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\def\dem{{\noindent\it Proof:\ \ }}
\def\lente{{$L^{2n+1}(2^e)$} }
\def\cepen{{$\cee P^n$} }
\def\axial{ {S^{2n+1}\times_e P^{2m-1}} }

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\newtheorem{teorema}{Theorem}[section]
\newtheorem{proposicion}[teorema]{Proposition}
\newtheorem{lema}[teorema]{Lemma}
\newtheorem{definicion}[teorema]{Definition}
\newtheorem{corolario}[teorema]{Corollary}
\newtheorem{notacion}[teorema]{Notation}    
\newtheorem{nota}[teorema]{Remark}
\newtheorem{ejemplo}[teorema]{Example}
\newtheorem{problema}[teorema]{Problem} 
\newtheorem{conjetura}[teorema]{Conjecture}

\title{Generalized axial maps and Euclidean immersions of lens spaces}
\author{\it Luis Astey, Donald M.~Davis and Jes\'us 
Gonz\'alez\footnote{The second author is supported by a grant from the Reidler 
Foundation. The third author is supported by CONACyT Grant No. 37296-E.}}
\date{\today}

\begin{document}
\maketitle

\begin{abstract}
Generalized axial maps are introduced and related to
the immersion problem for 2-torsion lens spaces to produce a characterization of this problem
in the metastable range. A direct analysis gives a complete solution of the immersion problem 
for the missing cases.
Other variations of the problem, which extend classical
constructions for projective spaces, are discussed.
\end{abstract}

{\smallskip
\small
\noindent
{\it Key words and phrases:} Axial maps, immersions of lens spaces, obstruction theory, modified Postnikov towers.

\noindent {\it 2000 Mathematics Subject Classification:} Primary 57R42. Secondary 55S45.}

\section{Introduction}
\label{sec:1}
The results in this paper are motivated by the
well known relationship
between the existence of Euclidean immersions of real projective spaces 
and the existence of axial maps of type $(n,k)$, that is, maps of the form
\begin{equation}
\label{axial}
P^n\times P^n\rightarrow P^{n+k}
\end{equation}
which are homotopically nontrivial over each axis.
B.~J.~Sanderson observed in~\cite{San} that an axial map as above 
can be obtained from a given immersion $P^n\subseteq \erre^{n+k}$
and that, at least in the metastable range (that is, when $n<2k$), such an immersion
can homotopically be recovered from~(\ref{axial}).
Sanderson's work relies on results of Hirsch and Haefliger~(\cite{HH,metaJames})
and takes advantage of the so-called {\it twisted normal bundle}
associated to an immersion $P^n\subseteq \erre^{n+k}$.
The success of the technique depends on the fact that
the canonical real line bundle over $P^n$ has multiplicative order 2.
However, this is precisely the main drawback in a first attempt to generalize the ideas for higher
2-torsion lens spaces. Indeed, on the one hand, Hirsch's basic result
on immersing manifolds~\cite{MR22:9980} implies that the codimension in an optimal immersion for
$L^{2n+1}(2^e)$ ---the ($2n+1$)-dimensional $2^e$-torsion lens space--- 
agrees with the geometric dimension of $-(n+1)\xi_{n,e}$,
where $\xi_{n,e}$ is the realification of the canonical 
complex line bundle over $L^{2n+1}(2^e)$;
but on the other hand, $\xi_{n,e}$ is not even a unit in KO$(L^{2n+1}(2^e))$.
We straighten out the situation by following a path,
first suggested in~\cite{AGJ} by Adem, Gitler and James, which 
naturally leads to the concept of
generalized $e$-axial maps (to be formalized in Definition~\ref{eaxial}).
Our main result is as follows.

\begin{teorema}\label{main}
If $L^{2n+1}(2^e)$ immerses in Euclidean codimension $k$, then 
there is an $e$-axial map
of the form $a\colon S^{2n+1}\times_e P^{2n+1}\rightarrow P^{2n+k+1}$. The converse holds
except perhaps for $n=2$, $3$ or $5$.
\end{teorema}

Theorem~\ref{main} is valid for $e=\infty$ 
(as long as we interpret $L^{2n+1}(2^\infty)$ as $\cee P^n$) and this can help to understand the close relationship
among Euclidean immersions of complex projective spaces and those for high 2-torsion lens 
spaces~(\cite{nonimm,shimkus}). 

