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\begin{document}

\title{Realising unstable modules as the cohomology of spaces and mapping spaces}
\author{G\'erald Gaudens and Lionel Schwartz} 
\date {UMR 7539 CNRS, 
 Universit\'e Paris 13\\
 LIA CNRS Formath Vietnam\\}

\maketitle

\begin{abstract}
This report discuss the question whether or not  a given unstable module is the mod-$p$ cohomology of a space. 
One first discuss shortly  the Hopf invariant $1$ problem and the Kervaire invariant $1$ problem  and give their relations to homotopy theory and geometric topology. Then 
one describes some more qualitative results, emphasizing the use of the space $\map(B\Z/p,X)$ and of the  structure of the category of unstable modules.
\end{abstract}



\begin{section}{Introduction}

Let $p$ be a prime number, in all the sequel $H^*X$ will denote the the mod $p$ singular cohomology of the topological space $X$. All spaces $X$ will be supposed $p$-complete
and connected.

The singular mod-$p$ 
cohomology is endowed with various  structures:

\begin{itemize}
\item it is a graded $\Fp$-algebra, commutative in the graded sense,
\item it is naturally a module over the algebra of 
stable cohomology operations which is  known as the mod $p$ Steenrod algebra and denoted by $\A$.
\end{itemize}



The Steenrod algebra is generated by element $Sq^i$ of degree $i>0$ if $p=2$, $\beta$ and $P^i$ of degree $1$ and $2i(p-1)>0$ if $p>2$. These elements  satisfy  the Adem relations. 

In  the mod-$2$ case the relations write
$$
Sq^aSq^b=\sum _{0}^{[a/2]} \binom{b-t-1}{a-2t}Sq^{a+b-t}Sq^t
$$
There are two types of relations for $p>2$
$$
     {P}^{a}  {P}^{b}= \sum _{0}^{[a/p]}(-1)^{a+t} \binom{(p-1)(b-t)-1
}{a-pt}    P^{a+b-t}  {P}^{t}
$$
for $a,b>0$,
and
$$
   {P}^{a}{\beta}    {P}^{b}=\sum _{0}^{[a/p]}(-1)^{a+t}
\binom{(p-1)(b-t)}{a-pt}{\beta} P^{a+b-t} 
{P}^{t} + \sum _{0}^{[(a-1)/p]}(-1)^{a+t-1}
\binom{(p-1)(b-t)-1}{a-pt-1} {P}^{a+b-t}{\beta}
{P}^{t}
$$
 for $a,b>0$.

An easy consequence of the relations is

\begin{thm}\label{gen} The elements $Sq^{2^i}$ if $p=2$; and $\beta$ and $P^{p^i}$ if $p>2$ form a minimal set of multiplicative generators.
\end{thm}

There is more structure. The  cohomology  is an unstable module,  which means that for any cohomology class $x$


\begin{itemize}
\item 
$Sq^i(x)=0$ if $i> \vert x \vert$ if $p=2$, 
\item $\beta^\epsilon P^i (x)=0$ if $\epsilon +2i > \vert x \vert$, if $p>2$,
\end{itemize}

One denotes by $\U$ the abelian category of unstable modules.

\saut


As an example consider $H^*B\Z/2 \cong \F2[u]$, $\vert u \vert =1$;  $H^*B\Z/p \cong E(t) \otimes \Fp[x]$, $\vert t \vert =1$ and $\vert x \vert =2$.

 One has $Sq^1(u)= u^2$; resp.  $\beta (t) =u$ and $P^1(x)=x^p$. 
 
 These relations,  the Cartan formula that gives the action on products, the restriction axiom which tells that
 \begin{itemize}
 \item $Sq^d=x^2$ if $\vert x \vert =d$ ($p=2$);
 \item $P^ix= x^p$ if $\vert x \vert =2i$ ($p>2$.
 \end{itemize}  
 
 and the instability  completely determine  the action.
 \saut
 
 The definition of the suspension of an unstable module is central in the theory. This is motivated by the suspension theorem for the cohomology of $\Sigma X$:
 $$
\tilde  H^* \Sigma X \cong \Sigma \tilde H^*X
 $$
 
 with $\Sigma M$ defined by:
 \begin{itemize}
 \item $(\Sigma M)^n \cong M^{n-1}$,
 \item for any $\theta \in \A$, $\theta (\Sigma x) =\Sigma \theta (x)$.
 \end{itemize}
 or $$\Sigma M \cong M \otimes \Sigma \Fp$$
 \saut
 
 The category of algebras that are unstable modules, and such that the above properties relating the two structures hold is called the category of unstable algebras and denoted by
 $\cal K$.
 
 \saut

It is  a very classical question in homotopy theory to ask whether or not a certain unstable $\A$-algebra $K$ is the mod $p$ cohomology of a space. If $K$ is the cohomology of a space one also would like  to read off the structure of the cohomology some  homotopical properties of  the associated space. This could be turned -using the Kan-Thurston theorem- into a (discrete) group theoretic question, however very little can said from this point of view. 

Characterising unstable modules or algebras that are the cohomology of a space is out of reach in full generality.   But there are interesting types of questions which are accessible :

\begin{itemize}

\item  Questions about specific modules. One gives two very classical examples in the second section and another one later.
\item  Necessary conditions  for a  module to be possibly the cohomology of a space, this will be done in the last sections.

\end{itemize}

There are two ways (at least) to get results on these questions. The first one is to use deeper structure in singular cohomology, for example secondary operations which give factorisations of primary operations, or to use extraordinary cohomology theories.

