In "A Hosse diagram for rational toral tanks," we see a CW complex ${\mathcal
T}(X)$, 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 {\it
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.Comment: 8 page

Yamaguchi, Toshihiro

English

arXiv

There is a CW complex (), 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

Toshihiro Yamaguchi

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International Journal of Mathematics and Mathematical Sciences

Examples of Rational Toral Rank Complex

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Hindawi Publishing CorporationInternational Journal of Mathematics and Mathematical SciencesVolume 2012, Article ID 867247, 8 pagesdoi:10.1155/2012/867247Research ArticleExamples of Rational Toral Rank ComplexToshihiro YamaguchiFaculty of Education, Kochi University, 2-5-1 Akebono-Cho, Kochi 780-8520, JapanCorrespondence should be addressed to Toshihiro Yamaguchi, tyamag@kochi-u.ac.jpReceived 21 December 2011; Accepted 6 March 2012Academic Editor: Frank WernerCopyright q 2012 Toshihiro Yamaguchi. This is an open access article distributed under theCreative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.There is a CW complex TX, which gives a rational homotopical classification of almost free toralactions on spaces in the rational homotopy type of X associated with rational toral ranks and alsopresents certain relations in them. We call it the rational toral rank complex of X. It represents avariety of toral actions. In this note, we will give eﬀective 2-dimensional examples of it when X isa finite product of odd spheres. This is a combinatorial approach in rational homotopy theory.1. IntroductionLet X be a simply connected CW complex with dimH∗X;Q < ∞ and r0X be the rationaltoral rank of X, which is the largest integer r such that an r-torus Tr  S1 × · · · × S1 r-factorscan act continuously on a CW-complex Y in the rational homotopy type of X with all itsisotropy subgroups finite such an action is called almost free 1. It is a very interestingrational invariant. For example, the inequalityr0X  r0X  r0(S2n)< r0(X × S2n)∗can hold for a formal space X and an integer n > 1 2. It must appear as one phenomenonin a variety of almost free toral actions. The example ∗ is given due to Halperin by usingSullivan minimal model 3.Put the Sullivan minimal model MX  ΛV, d of X. If an r-torus Tr acts on X byμ : Tr ×X → X, there is a minimal KS extension with |ti|  2 for i  1, . . . , rQt1, . . . , tr, 0 −→ Qt1, . . . , tr ⊗ ∧V,D −→ ∧V, d 1.12 International Journal of Mathematics and Mathematical Scienceswith Dti  0 and Dv ≡ dv modulo the ideal t1, . . . , tr for v ∈ V which is induced from theBorel fibration 4X −→ ETr ×μTr X −→ BTr. 1.2According to 1, Proposition 4.2, r0X ≥ r if and only if there is a KS extension of abovesatisfying dimH∗Qt1, . . . , tr⊗∧V,D < ∞. Moreover, then Tr acts freely on a finite complexthat has the same rational homotopy type as X. So we will discuss this note by Sullivanmodels.We want to give a classification of rationally almost free toral actions on X associatedwith rational toral ranks and