\smallskip
In Section~\ref{sec:2} we adapt the techniques 
in~\cite{AGJ} to show that an $e$-axial 
map as in Theorem~\ref{main} must exist
whenever $L^{2n+1}(2^e)$ 
admits a codimension-$k$ immersion. We also show that the converse holds, at least, in the metastable range.
This last restriction misses only the cases with $n<8$; they are studied in
Section~\ref{sec:3}, where the validity of Theorem~\ref{main} is verified for 
$n\notin\{2,3,5\}$\footnote{We do not know whether $n=2$, $3$ or $5$ 
are actual exceptional cases for Theorem~\ref{main}.}. 
In any case, we settle the immersion problem for $L^{2n+1}(2^e)$ when $n< 9$.
The situation is summarized in~(\ref{tabla1}), where we tabulate the optimal  
Euclidean dimensions in which $L^{2n+1}(2^e)$ can be immersed ---the last (well known) row, with $e=\infty$, 
is included for comparison purposes. 
{\small
\begin{equation}\label{tabla1}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}\hline
 & $n=0$ & $n=1$ & $n=2$ & $n=3$ & $n=4$ & $n=5$ & $n=6$ & $n=7$ & $n=8$ \\\hline
 $e=1$ & $2$ & $4$ & $7$ & $8$ & $15$ & $16$ & $22$ & $22$ & $31$ \\\hline
 $e=2$ & $2$ & $4$ & $8$ & $9$ & $16$ & $17$ & $23$ & $23$ & $32$ \\\hline
 $2<e<\infty$ & $2$ & $4$ & $8$ & $10$ & $16$ & $18$ & $23$ & $23$ & $32$ \\\hline
 $e=\infty$ & $\star$ & $3$ & $7$ & $9$ & $15$& $17$ & $22$ & $22$ & $31$ \\\hline
\end{tabular}
\end{equation}}

The cases $n=2,4,8$ are part of a general situation: with $n=2^m$ ($m\geq 1$),
it is well known that both $P^{2n+1}$ and $\cee P^n$ admit optimal immersions in $\erre^{4n-1}$. In particular,
any lens space $L^{2n+1}(2^e)$ immerses in $\erre^{4n}$, which is optimal for $e\geq 2$ (see Section~\ref{sec:3}).
%Likewise, the techniques of obstruction theory used in Section~\ref{sec:3} for the case $n=5$ in~(\ref{tabla1})
%could yield the optimal immersion for $n=2^m+1$ too.

%\smallskip
%\centerline{\bf The following parragraph might be removed}
%\hrule\smallskip 
%In Section~\ref{postnikov} we show that Theorem~\ref{main} does fail for $n=2$.
%Indeed, unlike the classical case of real projective spaces 
%($e=1$), we construct a 2-axial map $S^5\times_2 P^5\rightarrow P^7$ which, according 
%to~(\ref{tabla1}) above, has no corresponding immersion $L^5(4)\subset\erre^7$.
%The required axial map is obtained as a lifting through the first few stages 
%of the Postnikow tower for $P^7$.
%\hrule\smallskip

\smallskip
In Section~\ref{sec:4} we give a more familiar characterization of a 
generalized axial map and relate the concept to the existence of $2^e$-equivariant maps
from an odd dimensional sphere into a Stiefel manifold, and to the sectioning problem of bundles 
over lens spaces (all of these being classical constructions for $e=1$).
In view of Lemma~\ref{luissuggested} and Remarks~\ref{inducir} and~\ref{converse}, 
it would be interesting to know to what extent
a bilinear map technique can be used to improve a given immersion for $P^{2n+1}$
to one for $L^{2n+1}(2^e)$. 