A second approach is to consider mapping spaces. More precisely let $K$ be an unstable module, assume it is the reduced cohomology of a space $X$. Then consider
the mapping spaces $\map(S,X)$, pointed or not, and get restrictions by looking at the the cohomology of the mapping space. 
One option for the source space $S$ is to choose $S^n$. In this case one considers 
the space of pointed maps. If $n=1$, as tool to compute the cohomology of $\Omega X$,   one has at hand  the Eilenberg-Moore spectral sequence. 
More generally for any $n$ there is  a generalisation of the previous one induced by the Goodwillie-Arone tower. The first case is studied in \cite{S98}, the second one in \cite{K08}. A short description of  results   is given at the end of the next section.

Another option for $S$ is the classifying space $B\Z/p$. In this case  a theorem of Jean Lannes computes the cohomology of the mapping space. This will be described in more details in the last sections.


\saut
\saut

This talk was presented at the Vietnam Institute of Advanced Studies in Mathematics (VIASM) yearly meeting on August 2012. The second author thanks the VIASM, and in particular Ngo Bao Chau and Le Tuan Hoa, as well as the staff of the Institute,
for their invitation and warm hospitality.


\end{section}
\begin{section}{The Hopf invariant and the Kervaire invariant}

The Hopf invariant $1$ problem 
is a very famous example of the question,  it  follows. 
Let $f$ be a map $ S^{2n-1} \ra S^{n}$, consider the cone $C_f$ (homotopy cofiber). 
Here are two examples: the  self-map of $S^1$, $z \mapsto z^2$ whose cone is $\PR^2$, the Hopf map $S^3 \ra S^3/S^1 \cong S^2$ whose cone is $\PC^2$.

The reduced cohomology of $C_f$ is of dimension  $1$   in degree $n$ and $2n$, trivial elsewhere.
Denote by $g_{n}$ (resp. $g_{2n}$) a generator in degree $n$ (resp. $2n$). The Hopf invariant of $f$ is defined (up to a sign) by the equation $$g_{n}^2=H(f)g_{2n}$$ 
One can work either with integral cohomology or with mod-$p$ cohomology. If $n$ is odd one works with mod-$2$ cohomology. 

 The two examples alluded to above have both Hopf invariant $1$.
The question is to decide for which values of $n$  $H(f)$ can take the value
$1$.

\saut
Consider the case of mod-$2$ singular cohomology. The restriction axiom allows to rewrite the equation as
$$
Sq^{n}g_{n}=H(f) g_{2n}
$$

So one can reformulate the Hopf invariant $1$ question as follows.
Let $h$ be a given integer, suspend the map $h-n$ times. Does there exists a $2$-cells space, with one cell in degree $h$ a second one in degree $n+h$ related by the operation $Sq^{n}$
$$
\xymatrix{
&		&		h && n+ h \\
			&&		\F2 \ar@/^2pc/[rr]^{Sq^{n}}	& \ldots 0 \ldots&	\F2	&   &		\\
						}
$$
Doing that one modifies the question by going to the stable homotopy world. Because of \ref{gen} for such a complex to exist $n$ must be a power of $2$. So the problem reduces to look at complexes as:

$$
\xymatrix{
&		&		 h&& h+ 2^k \\
			&&		\F2 \ar@/^2pc/[rr]^{Sq^{2^k}}	& \ldots 0 \ldots&	\F2	&   &		\\
						}
$$
This is a non trivial element in the group $\Ext_{\mathcal A_2}^{1,2^k}(\F2,\F2)$ which is denoted by $h_k$.
If one works with the  Adams spectral sequence  for spheres the question is which  elements  $h_k$ of the first line of the $E_2$-term are infinite cycles.


The problem was solved by John Frank Adams using secondary operations in mod $2$ cohomology in a celebrated paper \cite{Ad60}, the only values of $k$  for which this holds are $0, 1,2,3$. The case of an odd prime was done by Arunas Liulevicius. Later Adams et Michael Atiyah gave an "unstable proof" based on Adams operations in $K$-theory \cite{AA66}.  This proof works  for $p=2$ and also for $p>2$ but is slightly more difficult in this
last case.  Some interesting generalisations are to be found in \cite{Hof69}.

In fact Adams theorem is stronger, it applies (for example)  to  modules which looks like 

$$
\xymatrix{
			&&		M \ar@/^2pc/[rr]^{Sq^{2^k}}	& \ldots 0 \ldots&	\F2	&   &		\\
						}
$$
where $M$ in an unstable module which is possibly non trivial only between degrees $h$ and $h+\ell$, and $\ell$ is small compared to $2^k$. It shows that it cannot be the cohomology of a space. But,  there is a  very important restriction: it is 
that the operation $Sq^1$ acts trivially.


The Hopf invariant $1$ problem  is strongly linked to  geometry: for which values of $k$ does there exists a Lie group structure (or a somewhat weaker structure, {\it e.g. } $H$-space structure) on the sphere $S^{k}$? Outside of $S^0$  the sphere needs to be of odd dimension by elementary differential geometry. The answer is that the only possible values are $1,3,7$.




\saut


 The Kervaire invariant $1$ problem is a case not covered by Adams' theorem. It is (roughly) equivalent to the existence of complexes  as  shown below. They  are not in the range of application of Adams' theorem because of the action of $Sq^1$

 
$$
\xymatrix{
		n &n+1& &&n+ 2^k& \\
			\F2  \ar@/_2pc/[rrrr]^{Sq^{2^k}} \ar@/^2pc/[r]^{Sq^{1}}   &		\F2 && \ldots 0 \ldots&	\F2&   &		\\
					}
$$

or

$$
\xymatrix{
		n &n+1& &&n+ 2^k&n+1+2^k \\
			\F2  \ar@/_2pc/[rrrr]^{Sq^{2^k}} \ar@/^2pc/[r]^{Sq^{1}}   &		\F2 \ar@/^2pc/[rrrr]^{Sq^{2^k}}	&& \ldots 0 \ldots&	\F2\ar@/_2pc/[r]^{Sq^{1}}	&  \F2 &		\\
					}
$$

The geometric question is the following. Given a stably framed manifold the Kervaire invariant is an obstruction to do surgery and to make it  cobordant (as stably framed manifold)
to a sphere with an exotic differentiable structure. William Browder \cite{B69} shows  this  is equivalent to decide whether or  not $h_j^2$
on the second line of the $E_2$-term of the Adams spectral sequence for spheres  is an infinite cycle.

