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.
1. Introduction
Let be a simply connected CW complex with and be the rational toral rank of , which is the largest integer such that an -torus (-factors) can act continuously on a CW-complex in the rational homotopy type of 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
can hold for a formal space and an integer [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 of . If an -torus acts on by , there is a minimal KS extension with for
with and modulo the ideal for which is induced from the Borel fibration [4]
According to [1, Proposition 4.2], if and only if there is a KS extension of above satisfying . Moreover, then acts freely on a finite complex that has the same rational homotopy type as . So we will discuss this note by Sullivan models.
We want to give a classification of rationally almost free toral actions on associated with rational toral ranks and also present certain relations in them. Recall a finite-based CW complex in [5, Section 5]. Put the set of isomorphism classes of KS extensions of such that . First, the set of 0-cells is the finite sets where the point of the coordinate exists if there is a model and . Of course, the model may not be uniquely determined. Note that the base point always exists by itself.
Next, 1-skeltons (vertexes) of the 1-skelton are represented by a KS-extension with for , where and . It is given as
where exists by , and exists by . The 2 cell is given if there is a (homotopy) commutative diagram of restrictions
which represents (a horizontal deformation of)
Here exists by , (or ) by , by , and (or ) by . Then we say that a 2 cell attaches to (the tetragon) . Thus, we can construct the 2-skelton .
Generally, an -cell is given by an -cube where a vertex of of height , -vertexes of height , , a vertex of height . Here is the symbol which removes the below element, and the differential is the restriction of .
We will call this connected regular complex the rational toral rank complex (r.t.r.c.) of . Since in our case, it is a finite complex. For example, when and , we have
which is an unusual case. Then, of course, . Recall that but [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 and for some products of odd spheres make certain different homotopy types of r.t.r.c., respectively. Remark that the above inequality (*) is a property on or as the example of Theorem 2.4(1). We see in Theorem 2.4(2) an example that but , which is a higher-dimensional phenomenon of (*).
2. Examples
In this section, the symbol means the tetragon, which is the cycle with vertexes , , , , and edges , , , .
In general, it is difficult to show that a point of does not exist on a certain coordinate. So the following lemma is useful for our purpose.
Lemma 2.1. If has the rational homotopy type of the product of finite odd spheres and finite complex projective spaces, then for any .
Proof. Suppose that has the rational homotopy type of the product of odd spheres and complex projective spaces. Put a minimal model with and odd. If , then is pure; that is, for all . Therefore, from [2, Lemma 2.12], . Thus, we have .
Theorem 2.2. Put and . Then . But is contractible and .
Proof. Let with and . Then
For example, they are given as follows.(0) is given by .(1) is given by with and .(2) is given by with , , and .(3) is given by with , , , and .(4) is given by with , , , , and .(5) is given by with , , , , and .(6) is given by with and (7) is given by with , , , and .(8) is given by with , , , and .(9) is given by with , , and .(10) is given by with and .(11) , that is, a point of the coordinate does not exist. Indeed, if it exists, it must be given by a model whose differential is and by degree reason. But, for any satisfying such conditions, we have for a KS extension
that is, . It contradicts the definition of . is given as
For example, the edges (1 simplexes)
are given as follows.(1) is given by the projection where , , , and and .(2) is given by , , and .(3) is given by , , and . is given as follows. (1) is attached by a 2 cell from , and . (Then is given by , , and is given by , .)(2) is attached by a 2 cell from , , and .(3) is attached by a 2 cell from , , and .(4) is attached by a 2 cell from , , , and .(5) is not attached by a 2 cell. Indeed, assume that a 2 cell attaches on it. Notice that is given by with and
where and is a regular sequence in . Since , both and must be contained in the ideal for some . Also they are not in by degree reason. Furthermore, since , we can put that both and are contained in the monogenetic ideal for some without losing generality. Then, for a KS extension
by putting for with and for . Thus, we have . It contradicts to the definition of . Notice there is no 3 cell since it must attach to a 3 cube (in graphs) in general. Thus, we see that is contractible. On the other hand, let with , and . Then we see that from same arguments. But, in , is attached by a 2 cell since we can put and
by degree reason. Here is given by , , and is given by , . Others are same as . Then three 2 cells on , , and in make the following:
to be homeomorphic to . Thus .
Theorem 2.3. Put and . Then . But and .
Proof. We see as the proof of Theorem 2.2 that
and both and are given as
For all tetragons in except the following 4 tetragons: (1) (2) (3), and (4) , 2 cells attach in . The proof is similar to it of Theorem 2.2. Thus we see that is homotopy equivalent to
which is homeomorphic to . For example, when with and , 2 cells attach , and from ,
respectively. In , 2 cells attach all tetragons in by degree reason. For example, when with , and , put and(1),
(2),
(3),
(4),
for (1)~(4) of above. Then we can check that ( cannot be embedded in ).
Theorem 2.4. Even when for the -dimensional complex projective space , it does not fold that in general. For example,(1)When and , then .(2)When and , then but .
Proof. Put with and for and . Put the model of a Borel space of . (1) and are given as
respectively. For with and . Here is given by for and . It is contained in both and . On the other hand, is given by for , , , , and . Then is given by for , , , and . They are contained only in . (2) Both and are same as one in Theorem 2.2. Notice that is attached by a 2 cell in from for , , , , and . So for .
Remark 2.5. The author must mention about the spaces and in [5, Examples 3.8 and 3.9] such that . We can check that 2 cells attach on both of them (compare [5, page 506]).
Remark 2.6. In [5, Question 1.6], a rigidity problem is proposed. It says that does with coordinates determine ? For , it is false as we see in above examples. But it seems that there are certain restrictions. For example, is simply connected?
S. Halperin, βRational homotopy and torus actions,β in Aspects of Topology, vol. 93 of London Math. Soc. Lecture Note Ser., pp. 293β306, Cambridge Univ. Press, Cambridge, UK, 1985.
B. Jessup and G. Lupton, βFree torus actions and two-stage spaces,β Mathematical Proceedings of the Cambridge Philosophical Society, vol. 137, no. 1, pp. 191β207, 2004.
