≤nof heightrn−1,. . ., a

vertexΛW, dWof heightr. Here∨is the symbol which removes the below element, and the

diﬀerentialDiis the restriction ofD.

We will call this connected regular complexTX ∪n≥0TnXthe rational toral rank

complexr.t.r.c.ofX. Sincer0X <∞in our case, it is a finite complex. For example, when

XS3_×_S3_and_Y _S5_{, we have}

TX∨ TY T1X∨ T1Y T1X×Y TX×Y, 1.3

which is an unusual case. Then, of course,r0X r0Y r0X×Y. Recall thatr0S3×S3

r0S7 r0S3×S3×S7butT1S3×S3∨T1S7T1S3×S3×S7 5, Example 3.5. In Section2,

we see that r.t.r.c. is not complicated as a CW complex but delicate. We see in Theorems2.2

and2.3that the diﬀerences betweenXZ×S7_and_Y _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 onT0XorT1Xas the example of Theorem2.41. We see in

Theorem2.42an example thatT1X T1X×CPnbutT2XT2X×CPn, which is a

higher-dimensional phenomenon of∗.

2. Examples

In this section, the symbolPiPjPkPlmeans the tetragon, which is the cycle with vertexesPi,Pj,

Pk,Pl, and edgesPiPj,PjPk,PkPl,PlPi.

In general, it is diﬃcult to show that a point ofT0Xdoes not exist on a certain

coor-dinate. So the following lemma is useful for our purpose.

Lemma 2.1. IfXhas the rational homotopy type of the product of finite odd spheres and finite complex projective spaces, then1, r∈ T/ 0Xfor anyr.

Proof. Suppose that X has the rational homotopy type of the product of n odd spheres and m complex projective spaces. Put a minimal model A Qt1, . . . , tn−1, x1, . . . , xm ⊗

Λv1, . . . , vn, y1, . . . , ym, Dwith|t1| · · · |tn−1| |x1| · · · |xm| 2 and|vi|,|yi|odd. If

dimH∗A < ∞, then A is pure; that is, Dvi, Dyi ∈ Qt1, . . . , tn−1, x1, . . . , xm for all i.

Therefore, from 2, Lemma 2.12,r0A 1. Thus, we have 1, r0X−1 1, n−1 ∈/

T0X.

Theorem 2.2. PutXS3×S3×S3×S7×S7andY S3×S3×S3×S7×S9. ThenT1X T1Y.

ButTXis contractible andTYS2_.

Proof. LetMX ΛV,0 Λv1, v2, v3, v4, v5,0with|v1||v2||v3|3 and|v4||v5|

7. Then

T0X {P0,0, P0,1, P0,2, P0,3, P0,4, P0,5, P2,1, P2,2, P2,3, P3,1, P3,2}. 2.1

For example, they are given as follows. 0P0,0is given byΛV,0.

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2P0,2 is given byQt1, t2⊗ΛV, DwithDv1 t2₁,Dv2 t2₂, andDv3 Dv4

Dv5 0.

3P0,3 is given byQt1, t2, t3⊗ΛV, Dwith Dv1 t2₁,Dv2 t2₂,Dv3 t2₃, and

Dv4 Dv50.

4P0,4 is given byQt1, t2, t3, t4⊗ΛV, D with Dv1 t2₁,Dv2 t2₂, Dv3 t2₃,

Dv4 t4₄, andDv50.

5P0,5 is given byQt1, t2, t3, t4, t5⊗ΛV, DwithDv1 t2₁,Dv2 t2₂,Dv3 t2₃,

Dv4 t4₄, andDv5t4₅.

6P2,1is given byQt1⊗ΛV, DwithDv1 Dv2 Dv3 Dv5 0 andDv4

v1v2t1t4₁

7P2,2is given byQt1, t2⊗ΛV, DwithDv1Dv20,Dv3t2₂,Dv4v1v2t1t2₁,

andDv5 0.