 Thus, such examples of manifolds can only occur in dimension $2^j-2$; examples of stably framed manifolds that are not cobordant to exotic spheres  are $S^1 \times S^1$, $S^3 \times S^3$,  $S^7 \times S^7$ with the framing induced by the Lie group structure , or the octonions.  
 
 Such complexes are known to exist if $k=0,1,2,3,4,5,6$ . They do not exist if $n >7$ after the recent (stable) work of Michael Hill, Michael Hopkins and Douglas Ravenel \cite{HHR}. Their proof depends on a $Z/8Z$-equivariant cohomology theory 
linked to a Lubin-Tate formal $\Z_2[\zeta_8]$-module, $\Z_2$ being the $2$-adic integers, $\zeta_8$ a primitive $8$-root of unity and $\pi=\zeta_8-1$,   whose logarithm writes:
$$
  log_F (x) = x + \sum_{k>0} \frac{x^{2^k}}{\pi^k}
$$

   The case $n=7$ remains unsolved.

\saut
 
  Let $M_1$ and $M_2$ be two unstable modules.
 Assume $M_i$ is the reduced cohomology of a space $X_i$, and that one is given a map $f \colon \Sigma^k  X_2 \ra X_1$ that induces the trivial 
 map in cohomology. Then the long exact sequence splits and the cohomology of the cone of $f$ is an  element in $\Ext^1_\U (M_1 , \Sigma^{k+1} M_2)$. The most famous examples
 have been described above as $\tilde H^*\PR^2$ and $ \tilde H^*\PC^2$.
 
 
 This construction  can be generalised as follows. Suppose given a map $f \colon X_2 \ra X_1$ and assume it can be factored as a composition of $n$-maps $g_i$, $1 \leq i \leq n$, inducing the trivial map in reduced cohomology. One says that $f$ has Adams filtration at least $n$.
 Splicing together the extensions obtained from the maps $g_i$ one gets an element in $Ext^n_\U(\tilde H^*X_1 , \tilde \Sigma^nH^*X_2)$. This  a way to construct the Adams spectral sequence.


\saut
 The preceding examples, as well as others, give evidences for the  "Local Realisation Conjecture" (LCR)
 done in a slightly more restricted form by Nick Kuhn \cite{K95}.


 
 \begin{conjecture} 
 Let $M_1$ and $M_2$ be two finite unstable modules. Let $k$ be an integer that is large enough. Then, for any $h$,  any non-trivial extension
 $$
 E \in \Ext^1_\U(\Sigma^{h}M_1, \Sigma^{h+k} M_2)
 $$
 is not the cohomology of a space.
 \end{conjecture}
 
 As such elements are common  for $h$ large enough this implies the existence of differentials in the Adams spectral sequence computing homotopy classes of stable maps from $X_2$ to $X_1$.
 
 

 
 
 

\saut

Consider a certain unstable module $M$, assume it is the reduced cohomology of a space $X$, and then consider
mapping spaces $\map(S,X)$, may be pointed. One can get restrictions on $M$ by looking at the the cohomology of the mapping space. One option for the space $S$ is to choose $S^n$. In this case one  considers
the space of pointed maps. If $n=1$  one uses  the Eilenberg-Moore spectral sequence to evaluate the cohomology of the space of pointed loops. More generally for any $n$ one uses  the generalisation  induced by the Goodwillie-Arone tower. The first case is studied in \cite{S98}, the second one in \cite{K08}.  It is possible to use this tools because of their  nice behaviour  with respect to the action of the Steenrod algebra. 

For example the Eilenberg-Moore spectral sequence has $E_2$-term 
$$
{\mathrm Tor}_{H^*X}^{-i,*}(\Fp,\Fp)
$$ 
which as $\A$-module is well understood. One gets contradiction by showing that certain Adem relations cannot be satisfied in the cohomology of the (iterated)-loop space. 

One will not describe the results in details here, but here are some:

\begin{thm}
Let $k$ be large enough. There does not exist a complex having as reduced cohomology "looking like"
{\tiny
$$
\xymatrix{
		n &n+1& &&n+ 2^k&n+1+2^k &&&n+2^{k+1}& n+1+2^{k+1}\\
			\F2  \ar@/_2pc/[rrrr]^{Sq^{2^k}} \ar@/^1pc/[r]^{Sq^{1}}   &		\F2 \ar@/^2pc/[rrrr]^{Sq^{2^k}}	&& \ldots 0 \ldots&	\F2  \ar@/^3pc/[rrrr]^{Sq^{2^{k+1}}} \ar@/_1pc/[r]^{Sq^{1}}&\F2 \ar@/_3pc/[rrrr]^{Sq^{2^{k+1}}}& \ldots 0 \ldots&&\F2 \ar@/^1pc/[r]^{Sq^{1}}	&  \F2 &		\\
					}
$$
}
\end{thm}

More generally, one would like to have a result as described informally below (at $p=2$):

\begin{conjecture}
Let $M_1$, $M_2$, $M_3$ be given finite modules. Let $k$ be large enough. As soon as there exists $x \in M_1$ such that 
$Sq^{2^{k+1}}Sq^{2^k} x  \not = 0$ there does not exist a complex which has as reduced cohomology "looking like" the following:
{\tiny
$$
\xymatrix{
			M_1 \ar@/_2pc/[rrrr]^{Sq^{2^k}}    &			& \ldots 0 \ldots&&	\Sigma^{2^k}M_2  \ar@/^2pc/[rrrr]^{Sq^{2^{k+1}}}  && \ldots 0 \ldots&&\Sigma^{2^k+2^{k+1}}M_3 & &		\\
					}
$$
}
\end{conjecture}

Lot of special cases are known, and the result looks to be within to reach.