also present certain relations in them. Recall a finite-based CWcomplex TX in 5, Section 5. Put Xr  {Qt1, . . . , tr ⊗ ∧V,D} the set of isomorphismclasses of KS extensions ofMX  ΛV, d such that dimH∗Qt1, . . . , 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 thecoordinate s, r exists if there is a model ΛW,dW ∈ Xr and r0ΛW,dW  r0X − s − r. Ofcourse, the model may not be uniquely determined. Note that the base point P0,0  0, 0always exists by X itself.Next, 1-skeltons vertexes of the 1-skelton T1X are represented by a KS-extensionQt, 0 → Qt⊗ΛW,D → ΛW,dWwith dimH∗Qt⊗∧W,D < ∞ for ΛW,dW ∈ Xr ,whereW  Qt1, . . . , tr ⊕ V and dW |V  d. It is given asor orQ Q Qor · · · ,PPPwhere P exists by ΛW,dW, and Q exists by Qt ⊗ ΛW,D. The 2 cell is given if there is ahomotopy commutative diagram of restrictions( W,dW )(Q[tr+1]⊗W,Dr+1)ΛΛ(Q[tr+2]⊗ΛW,Dr+2)(Q[tr+1, tr+2]⊗ΛW,D)which represents a horizontal deformation of.PcPbPaPdHere Pa exists by ΛW,dW, Pbor Pd by Qtr1 ⊗ ΛW,Dr1, Pc by Qtr1, tr2 ⊗ ΛW,D,and Pdor Pb by Qtr2 ⊗ ΛW,Dr2. Then we say that a 2 cell attaches to the tetragonPaPbPcPd. Thus, we can construct the 2-skelton T2X.International Journal of Mathematics and Mathematical Sciences 3Generally, an n-cell is given by an n-cubewhere a vertex of Qtr1, . . . , trn⊗ΛW,D ofheight r  n, n-vertexes {Qtr1, . . . ,∨tri, . . . , trn ⊗ ΛW,Di}1≤i≤n of height r  n − 1, . . ., avertex ΛW,dW of height r. Here ∨ is the symbol which removes the below element, and thediﬀerential Di is the restriction of D.We will call this connected regular complex TX  ∪n≥0TnX the rational toral rankcomplex r.t.r.c. of X. Since r0X < ∞ in our case, it is a finite complex. For example, whenX  S3 × S3 and Y  S5, we haveTX ∨ TY   T1X ∨ T1Y   T1X × Y   TX × Y , 1.3which is an unusual case. Then, of course, r0X  r0Y   r0X × Y . Recall that r0S3 × S3 r0S7  r0S3×S3×S7 butT1S3×S3∨T1S7  T1S3×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 2.2and 2.3 that the diﬀerences between X  Z × S7 and Y  Z × S9 for some products Z of oddspheres make certain diﬀerent homotopy types of r.t.r.c., respectively. Remark that the aboveinequality ∗ is a property on T0X or T1X as the example of Theorem 2.41. We see inTheorem 2.42 an example that T1X  T1X × CPn but T2X  T2X × CPn, which is ahigher-dimensional phenomenon of ∗.2. ExamplesIn this section, the symbol PiPjPkPl means the tetragon, which is the cycle with vertexes Pi, Pj ,Pk, Pl, and edges PiPj , PjPk, PkPl, PlPi.In general, it is diﬃcult to show that a point of T0X does not exist on a certain coor-dinate. So the following lemma is useful for our purpose.Lemma 2.1. IfX has the rational homotopy type of the product of finite odd spheres and finite complexprojective spaces, then 1, r /∈ T0X for any r.Proof. Suppose that X has the rational homotopy type of the product of n odd spheresand m complex projective spaces. Put a minimal model A  Qt1, . . . , tn−1, x1, . . . , xm ⊗Λv1, . . . , vn, y1, . . . , ym, D with |t1|  · · ·  |tn−1|  |x1|  · · ·  |xm|  2 and |vi|, |yi| odd. IfdimH∗A < ∞, then A is pure; that is, Dvi,Dyi ∈ Qt1, . . . , 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. PutX  S3 ×S3 ×S3 ×S7 ×S7 and Y  S3 ×S3 ×S3 ×S7 ×S9. Then T1X  T1Y .But TX is contractible and TY   S2.Proof. LetMX  ΛV, 0  Λv1, v2, v3, v4, v5, 0 with |v1|  |v2|  |v3|  3 and |v4|  |v5| 7. ThenT0X  {P0,0, P0,1, P0,2, P0,3, P0,4, P0,5, P2,1, P2,2, P2,3, P3,1, P3,2}. 