8P2,3 is given byQt1, t2, t3⊗ΛV, D with Dv1 Dv2 0,Dv3 t2₂,Dv4

t2

1v1v2t1, andDv5t43.

9P3,1is given byQt1⊗ΛV, DwithDv1 Dv2 Dv3 0,Dv4 v1v2t1t4₁,

andDv5 v1v3t1.

10P3,2is given byQt1, t2⊗ΛV, DwithDv4v1v2t1t4₁andDv5v1v3t1t4₂.

11P4,1, that is, a point of the coordinate4,1does not exist. Indeed, if it exists, it

must be given by a modelQt1⊗ΛV, Dwhose diﬀerential isDv1Dv2Dv30

andDv4, Dv5∈Qt1⊗Λv1, v2, v3by degree reason. But, for anyDsatisfying such

conditions, we have dimH∗Qt1, t2⊗ΛV,D<∞for a KS extension

Qt2,0−→

Qt1, t2⊗ΛV,D

−→Qt1⊗ΛV, D, 2.2

that is,r0Qt1⊗ΛV, D>0. It contradicts the definition ofP4,1.

T1Xis given as

.

P0,0

P0,4

P0,2

P0,1

P2,3

P2,2

P2,1 P3,1

P3,2

P0,3

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For example, the edges1 simplexes

{P0,0P0,1, P0,1P0,2, P0,2P0,3, P0,3P0,4, . . . , P0,0P3,1, P3,1P3,2} 2.3

are given as follows.

1P0,1P3,2 is given by the projection Qt1, t2⊗ΛV, D → Qt1⊗ΛV, D1 where

Dv1 Dv2 Dv3 0,Dv4 v1v2t2t4₁,Dv5 v1v3t2 t₂4, andD1v1 D1v2

D1v3D1v50 andD1v4t41.

2P2,1P3,2is given byDv1Dv2 Dv30,Dv4 v1v2t1t41, andDv5v1v3t2t42.

3P3,1P3,2is given byDv1Dv2 Dv30,Dv4 v1v2t1t4₁, andDv5v1v3t1t4₂.

T2Xis given as follows.

1P0,0P2,1P3,2P3,1is attached by a 2 cell fromDv1 Dv2 Dv3 0,Dv4 v1v2t1

t2 t4₁andDv5v1v3t2t4₂.ThenP2,1is given byD1v4v1v2t1t4₁,D1v5 0, and

P3,1is given byD2v4v1v2t2,D2v5v1v3t2t4₂.

2P0,0P0,1P3,2P3,1is attached by a 2 cell fromDv1 Dv2 Dv3 0,Dv4 v1v2t2t4₁,

andDv5 v1v3t2t42.

3P0,0P0,1P2,2P2,1is attached by a 2 cell fromDv1 Dv2 Dv3 0,Dv4 v1v2t2t42,

andDv5 t4₁.

4P0,1P0,2P2,3P2,2is attached by a 2 cell fromDv1Dv20,Dv3t2₃,Dv4v1v2t2t4₂,

andDv5 t4₁.

5P0,0P0,1P3,2P2,1is not attached by a 2 cell. Indeed, assume that a 2 cell attaches on it.

Notice thatP3,2is given byQt1, t2⊗ΛV, DwithDv1 Dv2Dv30 and

Dv4αv1, v2, v3 f, Dv5βv1, v2, v3 g, 2.4

whereα, β∈v1, v2, v3and{f, g}is a regular sequence inQt1, t2. SinceP0,1P3,2 ∈

T1X, bothαandβmust be contained in the idealtifor somei. Also they are not

int1t2by degree reason. Furthermore, sinceP2,1P3,2 ∈ T1X, we can put that both

αandβare contained in the monogenetic idealvivjfor some 1≤i < j≤3 without

losing generality. Then, dimH∗Qt1, t2, t3⊗ΛV,D<∞for a KS extension

Qt3,0−→

Qt1, t2, t3⊗ΛV,D

−→Qt1, t2⊗ΛV, D, 2.5

by puttingDv kt23fork∈ {1,2,3}withk /i, jandDv nDvnforn /k. Thus, we

haver0Qt1, t2⊗ΛV, D>0. It contradicts to the definition ofP3,2.