\saut

One can also consider  the space of all maps with $S=B\Z/p$. in this case the Bousfield-Kan spectral sequence for the cohomology of the mapping space degenerates because of the properties
of $H^*B\Z/p$ as an object of the categories $\U$ and $\calK$. This is what one is going to do, and show how information about the algebraic structure of the category $\U$ allows to get substantial results.



\saut


The results obtained using this approach are   of a more qualitative nature. As an example  the following was  conjectured by Kuhn \cite{K95} . 




\begin{thm}[G. Gaudens, L. Schwartz]\label{fg}
Let $X$ be a space such that $H^*X$ is finitely generated as an $\A$-module. Then $H^*X$ is finite.
\end{thm}


This result is also  a consequence of  \ref{cone} which is going to be given in the next section.
\saut



In the next section one describes $m$-cones and discuss some results that motivates interest for spaces  so that any element in $\tilde H^*X$ is nilpotent (one will say by abuse that the cohomology is nilpotent). In section \ref{Krull} one describes  a first filtration of the category  $\U$, a second one is described in section \ref{Nil}.




\end{section}
\begin{section}{$m$-cones and finite Postnikov systems}

This section is a  short digression  before the description of the algebraic structure of the category $\U$.  It also gives some motivations to consider spaces with nilpotent cohomology. It shows that such spaces naturally occur  in topology as $m$-cones. In \cite{FHLT} Yves F\'elix, Stephen Halperin, Jean-Michel Lemaire, Jean-Claude Thomas get some informations about the topological structure by  looking at  the homology of the loop space. This uses rather different techniques than the ones considered later in this report.


There are two, dual to some extent, ways to construct spaces in homotopy theory. The first one is by attaching cells. One says that a space is $0$-cone if it is contractible, an $m$-cone, $m \geq 1$, is the homotopy cofiber
(the cone) of any continuos map from a space $A$ to an $(m-1)$-cone, an $m$-cone has cup-length less than $m+1$. This means that any $(m+1)$-fold product of cohomology classes of positive degree
is trivial. In particular any element of positive degree is nilpotent. The following theorem \cite{FHLT} gives restrictions on the cohomology of an $n$-cone.




\begin{thm} [Y. F\'elix, S. Halperin, J.-M. Lemaire, J.-C.Claude Thomas]
{ If $X$ is $1$-connected and the homology is finite dimensional  in each degree  then
$$depth (H_*(\Omega X; \Fp)) \leq cat (X)$$}
\end{thm}

The depth of a graded connected $k$-algebra $R$ (possibly infinity) is the largest $n$ such that $Ext^i_R(k, R)=\{0\}$,  $i< n$, $cat(X)$ denotes the Lusternick-Schnirelman category of $X$. This  is the minimum
number of elements of covering of $X$ by contractible subspaces.

\saut
Here is the second way to construct spaces: a $1$-Postnikov system or GEM (generalized Eilenberg-Mac Lane space) is a product (may be infinite) of usual Eilenberg-Mac Lane  $K(\pi,n)$-spaces. An $m$-Postnikov system is the 
homotopy fiber of an $(m-1)$-Postnikov system into a GEM. The $m$-th Postnikov tower $P_m(X)$ of a space $X$ is a particular case of an $m$-system.


\begin{cor}
\label{theorem}{\it Let  $X$ be a $1$-connected  $m$-cone, assume that the cohomology is finite dimensional in any degree. Then the $p$-localisation of $X$ is never a finite $p$-local Postnikov system.}
\end{cor}

This is to be compared with \cite{LS89}:

\begin{thm}[J. Lannes, L. Schwartz]
{\it Let $P_n(X)$ be a $1$-connected $n$-Postnikov tower such that $H^*P_n(X)$ is non-trivial. Then  the reduced cohomology $\tilde H^*P_nX$  contains a non nilpotent element.}
\end{thm}



A finite $1$-connected Postnikov tower is never an $m$-cone, because the cup length of an $m$-cone is bounded by $m+1$.
Nevertheless, Jiang Dong Hua \cite{JDH} has shown  there exists a $3$-stage Postnikov system with nilpotent cohomology.

The following result applies to $m$-cones and more generally to spaces with nilpotent cohomogy. The proof uses the Krull filtration on the category $\U$  (see section \ref{Krull}. Here  $Q(K)=\Ker (\epsilon)/ \Ker(\epsilon)^2$ denotes as usual the indecomposable functor of an augmented unstable algebra $\epsilon \colon K \ra \Fp$.

 
 \begin{thm}  [G. Gaudens, Nguyen T. Cuong, L. Schwartz]  \label{cone}
{\it Let  $X$ be an $m$-cone, for  some $m$. If    $QH^* X \in\U_n$, then $QH^*X \in \U_0$}
\end{thm}


The proofs depends on the algebraic 
structure of the category $\U$, and as said above,  on the cohomology of mapping spaces.