2.1For example, they are given as follows.0 P0,0 is given by ΛV, 0.1 P0,1 is given by Qt1 ⊗ΛV,D with Dv1  t21 and Dv2  Dv3  Dv4  Dv5  0.4 International Journal of Mathematics and Mathematical Sciences2 P0,2 is given by Qt1, t2 ⊗ ΛV,D with Dv1  t21, Dv2  t22, and Dv3  Dv4 Dv5  0.3 P0,3 is given by Qt1, t2, t3 ⊗ ΛV,D with Dv1  t21, Dv2  t22, Dv3  t23, andDv4  Dv5  0.4 P0,4 is given by Qt1, t2, t3, t4 ⊗ ΛV,D with Dv1  t21, Dv2  t22, Dv3  t23,Dv4  t44, and Dv5  0.5 P0,5 is given by Qt1, t2, t3, t4, t5 ⊗ ΛV,D with Dv1  t21, Dv2  t22, Dv3  t23,Dv4  t44, and Dv5  t45.6 P2,1 is given by Qt1 ⊗ ΛV,D with Dv1  Dv2  Dv3  Dv5  0 and Dv4 v1v2t1  t417 P2,2 is given by Qt1, t2⊗ΛV,DwithDv1  Dv2  0,Dv3  t22,Dv4  v1v2t1t21,and Dv5  0.8 P2,3 is given by Qt1, t2, t3 ⊗ ΛV,D with Dv1  Dv2  0, Dv3  t22, Dv4 t21  v1v2t1, and Dv5  t43.9 P3,1 is given by Qt1 ⊗ ΛV,D with Dv1  Dv2  Dv3  0, Dv4  v1v2t1  t41,and Dv5  v1v3t1.10 P3,2 is given by Qt1, t2 ⊗ΛV,DwithDv4  v1v2t1  t41 andDv5  v1v3t1  t42.11 P4,1, that is, a point of the coordinate 4, 1 does not exist. Indeed, if it exists, itmust be given by amodel Qt1⊗ΛV,Dwhose diﬀerential isDv1  Dv2  Dv3  0andDv4, Dv5 ∈ Qt1⊗Λv1, v2, v3 by degree reason. But, for anyD satisfying suchconditions, we have dimH∗Qt1, t2 ⊗ΛV, D˜ < ∞ for a KS extensionQt2, 0 −→(Qt1, t2 ⊗ΛV, D˜)−→ Qt1 ⊗ΛV,D, 2.2that is, r0Qt1 ⊗ΛV,D > 0. It contradicts the definition of P4,1.T1X is given as.P0,0P0,4P0,2P0,1P2,3P2,2P2,1 P3,1P3,2P0,3P0,4International Journal of Mathematics and Mathematical Sciences 5For example, the edges 1 simplexes{P0,0P0,1, P0,1P0,2, P0,2P0,3, P0,3P0,4, . . . , P0,0P3,1, P3,1P3,2} 2.3are given as follows.1 P0,1P3,2 is given by the projection Qt1, t2 ⊗ ΛV,D → Qt1 ⊗ ΛV,D1 whereDv1  Dv2  Dv3  0, Dv4  v1v2t2  t41, Dv5  v1v3t2  t42, and D1v1  D1v2 D1v3  D1v5  0 and D1v4  t41.2 P2,1P3,2 is given by Dv1  Dv2  Dv3  0, Dv4  v1v2t1  t41, and Dv5  v1v3t2  t42.3 P3,1P3,2 is given by Dv1  Dv2  Dv3  0, Dv4  v1v2t1  t41, and Dv5  v1v3t1  t42.T2X is given as follows.1 P0,0P2,1P3,2P3,1 is attached by a 2 cell from Dv1  Dv2  Dv3  0, Dv4  v1v2t1 t2 t41 andDv5  v1v3t2  t42. Then P2,1 is given byD1v4  v1v2t1  t41,D1v5  0, andP3,1 is given by D2v4  v1v2t2, D2v5  v1v3t2  t42.2 P0,0P0,1P3,2P3,1 is attached by a 2 cell from Dv1  Dv2  Dv3  0, Dv4  v1v2t2  t41,and Dv5  v1v3t2  t42.3 P0,0P0,1P2,2P2,1 is attached by a 2 cell from Dv1  Dv2  Dv3  0, Dv4  v1v2t2  t42,and Dv5  t41.4 P0,1P0,2P2,3P2,2 is attached by a 2 cell fromDv1  Dv2  0,Dv3  t23,Dv4  v1v2t2t42,and Dv5  t41.5 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 Qt1, t2 ⊗ΛV,D with Dv1  Dv2  Dv3  0 andDv4  αv1, v2, v3  f, Dv5  βv1, v2, v3  g, 2.4where α, β ∈ v1, v2, v3 and {f, g} is a regular sequence in Qt1, t2. Since P0,1P3,2 ∈T1X, both α and βmust be contained in the ideal ti for some i. Also they are notin t1t2 by degree reason. Furthermore, since