Examples of Rational Toral Rank Complex

Hindawi Publishing Corporation International Journal of Mathematics and Mathematical Sciences Volume 2012, Article ID 867247, 8 pages 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, Received 21 December 2011; Accepted 6 March 2012 Academic Editor: Frank Werner Copyright q 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. There is a CW complex TX, 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 dim H ∗ X; Q < ∞ and r0 X be the rational toral rank of X, which is the largest integer r such that an r-torus T r  S1 × · · · × S1 r-factors can act continuously on a CW-complex Y 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     r0 X  r0 X  r0 S2n < r0 X × S2n ∗ 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 MX  ΛV, d of X. If an r-torus T r acts on X by r μ : T × X → X, there is a minimal KS extension with |ti |  2 for i  1, . . . , r Qt1 , . . . , tr , 0 −→ Qt1 , . . . , tr  ⊗ ∧V, D −→ ∧V, d 1.1 2 International Journal of Mathematics and Mathematical Sciences with Dti  0 and Dv ≡ dv modulo the ideal t1 , . . . , tr  for v ∈ V which is induced from the Borel fibration 4 μ 1.2 X −→ ET r ×T r X −→ BT r . According to 1, Proposition 4.2, r0 X ≥ r if and only if there is a KS extension of above satisfying dim H ∗ Qt1 , . . . , tr  ⊗ ∧V, D < ∞. Moreover, then T r 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 TX in 5, Section 5. Put Xr  {Qt1 , . . . , tr  ⊗ ∧V, D} the set of isomorphism classes of KS extensions of MX  ΛV, d such that dim H ∗ Qt1 , . . . , tr  ⊗ ∧V, D < ∞. First, the set of 0-cells T0 X is the finite sets {s, r ∈ Z≥0 × Z≥0 } where the point Ps,r of the coordinate s, r exists if there is a model ΛW, dW  ∈ Xr and r0 ΛW, dW   r0 X − s − r. 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 Qt, 0 → Qt⊗ΛW, D → ΛW, dW  with dim H ∗ Qt⊗∧W, D < ∞ for ΛW, dW  ∈ Xr , where W  Qt1 , . . . , tr  ⊕ V and dW |V  d. It is given as Q P or or Q P Q or · · · , P where P exists by ΛW, dW , and Q exists by Qt ⊗ ΛW, D. The 2 cell is given if there is a homotopy commutative diagram of restrictions ⊗ ( Λ W, dW ) (Q[tr+2 ] ⊗ (Q[tr+1 , tr+2 ] (Q[tr+1 ] Λ W, Dr+1 ) ΛW,Dr+2 ) ⊗ ΛW,D) which represents a horizontal deformation of Pc Pb Pd Pa . Here Pa exists by ΛW, dW , Pb or Pd  by Qtr1  ⊗ ΛW, Dr1 , Pc by Qtr1 , tr2  ⊗ ΛW, D, and Pd or Pb  by Qtr2  ⊗ ΛW, Dr2 . Then we say that a 2 cell attaches to the tetragon Pa Pb Pc Pd . Thus, we can construct the 2-skelton T2 X. International Journal of Mathematics and Mathematical Sciences 3 Generally, an n-cell is given by an n-cube where a vertex of Qtr1 , . . . , trn ⊗ΛW, D of ∨ height r  n, n-vertexes {Qtr1 , . . . , tri , . . . , trn  ⊗ ΛW, Di }1≤i≤n of height r  n − 1, . . ., a vertex ΛW, dW  of height r. Here ∨ is the symbol which removes the below element, and the differential Di is the restriction of D. We will call this connected regular complex TX  ∪n≥0 Tn X the rational toral rank complex r.t.r.c. of X. Since r0 X < ∞ in our case, it is a finite complex. For example, when X  S3 × S3 and Y  S5 , we have TX ∨ TY   T1 X ∨ T1 Y   T1 X × Y   TX × Y , 1.3 which is an unusual case. Then, of course, r0 X  r0 Y   r0 X × Y . Recall that r0 S3 × S3   r0 S7   r0 S3 ×S3 ×S7  but T1 S3 ×S3 ∨T1 S7   T1 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 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 on T0 X or T1 X as the example of Theorem 2.41. We see in Theorem 2.42 an example that T1 X  T1 X × CP n  but T2 X  T2 X × CP n , which is a higher-dimensional phenomenon of ∗. 2. Examples In this section, the symbol Pi Pj Pk Pl means the tetragon, which is the cycle with vertexes Pi , Pj , Pk , Pl , and edges Pi Pj , Pj Pk , Pk Pl , Pl Pi . 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, r ∈ / T0 X for any r. 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 , D with |t1 |  · · ·  |tn−1 |  |x1 |  · · ·  |xm |  2 and |vi |, |yi | odd. If dim H ∗ A < ∞, then A is pure; that is, Dvi , Dyi ∈ Qt1 , . . . , tn−1 , x1 , . . . , xm  for all i. / Therefore, from 2, Lemma 2.12, r0 A  1. Thus, we have 1, r0 X − 1  1, n − 1 ∈ T0 X. Theorem 2.2. Put X  S3 × S3 × S3 × S7 × S7 and Y  S3 × S3 × S3 × S7 × S9 . Then T1 X  T1 Y . But TX is contractible and TY   S2 . Proof. Let MX  ΛV, 0  Λv1 , v2 , v3 , v4 , v5 , 0 with |v1 |  |v2 |  |v3 |  3 and |v4 |  |v5 |  7. Then T0 X  {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. 0 P0,0 is gi (...truncated)