\end{section}

\begin{section}{The Krull filtration on $\U$}\label{Krull}
The category of unstable modules
  $\U$, as any abelian category,  has a natural   filtration: the Krull filtration, by thick subcategories stable under colimits 
$$
\U _0 \subset \U _1 \subset \U _2  \subset \ldots \subset \U
$$

Because of the degree filtration on any unstable module the simple objects in $\U$ are the  modules $\Sigma^n \Fp$.

The subcategory $\U_0$ is the  largest thick sub-category generated by simple objects and stable under colimits. It is the subcategory of locally finite modules. 
 An unstable module is locally finite if  the span over $\A$ of any $x \in M$ is finite. Indeed this last condition is obviously  satisfied when taking extensions and is preserved by colimits.
 
  Having defined by induction $\U_n$ one defines $\U_{n+1}$ as follows. One first introduces the quotient category $\U/\U_n$ whose objects are the same of those of $\U$ but where morphisms in $\U$ that have kernel and cokernel in $\U_0$ are formally inverted.
 Then  $(\U/\U_n)_0$ is defined  as above and $\U_{n+1}$ is the pre-image of this subcategory in $\U$ {\it via} the canonical projection functor.
  As said above this construction works for any abelian category. One refers to \cite{Gab} for details. 
 This induces a filtration on any unstable module $M$, one has \cite{S94}: 
 
 
\begin{thm} Let $M \in \U$ and $K_n(M)$ be the largest sub-object of $M$ that is in $\U_n$, then $$M = \cup_n K_n(M{})$$
\end{thm}

As examples one has 
\begin{itemize}
\item
$\Sigma^kF(n) \in \U_n \setminus\U_{n-1}$, the unstable modules $F(n)$ are the canonical generators of $\U$, generated in degree $n$ by $\iota_n$ and $\Fp$-basis 
$P^I\iota_n$, $I$ an admissible multi-index of excess less than $n$;


\item  $H^* B\Z/2\cong \F2[u]$, does not belong to $\U _n$ any $n$ but,
\item $H^*B\Z/2$ is a Hopf algebra and the $n$-th step of the primitive filtration $P_nH^*B\Z/2$ is in $\U_n$.  The analogous result holds for $H^*\Z/p$.
 \end{itemize}


\saut

There is a characterisation of the Krull filtration in terms of a functor introduced by Lannes and denoted $T$.

\begin{defi}
The functor $T\colon \U \ra  \U$ is left adjoint to the functor $M \mapsto H^*B\Z/p \otimes M$. As the unstable module splits up as the direct sum $\Fp \oplus \tilde H^*B\Z/p$. 
the functor $T$
is isomorphic to the direct sum of the identity functor and of the functor $\bar T$ left adjoint of  $M \mapsto \tilde H^*B\Z/p \otimes M$.
\end{defi}

Direct computation shows that $T(\Sigma^n \Fp)$ is isomorphic to $\Sigma^n \Fp$. The functor $T$ has wonderful properties. 
It generalizes directly to a functor $T_V$, $V$ elementary abelian $p$-group, left adjoint to $M \mapsto H^*BV \otimes M$, $T_V \cong T^{dim(V)}$.
Below are the main properties of  $T_V$, \cite{La92}, \cite{S94}.

\begin{thm}[J. Lannes] \label{Lannes1} The functor $T_V$ commutes with colimits (as a left adjoint). It  is exact. Moreover there is a canonical isomorphism
$$
T_V(M_1 \otimes M_2) \cong T_V(M_1) \otimes T_V(M_2)
$$
\end{thm}

A special case of the last property,  if $M_1 = \Sigma \Fp$ it  writes as

$$
T_V(\Sigma M) \cong \Sigma T_V(M)
$$



One gets \cite{S94}
 %$\U \not = \cup_n\U_n$, but  each object  $M \in \U$ is colimit of sub-objects  $K_n(M)$, $K_n(M)$: the largest sub-object of $M$ in $\U_n$ 
 \begin{thm}\label{krull} The following two conditions are equivalent:
 \begin{itemize}
 
 \item$M \in \U_n$ ,
 \item $\bar T^{n+1} (M)=\{0\}$.
 \end{itemize}
\end{thm}

It follows that

\begin{cor}\label{krull-t}

If $M \in \U_m$ and $N \in \U_n$ then $ M \otimes N \in \U_{m+n}$

\end{cor}


There is also a characterisation of objects in $\U_n$ of combinatorial nature  for $p=2$
 \cite{S06}:

\begin{thm}\label{comb}
A finitely generated unstable ${\mathcal A}_2$-module $M$ is in $\U_n$ if and only if its Poincar\'e series $\Sigma_n  a_nt^n$ has the following property. There exists an integer $k$ so that the coefficient 
$a_d$ is posibly non trivial only for those values of $d$ such that   if $\alpha (d-i) \leq n$, for some  $0 \leq i \leq k$. 
\end{thm}



In this statement $\alpha(k)$ is the number of $1$ in the $2$-adic expansion of $k$. This holds in case of finite dimension in any degree.
A similar statement holds for $p>2$.