Notice there is no 3 cell since it must attach to a 3 cubein graphsin general. Thus,

we see thatTX T2Xis contractible.

On the other hand, letMY ΛW,0 Λw1, w2, w3, w4, w5,0with|w1||w2|

|w3|3,|w4|7 and|w5|9. Then we see thatT1X T1Yfrom same arguments. But, in

T2Y,P0,0P0,1P3,2P2,1is attached by a 2 cell since we can putDw1Dw2Dw3 0 and

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by degree reason. HereP0,1 is given byD1w4 0,D1w5 t5₁, andP2,1 is given by D2w4

w1w2t2 t₂4, D2w5 0. Others are same as T2X. Then three 2 cells on P0,0P0,1P3,2P2,1,

P0,0P2,1P3,2P3,1, andP0,0P0,1P3,2P3,1inT2Ymake the following:

P0,1

P0,0

P2,1 P3,1

P3,2

to be homeomorphic toS2_{. Thus}_T_Y _T

2YS2.

Theorem 2.3. PutX S3_×_S3_×_S3_×_S3_×_S7_×_S7 _and_Y _S3_×_S3_×_S3_×_S3_×_S7_×_S9_{. Then}

T1X T1Y. ButTXS2andTY ∨6_i₁S2_i.

Proof. We see as the proof of Theorem2.2that

T0X {P0,0, 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, P4,2} 2.7

and bothT1XandT1Yare given as

.

P0,5

P0,6

P0,3

P0,2

P0,1

P2,3

P2,2

P2,1 P3,1 P4,1

P3,2 P4,2

P0,4 P2,4

P3,3

P0,0

For all tetragons inT1Xexcept the following 4 tetragons:

1P0,0P0,1P3,2P2,1, 2P0,1P0,2P3,3P2,2, 3P0,0P0,1P4,2P2,1, and4P0,0P0,1P4,2P3,1, 2 cells attach

in T2X. The proof is similar to it of Theorem2.2. Thus we see that T2X is homotopy

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P2,1 P3,1 P4,1

P0,0

P4,2

which is homeomorphic toS2_{. For example, when}_M_X_Λ_V,_{0 Λ}_v

1, v2, v3, v4, v5,

v6,0 with |v1| |v2| |v3| |v4| 3 and |v5| |v6| 7, 2 cells attach P0,0P2,1P4,2P3,1,

P0,0P3,1P4,2P4,1andP0,0P2,1P4,2P4,1fromDv1· · ·Dv4 0,

Dv5v1v2t1t41, Dv6v1v3t1v2v4t2t42,

Dv5v1v2t1t41, Dv6v1v3t1t2 v2v4t2t42,

Dv5v1v2t1t41, Dv6v1v3t2v2v4t2t42,

2.8

respectively.

InT2Y, 2 cells attach all tetragons inT1Yby degree reason. For example, when

MY ΛW,0 Λw1, w2, w3, w4, w5, w6,0with|w1| |w2||w3| |w4|3,|w5|7

and|w6|9, putDw1Dw2Dw30 and

1Dw40, Dw5w1w3t2t₂4, Dw6w2w3t1t2t5₁,

2Dw4t23, Dw5w1w3t2t42, Dw6w2w3t1t2t5₁,

3Dw40, Dw5w1w2t2t₂4, Dw6w3w4t1t2t5₁,

4Dw40, Dw5w1w3t2t42, Dw6w1w4t22w2w3t1t2t5₁,

for1∼4of above. Then we can check thatTY ∨_i6₁S2_i TYcannot be embedded in

R3_.