P2,1P3,2 ∈ T1X, we can put that bothα and β are contained in the monogenetic ideal vivj for some 1 ≤ i < j ≤ 3 withoutlosing generality. Then, dimH∗Qt1, t2, t3 ⊗ΛV, D˜ < ∞ for a KS extensionQt3, 0 −→(Qt1, t2, t3 ⊗ΛV, D˜)−→ Qt1, t2 ⊗ΛV,D, 2.5by putting D˜vk  t23 for k ∈ {1, 2, 3} with k / i, j and D˜vn  Dvn for n/ k. Thus, wehave r0Qt1, t2 ⊗ΛV,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 TX  T2X is contractible.On the other hand, let MY   ΛW, 0  Λw1, w2, w3, w4, w5, 0 with |w1|  |w2| |w3|  3, |w4|  7 and |w5|  9. Then we see that T1X  T1Y  from same arguments. But, inT2Y , P0,0P0,1P3,2P2,1 is attached by a 2 cell since we can put Dw1  Dw2  Dw3  0 andDw4  w1w2t2  t42, Dw5  w1w3t1t2  t51, 2.66 International Journal of Mathematics and Mathematical Sciencesby degree reason. Here P0,1 is given by D1w4  0, D1w5  t51, and P2,1 is given by D2w4 w1w2t2  t42, D2w5  0. Others are same as T2X. Then three 2 cells on P0,0P0,1P3,2P2,1,P0,0P2,1P3,2P3,1, and P0,0P0,1P3,2P3,1 in T2Y make the following:P0,1P0,0P2,1 P3,1P3,2to be homeomorphic to S2. Thus TY   T2Y   S2.Theorem 2.3. Put X  S3 × S3 × S3 × S3 × S7 × S7 and Y  S3 × S3 × S3 × S3 × S7 × S9. ThenT1X  T1Y . But TX  S2 and TY   ∨6i1S2i .Proof. We see as the proof of Theorem 2.2 thatT0X  {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} 2.7and both T1X and T1Y  are given as.P0,5P0,6P0,3P0,2P0,1P2,3P2,2P2,1 P4,1P3,1P3,2 P4,2P0,4 P2,4P3,3P0,0For all tetragons in T1X 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 attachin T2X. The proof is similar to it of Theorem 2.2. Thus we see that T2X is homotopyequivalent toInternational Journal of Mathematics and Mathematical Sciences 7P2,1 P4,1P3,1P0,0P4,2which is homeomorphic to S2. For example, when MX  ΛV, 0  Λv1, 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  · · ·  Dv4  0,Dv5  v1v2t1  t41, Dv6  v1v3t1  v2v4t2  t42,Dv5  v1v2t1  t41, Dv6  v1v3t1  t2  v2v4t2  t42,Dv5  v1v2t1  t41, Dv6  v1v3t2  v2v4t2  t42,2.8respectively.In T2Y , 2 cells attach all tetragons in T1Y  by degree reason. For example, whenMY   ΛW, 0  Λw1, w2, w3, w4, w5, w6, 0 with |w1|  |w2|  |w3|  |w4|  3, |w5|  7and |w6|  9, put Dw1  Dw2  Dw3  0 and1 Dw4  0, Dw5  w1w3t2  t42, Dw6  w2w3t1t2  t51,2 Dw4  t23, Dw5  w1w3t2  t42, Dw6  w2w3t1t2  t51,3 Dw4  0, Dw5  w1w2t2  t42, Dw6  w3w4t1t2  t51,4 Dw4  0, Dw5  w1w3t2  t42, Dw6  w1w4t22 w2w3t1t2  t51,for 1∼4 of above. Then we can check that TY   ∨6i1S2i TY  cannot be embedded inR3.Theorem 2.4. Even when r0X  r0X×CPn for the n-dimensional complex projective space CPn,it does not fold that TX  TX × CPn in general. For example,1 When X  S3 × S3 × S3 × S3 × S7 and n  4, then T1X  T1X × CP 4.2 When X  S3 × S3 × S3 × S7 × S7 and n  4, then T1X  T1X × CP 4 but T2X T2X × CP 4.Proof. Put MCPn  Λx, y, d with dx  0 and dy  xn1 for |x|  2 and |y|  2n  1. PutQt1, . . . , tr ⊗ΛV ⊗Λx, y, D the model of a Borel space ETr×Tr X × CPn of X × CPn.1 T1X and T1X × CP 4 are given as8 International Journal of Mathematics and Mathematical SciencesP0,5P0,4P0,3P0,2P0,1P0,0P2,3P2,2P2,1 P4,1P0,5P0,4P0,3P0,2P0,1P0,0P2,3P2,2P2,1 P4,1P3,1P3,2andrespectively. For MX  ΛV, 0  