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.

**Acknowledgments**

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 :

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

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
.

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*(x

_{0}, … , x

_{n}) of

**P**

^{n}: V = {(x

_{0}, … , x

_{n}) | |

*fi*(x

_{0}, … , x

_{n}) = 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 2^{r} [^{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 .