\saut

Let $\calF$ be the category of functors from finite dimensional $\Fp$-vector spaces to all vector spaces. Define, ( \cite{HLS},),  a functor  $f \colon  \U \ra \calF$  by  (here $V$ is a finite dimensional $\Fp$-vector space)
$$f(M)(V)=\Hom_{\U}(M,H^*(BV))^* \cong T_V(M)^0$$

Let $\calF_n$ be the sub-category of polynomial functors of degree less than $n$. It  is defined as follows. Let $F \in \calF$, let $\Delta(F) \in \calF$
defined by
$$
\Delta(F)(V)=\Ker(F(V \oplus \Fp) \ra F(V))
$$
Then by definition $F \in \calF_n$ if and only if $\Delta^{n+1}(F)=0$. 
As an an example $V \mapsto V^{\otimes n}$ is in $\calF_n$. The followwing holds for any $M$ :
$$
\Delta(f(M)) \cong f ( \bar T(M))
$$


Thus,  the  diagram commutes:

%\begin{equation*}
%\begin{CD}
%\U_0@>{}>>\cdots@>{}>>  \U_n @>{}>> \U_{n+1}@>{}>>\cdots\\
%@VVV @VVV @VVV @ VVV \\
%\calF_0@>{}>>\cdots@>{}>>  \calF_{n} @>{}>>\calF_{n+1} @> >> \cdots\\
%\end{CD}
%\end{equation*}
$${
\xymatrix{
			 		&\U_0  \ar@{>}[d]^{f}&  \ldots &	\U_{n-1}  \ar@{>}[d]^{f} 	    &\hookrightarrow	&	\U_n   \ar@{>}[d]^{f}   &\hookrightarrow& \U  \ar@{>}[d]^{f} \\
					&\calF_0& \ldots &	\calF_{n-1}   	    &\hookrightarrow	&	\calF_n     &\hookrightarrow & \cal F\\	
					}}$$


\saut



\saut

One can now state a conjecture of  Kuhn conjectures and give results.
The first conjecture  is as follows. 


\begin{thm}[Gaudens, Schwartz]\label{fk} 
{\it Let  $X$ be a space such that  $H^* X \in \U_n$ then   $H^* X \in\U_0$.}
\end{thm}



As said above the following corollary was also conjectured by Kuhn.

 
 \begin{cor}
 {\it Let $X$ be a space.  If $H^* X$ has finitely many generators as unstable module it is finite.}

\end{cor}

Indeed, if an unstable module has finitely many generators it is in $\U_n$ for some $n$, because it is a quotient of a finite direct sum of $F(k)'s$. Then  by \ref{fk}
it is in $\U_0$. But a finitely generated locally finite unstable module is finite.
\saut


In  [K95] Kuhn proved the corollary under additional hypothesis,  using the Hopf invariant $1$ theorem. One key step is a reduction depends on Lannes' mapping space theorem which is going to be described in section \ref{KT}.
In  [S98] the corollary  is proved for $p=2$ using  the Eilenberg-Moore spectral sequence, the argument is claimed to extend to all primes. However it is observed that one has to take care of a differential
$d_{p-1}$ in the Eilenberg-Moore spectral sequence.
As Gaudens observed the method of  [S98]  does not work without some more hypothesis, alike the triviality of the Bockstein homomorphism, see \cite{S10}.
\saut

For $p=2$ in [K08] Kuhn gives a proof depending on the Goodwillie-Arone spectral sequence. S. B\"{u}scher F. Hebestreit, O. Rondig, and M. Stelzer get partial results for $p>2$, \cite{BHRS}.

The theorem is proved now  using only the Bott-Samelson theorem and Lannes' mapping space theorem \cite{GS12}.



\saut
Let  $QH^*X$ the quotient of indecomposable elements  of$H^*X$:
$$
QH^*X \cong \tilde H^*X /(\tilde H^*X)^2
$$

Theorem \ref{fk} is a consequence of:

 \begin{thm}
\label{NGS} {\it Let  $X$ be an $m$-cone, for  some $m$ or more generally a space so that any element in $\tilde H^*X$ is nilpotent.
If    $QH^* X \in\U_n$, then $QH^*X \in \U_0$}
\end{thm}

As observed above  the reduced cohomology of an $n$-cone is nilpotent, by that one means that any element is nilpotent.

\saut

There are two cases to distinguish. 

In the first one there is a non-nilpotent element in the cohomology of $X$. Then, as $\tilde H^*X$ contains non nilpotent element in degrees $d$ with $\alpha(d)$ arbitrary large,  section  \ref{Nil} and \ref{comb}
imply that  $H^*X \not \in \U_n$ for all $n$.




In the second case, $\tilde H^*X$ is nilpotent in the sense defined above. In particular there are non trivial algebra maps from $H^*X$ into a polynomial algebra. So
$$\Hom_\calK (H^*X, H^*BV) =*$$
The argument works  for $p=2$. The case $p>2$ is  needs more care because of the exterior algebra factor, see \cite{LZ86} and the action of $\A$. 

If $H^*X  \in \U_n$, then $QH^*X  \in \U_n$ and by the theorem $QH^*X  \in \U_0$. 

Under this hypothesis proposition 3.9.7 of \cite{S94} (se also \cite{DW})  implies that $H^*X \in \U_0$, in fact there is even an equivalence proposition 6.4.5 of the same reference). 








\end{section}
\begin{section}{The nilpotent filtration}\label{Nil}

Above one has considered spaces so that any element in $\tilde H^*X$
is nilpotent and introduced the terminology "nilpotent" for the cohomology. The restriction axiom allows to express this in term of the action of the Steenrod algebra. More precisely (for $p=2$) it is equivalent to ask that the operation $Sq_0\colon x \mapsto Sq^{\vert x \vert}x$ is "nilpotent" on any element. 
It makes it possible  to extend this definition to any unstable module.

\begin{defi} One says that an unstable module $M$ is nilpotent if for any $x \in M$ there exists $k$ such that $Sq_0^kx=0$.
\end{defi}
In particular a nilpotent module is $0$-connected. A suspension is nilpotent. In fact one has the following:

\begin{prop} An unstable module $M$ is nilpotent if and only if it is the colimit of unstable modules  which have  a finite filtration whose quotients are suspensions.
\end{prop}

This allows to extend  the definition for $p>2$.