Λv1, v2, v3, v4, v5, 0 with |v1|  |v2|  |v3|  |v4|  3and |v5|  7. Here P4,1 is given by Dvi  0 for i  1, 2, 3, 4 and Dv5  v1v2t1  v3v4t1  t41. Itis contained in both T0X and T0X × CP 4. On the other hand, P3,2 is given by Dvi  0 fori  1, 2, 3, Dv4  t22, Dv5  v1v2t1  t41, Dx  0, and Dy  x5  v1v3t21. Then P3,1 is given byDvi  0 for i  1, 2, 3, 4, Dv5  v1v2t1  t41, Dx  0, and Dy  x5  v1v3t21. They are containedonly in T0X × CP 4.2 Both T1X and T1X × CP 4 are same as one in Theorem 2.2. Notice thatP0,0P0,1P3,2P2,1 is attached by a 2 cell inT2X×CP 4 fromDvi  0 for i  1, 2, 3,Dv4  v1v2t1t41,Dv5  t42,Dx  0, andDy  x5v1v3t1t2. SoTX×CP 4  TY  for Y  S3×S3×S3×S7×S9.Remark 2.5. The author must mention about the spacesX1 andX2 in 5, Examples 3.8 and 3.9such that T1X1  T1X2. We can check that 2 cells attach on both P0P5P9P8 of them com-pare 5, page 506.Remark 2.6. In 5, Question 1.6, a rigidity problem is proposed. It says that does T0X withcoordinates determine T1X? For TX, it is false as we see in above examples. But it seemsthat there are certain restrictions. For example, is T2X simply connected?AcknowledgmentThis paper is dedicated to Yves Fe´lix 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 theCambridge Philosophical Society, vol. 137, no. 1, pp. 191–207, 2004.3 Y. Fe´lix, S. Halperin, and J.-C. Thomas, Rational Homotopy Theory, Springer, Berlin, Germany, 2001.4 Y. Fe´lix, J. Oprea, and D. Tanre´, 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—Simon Stevin, vol. 18, pp. 493–508, 2011.Submit your manuscripts athttp://www.hindawi.comHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014MathematicsJournal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Mathematical Problems in EngineeringHindawi Publishing Corporationhttp://www.hindawi.comDifferential EquationsInternational Journal ofVolume 2014Applied MathematicsJournal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Probability and StatisticsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Journal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Mathematical PhysicsAdvances inComplex AnalysisJournal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014OptimizationJournal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014CombinatoricsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014International Journal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Operations ResearchAdvances inJournal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Function SpacesAbstract and Applied AnalysisHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014International Journal of Mathematics and Mathematical SciencesHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014The Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014AlgebraDiscrete Dynamics in Nature and SocietyHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Decision SciencesAdvances inDiscrete MathematicsJournal ofHindawi Publishing Corporationhttp://www.hindawi.comVolume 2014 Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Stochastic AnalysisInternational Journal of

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