\section{Generalized axial maps}
\label{sec:2}
For a real vector bundle $\alpha$ over a space $X$, let $S(\alpha)$ and $P(\alpha)$ stand, 
respectively, 
for the sphere and projectivized bundles associated to $\alpha$, 
and let $h_\alpha$ denote the canonical line bundle
over $P(\alpha)$ splitting off $\pi^*(\alpha)$, where $\pi\colon P(\alpha)\rightarrow X$
is the projection.

\begin{definicion}\label{basica}
Let $P^k$ ($k\leq \infty$) be the $k$-dimensional real projective space.
We say that a map $a\colon P(\alpha)\rightarrow P^N$ is axial when the composite 
$P(\alpha)\stackrel{a}{\rightarrow} P^N\hookrightarrow P^\infty$
classifies $h_\alpha$.
\end{definicion}

We are interested in the following particular situation.
Let \cepen be the $n$-dimensional complex projective space and, for $e\geq 1$, let
\lente be the ($2n+1$)-dimensional $2^e$-torsion lens space, that is, the quotient 
of $S^{2n+1}$ by the restriction to $\zet/2^e$ of the diagonal action of $S^1$ in $\cee^{n+1}$.
Let $\eta_n$ (also denoted by $\eta_{n,\infty}$) 
be the canonical complex line bundle over $\cee P^n$ and let $\eta_{n,e}$ be its pull back 
under the projection $L^{2n+1}(2^e)\rightarrow \cee P^n$. We set $\xi_{n,e}=r(\eta_{n,e})$, 
the realification of $\eta_{n,e}$.

\begin{definicion}\label{eaxial}
An $e$-axial map is any
axial map associated to a multiple of $\xi_{n,e}$ 
($e\leq\infty$). 
Thus, $a:P(m\xi_{n,e})\rightarrow P^N$
is $e$-axial provided the composite $P(m\xi_{n,e})\stackrel{a}{\rightarrow} 
P^N\hookrightarrow P^\infty$
classifies $h_{m\xi_{n,e}}$.
\end{definicion}

The following considerations provide us with a more manageable description for $P(m\xi_{n,e})$.
It is well known and simple to see that
the total space of the iterated $m$-fold Whitney sum of $\eta_{n,e}$ is the quotient
of $S^{2n+1}\times\cee^m$ by the equivalence relation which identifies $(wx,y)$ and $(x,wy)$,
for $(x,y)\in S^{2n+1}\times\cee^m$ and $w\in \zet/2^e$. We denote the resulting space 
and the corresponding associated sphere bundle by $S^{2n+1}\times_e\cee^{m}$ and 
$S^{2n+1}\times_e S^{2m-1}$, respectively.  
In these terms we see that the projectivization 
$P(m\xi_{n,e})$ is the quotient of
$S^{2n+1}\times P^{2m-1}$ by the free action of $\zet/2^e$ given by $w\cdot(x,y)=(wx,w^{-1}y)$.
We also use the more suggestive notation $S^{2n+1}\times_e P^{2m-1}$ for this space.
It is convenient to extend these observations to the case $e=\infty$ by setting
$\zet/2^\infty=S^1$ (we agree to interpret $L^{2n+1}(2^\infty)$ as $\cee P^n$).

\begin{nota}\label{accionlibre}{\em
In the situation above, the (nonfree) action of $\zet/2^e$ on $P^{2m-1}$ is induced through the 
quotient 
$S^{2m-1}\rightarrow P^{2m-1}$; in any case, this yields a {\it free} action of $\zet/2^{e-1}$ 
on $P^{2m-1}$.
For instance, when $e=\infty$, the free action is given by $w\cdot x=w^{1/2}y$, where $y\in 
S^{2m-1}$ is any 
representative for $x\in P^{2m-1}$, and $w^{1/2}$ is any square root of $w\in S^1$.
On the same lines, we note that
$S^{2n+1}\times_e P^{2m-1}$ has a more symmetric expression as the quotient of $P^{2n+1}\times 
P^{2m-1}$
by the action of $\zet/2^{e-1}$ given by $w(x,y)=(wx,w^{-1}y)$. Here $\zet/2^{e-1}$ is acting 
freely on each
projective space as above. 
For $e=1$ this will help us to relate the concepts of $1$-axial maps
and (regular) axial maps
(see Remark~\ref{converse}).
}\end{nota}