More generally  one can define  a filtration on  $\U$. It is filtered by  subcategories ${\mathcal N}il_s$, $s \geq 0$, ${\mathcal N}il_s$ is the smallest thick subcategory stable under colimits and containing all $s$-suspensions.
$$
\U = {\mathcal N}il_0 \supset {\mathcal N}il_1 \supset {\mathcal N}il_2  \supset \ldots \supset {\mathcal N}il_s \supset \ldots
$$
By the very definition {\it  any $M \in {\mathcal N}il_s$ is $(s-1)$-connected.}. 

\begin{prop} Any  $M$ has a convergent decreasing filtration
 $\{M_s\}_{s\geq 0}$ with $M_s/M_{s+1} \cong \Sigma^s R_s(M)$ 
 where $R_s(M)$ is a reduced unstable module, {\it i.e.}  does not contain a non trivial suspension.
\end{prop}

Only the second part of the proposition needs a small argument see \cite{S94}.
The following results are easy consequences of the commutation of  $T$ with suspension, the definition, and of \ref{Lannes1}. Just the last part 
 needs 
a small amount of additional care because $T$ does not commutes with limits.


\begin{prop}\label{nilp}
One has 

\begin{itemize}
\item 
 if $M \in \Nil_m$, $N \in \Nil_n$ then $ M \otimes N \in \Nil_{m+n}$;
 \item  if $M \in \Nil_m$ then $T(M) \in \Nil_m$,
\item $M \in \U_n$ if and only if for any $s$ $f(R_s(M)) \in \calF_n$.
\end{itemize}
\end{prop}







\begin{prop}
The indecomposable elements of an augmented unstable algebra are in $\Nil_1$. 
\end{prop}

This is a suspension if $p=2$.


Following Kuhn,    for an unstable module $M$, one defines a function $w_M \colon \N \ra \Z \cup \infty$ by:

$$
w_M(i) = \deg f(R_i(M)) 
$$

The following lemma is a consequence of \ref{krull-t} and \ref{nilp}:
\begin{lem}

$
M \in \Nil_s \; \Rightarrow \; T(M) \in \Nil_s 
$ 

Let $M$ be such that   $w_M(i) \leq i$, $w_{\mathbb{T}(M)}$ the tensor algebra on $M$ . Then  the function $w_{\mathbb{T}(M)}$ has the same property. 
\end{lem}

Below are two statements that imply  \ref{NGS}



\saut
























Let $X$ be  a space, define  $w_X=w_{H^*X}$ and $q_X=w_{QH^*X}$.



\begin{thm} [Gaudens, Nguyen T. Cuong, Schwartz]\label{GNS} Let $X$ be such that $\tilde H^*X \in \Nil_1$.
The function $q_X$ either is equal to $0$ or $q_X-Id$ takes at least one positive (non zero) value.
\end{thm}


\begin{thm} [Gaudens, Schwartz]\label{GS}  Let $X$ be such that $\tilde H^*X \in \Nil_1$.
The function $w_X$ either is equal to $0$ or $w_X-Id$ takes arbitrary large values.
\end{thm}

One would like to reformulate \ref{GNS} like \ref{GS} but there are some (presumably small) difficulties.



\end{section}
\begin{section}{Lannes' theorem and Kuhn's reduction, proof of \ref{GNS} and \ref{GS}}\label{KT}



  $X$  $p$-complete, $1$-connected,  assume that $TH^*X$ is finite dimensional in each degree.   
 The following theorem of Lannes is the major geometrical application of \ref{Lannes1}. The evaluation map:
  $$
  B\Z/p \times \map (B\Z/p, X) \ra X
  $$
induces a map in cohomology:

$$
H^*X \ra H^* B\Z/p \otimes H^* map(B\Z/p,X)
$$
and by adjunction
$$
TH^*X \ra H^* \map (B\Z/p,X)
$$
\begin{thm} [J. Lannes]\label{Lannes2}
Under the hypothesis mentionned above the natural map
$
T H^*X \to H^* \map (B\Z/p,X)
$
is an isomorphism of unstable algebras.
\end{thm}

Following Fran\c{c}ois Xavier Dehon and Gaudens  these conditions could be relaxed using Morel's machinery
  of pro-$p$ spaces \cite{Mo}. 




This theorem extends replacing $B\Z/p$ by $BV$, $V$ an elementary abelian $p$-group. It is linked to the Sullivan conjecture and has lot of applications, in particular in the theory of $p$-compact groups (William Dwyer and Clarence Wilkerson  \cite{DW90}) and the one of $p$-local groups or Bob Oliver  \cite{BLO03}. In particular it leads to a complete solution
of Steenrod's problem. The question being to determine which polynomial algebras are the cohomology of a space. One famous example is the algebra of modular invariants. Let $V$ be an elementary abelian $2$-group and $d=dim(V)$. They are the elements invariant under the action of $\GL(V)$
  in $$S^*(V^*) \cong H^*BV \cong \F2[x_1,\ldots , x_d]$$
 It is known (Dickson) to be polynomial in generators of degree
$2^d-2^{d-1},\ldots,2^d-1$. This is the cohomology of a space if and only if $d \leq 4$, the case $d=4$ is "exotic" \cite{DW93}, all this would deserve other reports and one will not proceed further.

\saut



Kuhn considers the homotopy cofiber $\Delta (X)$, of the natural map $X \ra \map (B\Z/p, X)$. reduction is to consider the cofiber  $\Delta (X)$  of $ X \rightarrow  {\rm \map}(B\Z/p,X)$. Then \ref{Lannes2} immediately yields:
  

 \begin{prop}
 $$H^* (\Delta(X) )\cong \bar T H^*X$$

$$w_{\Delta(X)}=w_X-1$$
\end{prop}
 
 

 
 As a consequence, if $H^*X \in \U_n \setminus \U_{n-1}$, then $H^*\Delta(X)  \in \U_{n-1} \setminus \U_{n-2}$.