Theorem 2.4. Even whenr0X r0X×CPnfor then-dimensional complex projective spaceCPn,

it does not fold thatTX TX×CPn_{in general. For example,}

1WhenXS3_×_S3_×_S3_×_S3_×_S7_and_n_{4, then}_T

1XT1X×CP4.

2WhenX S3_×_S3_×_S3_×_S7_×_S7_and_n_{4, then}_T

1X T1X×CP4butT2X

T2X×CP4.

Proof. PutMCPn_Λ_{x, y}_{, d}_with_dx_{0 and}_dy_xn1_for_|x|_{2 and}_|y|₂_n_{1. Put}

Qt1, . . . , tr⊗ΛV ⊗Λx, y, Dthe model of a Borel spaceETr×TrX×CPnofX×CPn.

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P0,5

P0,4

P0,3

P0,2

P0,1

P0,0

P2,3

P2,2

P2,1 _P₄_,₁

P0,5

P0,4

P0,3

P0,2

P0,1

P0,0

P2,3

P2,2

P2,1 P3,1 P4,1

P3,2

and

respectively. ForMX ΛV,0 Λv1, v2, v3, v4, v5,0with|v1| |v2| |v3| |v4| 3

and|v5| 7. HereP4,1is given byDvi 0 fori 1,2,3,4 andDv5 v1v2t1v3v4t1t4₁. It

is contained in bothT0XandT0X×CP4. On the other hand,P3,2is given byDvi0 for

i 1,2,3,Dv4 t₂2,Dv5 v1v2t1t4₁,Dx 0, andDy x5v1v3t2₁. ThenP3,1is given by

Dvi 0 fori1,2,3,4,Dv5 v1v2t1t4₁,Dx0, andDyx5v1v3t2₁. They are contained

only inT0X×CP4.

2 Both T1X and T1X × CP4 are same as one in Theorem 2.2. Notice that

P0,0P0,1P3,2P2,1is attached by a 2 cell inT2X×CP4fromDvi0 fori1,2,3,Dv4v1v2t1t4₁,

Dv5t4₂,Dx0, andDyx5v1v3t1t2. SoTX×CP4 TYforY S3×S3×S3×S7×S9.

Remark 2.5. The author must mention about the spacesX1andX2in5, Examples 3.8 and 3.9

such thatT1X1 T1X2. We can check that 2 cells attach on bothP0P5P9P8of them

com-pare5, page 506.

Remark 2.6. In5, Question 1.6, a rigidity problem is proposed. It says that doesT0Xwith

coordinates determineT1X? ForTX, it is false as we see in above examples. But it seems

that there are certain restrictions. For example, isT2Xsimply connected?

Acknowledgment

This paper is dedicated to Yves F´elix on his 60th birthday.

References

1 S. Halperin, “Rational homotopy and torus actions,” in Aspects of Topology, vol. 93 of London Math. Soc.

Lecture Note Ser., pp. 293–306, Cambridge Univ. Press, Cambridge, UK, 1985.

2 B. Jessup and G. Lupton, “Free torus actions and two-stage spaces,” Mathematical Proceedings of the

Cambridge Philosophical Society, vol. 137, no. 1, pp. 191–207, 2004.

3 Y. F´elix, S. Halperin, and J.-C. Thomas, Rational Homotopy Theory, Springer, Berlin, Germany, 2001.

4 Y. F´elix, J. Oprea, and D. Tanr´e, Algebraic Models in Geometry, Oxford University Press, Oxford, UK,

2008.

5 T. Yamaguchi, “A Hasse diagram for rational toral ranks,” Bulletin of the Belgian Mathematical Society—

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Examples of Rational Toral Rank Complex

Research Article

Examples of Rational Toral Rank Complex

Toshihiro Yamaguchi

1. Introduction

2. Examples

Acknowledgment

References

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