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Proyecciones (Antofagasta)

Print version ISSN 0716-0917

Proyecciones (Antofagasta) vol.36 no.2 Antofagasta June 2017 


On the toral rank conjecture and some consequences

Hassan Aaya1 

Mohamed Anas Hilali1 

Mohamed Rachid Hilali1 

Tarik Jawad1 

1 University Hassan II, Morocco


The aim of this work is to improve the lower bound of the Puppe inequality. His theorem [15,Theorem 1.1] states that the sum of all Betti numbers of a well-behaved space X is at least equal to 2n, where n is rank of an n-torus T n acting almost freely on X.

1. Introduction

The well-known Halperin conjecture [8, ] p. 271about torus actions on topological spaces is behind many works in mathematics like the Hilali conjecture [12] and the inequality of Puppe [15) ] Theorem 1.1:If X is a space on which an n-torus acts, we say the action is almost-free if each isotropy subgroup is finite. The largest integer for which X admits an almost free n-torus is called the toral rank of X and denoted rk(X). If X does not admit any almost free torus action, then rk(X) = 0. Unfortunately rk(X) is not a homotopy invariant and is quite difficult to compute. To obtain a homotopy invariant, we introduce the rational toral rank, rk 0 (X) that is, the maximum of rk(Y) among all finite CW complexes Y in the same rational homotopy type as X.

Conjecture (The Toral rank conjecture).

If X is simply connected, then dim .

Conjecture (The Hilali conjecture).

If X is elliptic and simply connected, then dim .

Theorem 1.1 (Puppe Inequality).

If X is simply connected, then dim .

In the present paper we give in section 2 an optimised proof of the theorem 1.1, and in section 3 we establish a new theorem with an improvement of the lower bound of the Puppe inequality.


The authors are grateful to Aziz El Kacimi for pointing out a mistake in the original version of the paper and for suggesting examples.

2. New proof of the theorem 1.1

Let X be a simply connected topological space with an almost free T n -action. We denote rk 0(X) = n and we suppose that

The Sullivan minimal model of the classifying space of the Lie group T n is a polynomial ring denoted here R, with the following form:

The associated Borel fibration [5 ] , p. 53 of this action is:

According to Brown [6], there exists a complex of differential R-modules with a quasi-isomorphism of R-modules :

Let be a basis of such that:

The differential can be written for

where P i is a homogeneous polynomial in (t 1 ,…,t n ).

Lets consider the p x p matrix over , and lets denote

I is an ideal of R, and we have the following diagram:

where i is the canonical injection and p is the canonical surjection, is induced by passing i to the quotient.

By passing to cohomology the diagram hereinabove induces the following diagram:

is injective because if such that then


Hence .

Therefore .

And dim is a free family in .

In another hand since M is a square matrix of order p over the ring R then I is a free R-module of dimension .

We have dim witch implies that dim .

3. An improvement of the lower bound of Puppe inequality

3.1. The main theorem

Let X be a simply connected topological space with an almost free -action, such that dim .

The associated Borel fibration [5 ] , p. 53 of this action is:

has the Hirsch-Brown minimal model

Module [2, ] Section 1.3

We define an increasing filtration F q on H*(X; C) by:

The length l(X) of H*(X, C) is defined by:


We use the evaluation at to define the new space:

where the structure of this H*(X, C)-module is defined by the map:

we know by [3] , theorem 4-1, that for we have:

The coboundary is given by evaluated at ([2] , p. 26),

Now for every , we define A q to be a complement of F q-1 in F q :

The main result of this article is an improvement of the lower bound of Puppe inequality expressed in the following theorem:

Theorem 3.1.Let X be a simply connected topological space with an almost free -action, we denote n = rk 0 (X) for we always have dim

The proof of this theorem is based on the following lemma:

Lemma 3.2. Under the same conditions as the theorem above one has: dim .

Definition 3.3.[17] ,vol 1, p. 90 By P n we denote n-dimensional projective space over C . A projective algebraic variety V is an algebraic subset of P n, that is, the zero-set of some homogeneous polynomials , in the homogeneous coordinates (x0, … , xn) of P n: V = {(x0, … , xn) | |fi (x0, … , xn) = 0, i I}

Proof of the lemma 3.2 According to Puppe [16] , p. 7, we know that , and dim , for every q, .

Let {a 1, …, a r} and {b 1 , …, b s } be two bases of A 1 and A 0. For each we can write:

where p i are homogeneous polynomials on t 1 , … , t n and is a linear composition of b 2, … , b s over . If we suppose that , then the algebraic variety V(p 1, …, p r) is different from {(0, …,0)}. Hence we can take such that:

This shows us that the cocyle b 1 is not a zero in which is absurd.

Proof of the Theorem 3.1 Let's denote . Then we have .

One has:

3.2. Examples

Remark 3.4. The theorem 3.1 gives a measurement of the obstruction of a manifold to have an almost free -action, for example a compact simply connected manifold M with the sum of it's Betti numbers < 3n - 2 can't have an almost free - action.

Example 3.5.The toral rank of the manifold M = (S 2n+1)r is equal to r and the sum of it's Betti numbers is equal to 2r [8] , p. 284, We have dim H*((S 2n+1)4; Q) = 16 < 3 x 7 - 2 = 19 so (S 2n+1)4 can't have an almost free - action.

Remark 3.6. In 2012 M. Amann [4] , Theorem A established the following result:

Theorem A. If an n-torus T acts almost freely on a finite-dimensional paracompact Hausdorf space X, then dim

X may be taken to be a finite CW-complex or a compact manifold.

It's clear that starting from n=7 the theorem 3.1 gives a greater lower bound than theorem A .


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Received: May 2016; Accepted: March 2017

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