Free torus actions and two-stage spaces

Jessup, Barry; Lupton, Gregory

doi:10.1017/s0305004103007242

Cited by 18 publications

(19 citation statements)

References 10 publications

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“…Refer the arguments of [4, 7.3.2] or [7] for the computations of toral ranks with minimal models. We put M (X) = (ΛV, d).…”

Section: A Halperin's Resultsmentioning

confidence: 99%

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A Hasse diagram for rational toral ranks

Yamaguchi¹

2011

Bull. Belg. Math. Soc. Simon Stevin

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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.

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“…Refer the arguments of [4, 7.3.2] or [7] for the computations of toral ranks with minimal models. We put M (X) = (ΛV, d).…”

Section: A Halperin's Resultsmentioning

confidence: 99%

“…It is a grafting of one on the other. By using a Sullivan model, S.Halperin indicates that rational toral rank does not preserve the product fomula r 0 (X × X ′ ) = r 0 (X) + r 0 (X ′ ) in general [7]([4, Ex.7.19]). Thus this embedding may be complicated in general (see Example 3.9 below).…”

Section: F4mentioning

confidence: 99%

A Hasse diagram for rational toral ranks

Yamaguchi¹

2011

Bull. Belg. Math. Soc. Simon Stevin

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“…Such an action is called almost free. Refer [12] for a relation with Gottlieb groups and rational toral ranks. It is known that r 0 (X) ≤ −χ π (X) for an elliptic space X [1].…”

Section: Rational Toral Rankmentioning

confidence: 99%

A fibre-restricted Gottlieb group and its rational realization problem

Yamaguchi

2015

Topology and its Applications

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“…On the other hand, P 3,2 is given by Dv i = 0 for i = 1, 2, 3, Dv 4 = t 2 2 , Dv 5 = v 1 v 2 t 1 + t 4 1 , Dx = 0 and Dy = x 5 + v 1 v 3 t 2 1 . Then P 3,1 is given by Dv i = 0 for i = 1, 2, 3, 4, Dv 5 = v 1 v 2 t 1 + t 4 1 , Dx = 0 and Dy = x 5 + v 1 v 3 t 2 1 . They are contained only in T 0 (X × CP 4 ).…”

Section: Proofmentioning

confidence: 99%

“…It is a very interesting rational invariant. For example, the inequality r 0 (X) = r 0 (X) + r 0 (S 2n ) < r 0 (X × S 2n ) ( * ) can hold for a formal space X and an integer n > 1 [4]. It must appear as one phenomenon in a variety of almost free toral actions.…”

Section: Introductionmentioning

confidence: 99%

Examples of Rational Toral Rank Complex

Yamaguchi

2012

International Journal of Mathematics and Mathematical Sciences

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There is 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. 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.

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Free torus actions and two-stage spaces

Cited by 18 publications

References 10 publications

A Hasse diagram for rational toral ranks

A Hasse diagram for rational toral ranks

A fibre-restricted Gottlieb group and its rational realization problem

Examples of Rational Toral Rank Complex

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