A Hasse diagram for rational toral ranks

Abstract

Let X be a simply connected CW complex with finite rational cohomology. For the finite quotient set of rationalized orbit spaces of X obtained by almost free toral actions, T 0 (X) = {[Y i ]}, induced by an equivalence relation based on rational toral ranks, we order asIt presents a variation of almost free toral actions on X. We consider about the Hasse diagram H(X) of the poset T 0 (X), which makes a based graph GH(X), with some examples. Finally we will try to regard GH(X) as the 1-skeleton of a finite CW complex T (X) with base point X Q .Here we put Y 1 := X Q if m = 0. Then (X , <) is a strict partially ordered set (poset).Notice that even if T m acts almost freely on X and r 0 (X) = n(> m), then there does not always exist an almost free action of T n−m on a complex in the rational homotopy type of the Borel space ET m × T m X. For example, when X = S 3 × S 3 × S 7 , we obtain r 0 (X) = 3 by standard T 3 -action (). But there exists a free S 1 -action µ :It is also rationally given as the total space of a non-trivial fibration with fiber CP 3 and base S 3 × S 3 . See Example 3.5 below for detail. Thus we stand on our starting point.Claim 1.2. The poset T 0 (X) = ({P i } i , <) is not totally ordered in general.

A rational realization problem in Gottlieb group

“…By the arguments of rationa toral poset structure in [27], we have in Example 4.8 a weaker result of Corollary 1.26.…”

Examples of Rational Toral Rank Complex

“…Then, of course, r 0 (X) + r 0 (Y ) = r 0 (X × Y ). Recall that r 0 (S 3 ×S 3 )+r 0 (S 7 ) = r 0 (S 3 ×S 3 ×S 7 ) but T 1 (S 3 ×S 3 )∨T 1 (S 7 ) T 1 (S 3 ×S 3 ×S 7 ) [5,Example 3.5]. In §2, we see that r.t.r.c.…”

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In [5, Appendix], we see 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.

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Pre-c-symplectic condition for the product of odd-spheres

“…(see[31, Examples 3.5, 3.6]) Suppose that X with r 0 (X) = 3 is pre-c-symplectic. When X = S k1 × S k2 × S k3 , from Theorem 1.7 and Proposition 3.2, the Hasse diagram H(X) is uniquely given as the point P 4 = (2, 1).…”

“…Recall the Hasse diagram H(X) of rational toral ranks for a simply connected space X [31], which is the Hasse diagram of a poset induced by ordering of the Borel fibrations of rationally almost free toral actions on X. When there exists a free t-toral action on a finite complex X ′ of same rational homotopy with X (Proposition 3.1), we can describe a point P = [ET t × T t X ′ ] rationally presented by the Borel space Y = ET t × T t X ′ in the lattice points of the quadrant I.…”