\saut

Given an augmented unstable algebra $K$ he indecomposable functor $Q$ does commute with $T$:
$$
T(Q(K)) \cong Q(TK)
$$ 
but this is not true with $\bar T$. 
However if  $Z$ be an  $H$-space, then \cite{CCS07}



 \begin{prop}\label{indec}{\it $Q H^* \map _*
(B\Z/p^{\wedge n}, Z )= \bar{T}^n QH^* Z$.}
\end{prop}

This follows from the homotopy equivalence:
$$
\map(B\Z/p,Z) \cong Z \times \map_*(B\Z/p,Z)
$$


On the way one note the beautiful result of these authors








\begin{thm}[N. Castellana, C. Crespo, J. Scherer]
\label{V} {\it Let  $X$ be an $H$-space such that      $QH^* X \in\U_n$, then $QH^*\Omega X \in \U_{n-1}$}
\end{thm}







In order to prove \ref{GNS} or \ref{GS} one shows a space cannot be such that $\tilde H^*X \in \Nil_1$ and such that $q_X$ is not $0$ and  less or equal to $Id$.
Kuhn's reduction allows us to suppose that the reduced  mod-$p$ cohomology is exactly $s$-nilpotent,  $s>0$ and that $R_s(H^*X) \in \U_1 \setminus \U_0$.  
\saut
 Let $Z$ be $\Omega \Sigma X$, then  
 $H^*Z \cong \mathbb{T}(\tilde H^* X)$, recall that $\mathbb T()$ denotes the tensor algebra.

\saut


The first part of the proof  consists of  the following chain of implications:

\begin{itemize}
\item
$q_X \leq Id \Rightarrow w_X \leq Id  $, in fact this holds for any unstable algebra $K$ so that $K^{>0}$ is nilpotent;

\item
\saut

 Kuhn's reduction plus the collapse of a low dimensional skeleton allows  to suppose that the reduced  mod-$p$ cohomology is exactly $s$-nilpotent,  $s>0$ and that $R_s(H^*X) \in \U_1 \setminus \U_0$, and that
 $w_X(s+i) \leq i+1$,
\saut
 Let $Z$ be $\Omega \Sigma X$, then  
 $H^*Z \cong \mathbb{T}(\tilde H^* X)$, the condition above implies for algebraic reasons that $\bar T ^n H^*Z$  is $(n+s-2)$-connected, it follows from \ref{indec} that

\saut

\item $map_*(B^{\wedge n},Z) \;\;(n+s-2)$-connected.

\saut\end{itemize}








\begin{prop}\label{connec}    $ \tilde H^* Z \in {\mathcal N}il_s$,
 $\bar T^{n} (H^*Z)$ is $(n+s-2)$-connected, thus  $\map_*(B^{\wedge n},Z) \;\;(n+s-2)$-connected.
\end{prop}





 Then,  one gets a non trivial  algebraic map of unstable algebras


$$
\varphi^*_s: H^* Z \to \Sigma ^s R_s H^* Z \to \Sigma ^s F(1)  \subset \Sigma^s  \tilde H^* B\Z/p~.
$$

\saut



It cannot factor through $H^*\Sigma^{s-1}K(\Z/p, 2)$, because there are no non trivial map  from an  $s$-suspension (and thus from an unstable module in $\Nil_s$) to an  $(s-1)$-suspension of a reduced module, and 

\begin{prop}
$H^*K(\Z/p,2)$ is reduced.
\end{prop}

\saut


Then comes the second part of the proof, the contradiction comes from the fact that using obstruction theory one can construct a factorisation.





The existence of a map realising $\varphi_s^*$ is a consequence (using Lannes' theorem) and of  the Hurewicz theorem because  ${\rm \map}_*(B\Z/p,Z)$ is $(s-1)$-connected.
 

\saut




$K(\Z/p,2)$ is built up, starting with $\Sigma B\Z/p$, as follows (Milnor's construction). There is a filtration   $* =C_0 \subset C_1= \Sigma B\Z/p \subset C_2 \subset \ldots \subset \cup_n C_n=K(\Z/p,2)$,  a diagram 

\begin{equation*}
\begin{CD}
\cdots@>{}>>  B^{*n+1} @>{}>>B^{*n+2}@>{}>>\cdots\\
@VVV @VVV @VVV @ VVV \\
\cdots@>{}>>  C_{n} @>{}>>C_{n+1} @> >> \cdots\\
\end{CD}
\end{equation*}










and  cofibrations, up to homotopy
$$
\Sigma^{n-1}B^{\wedge n}  \rightarrow  C_{n-1} \rightarrow C_{n} 
$$



$$
\Sigma^{n-2+s}B^{\wedge n}  \rightarrow  \Sigma^{s-1}C_{n-1} \rightarrow \Sigma^{s-1}C_{n} 
$$


The obstructions to extend  $\varphi_s \colon \Sigma^s B\Z/p \rightarrow Z$  to 
$\Sigma^{s-1} K(\Z/p,2)$ are in the groups 
$$[\Sigma ^{n+s-2}  (B\Z/p)^{\wedge n}, Z]= \pi_{n+s-2}
\map_* ( B\Z/p^{\wedge n},  Z)$$ but  they are trivial. It follows one can do the extension, this is a contradiction. 
\saut
To prove \ref{GS} it is now enough to use again  $w_{\Delta(X)}=w_X-1$.



\end{section}





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\begin{thebibliography}{}



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\end{document}