\begin{nota}\label{pullback}{\em
Back to the general situation in Definition~\ref{basica},
for any map $f\colon Y\rightarrow X$, 
the naturality of the construction for $h_\alpha$ shows that
the composite $P(f^*(\alpha))\rightarrow P(\alpha)\stackrel{a}{\longrightarrow} P^N$
(where the first map is induced by $f$) is axial provided $a$ is. 
}\end{nota}

More generally, for a sub-vector bundle $\beta$ of 
$f^*\alpha$, the restriction to $P(\beta)$ of the above composite yields 
an axial map $P(\beta)\rightarrow P^N$. This gives an easy but important way of constructing 
axial maps:
Assume $\alpha$ has a complement $\gamma$ so that $\alpha\oplus\gamma$ is a trivial 
$\ell$-dimensional (say) 
bundle over $X$. Then the composite $P(\alpha)\rightarrow P(\alpha\oplus\gamma)=X\times P^{\ell-1}
\stackrel{\pi_2}{\longrightarrow} P^{\ell-1}$
yields an axial map, where $\pi_2$ stands for projection onto the second factor (note that, 
as in the context above, starting with the trivial $\ell$-dimensional vector bundle over a point,
we can interpret $\pi_2$ as the map induced by $X\rightarrow \star$).

\smallskip
The relevance of the construction comes from
Hirsch's Theorem on immersions of manifolds which claims that an $m$-dimensional manifold $M$
admits an immersion in $\erre^{N}$ precisely when $\tau_M$ ---the tangent bundle of $M$--- admits
an ($N-m$)-dimensional complement $\nu$. In that case, the observations above yield axial maps
$P(\tau_M)\rightarrow P^{N-1}$ and $P(\nu)\rightarrow P^{N-1}$. The paper
\cite{AGJ} studies the converse 
of these constructions for $M$ a real projective space; we do a similar analysis for lens spaces. 
%In either of these cases, it is 
%convenient
%to have a concrete description of (the stable class of) $\tau_M$.

\begin{nota}\label{half}{\em
One half of Theorem~\ref{main} is now self evident; namely, 
each Euclidean immersion of $L^{2n+1}(2^e)$ in codimension $k$ gives rise 
to an axial map $S^{2n+1}\times_e P^{2n+1}\rightarrow P^{2n+k+1}$. Indeed, we only need to recall
the bundle isomorphisms
\begin{equation}\label{iso}
\tau_{L^{2n+1}(2^e)}\oplus 1=(n+1)\xi_{n,e}, \;\;\rm{for \ } e<\infty, \rm{\  and}\;\;
\tau_{\ceei P^n}\oplus 1_{\ceei}=(n+1)\eta_{n,\infty}.
\end{equation}
}\end{nota}

\begin{nota}\label{inducir}{\em
In view of the final considerations in Remark~\ref{accionlibre}, 
there are 2-fold covering projections
\begin{equation}\label{comparar}
P^{2n+1} \times P^{2m-1} \rightarrow S^{2n+1}\times_2 P^{2m-1} \rightarrow \cdots \rightarrow 
S^{2n+1}\times_e P^{2m-1} 
\rightarrow S^{2n+1}\times_{e+1} P^{2m-1}.
\end{equation}
Moreover, from Remark~\ref{pullback}, we see that the composition 
of the last map in~(\ref{comparar}) with an ($e+1$)-axial map 
yields a corresponding $e$-axial map.
Similarly, the composition of the obvious projection 
$S^{2n+1}\times_e P^{2m-1}\rightarrow S^{2n+1}\times_\infty P^{2m-1}$ with 
an $\infty$-axial map yields an $e$-axial map.
In Section~\ref{sec:4} we
get a partial converse for these observations~(Remark~\ref{converse}).
In view of Theorem~\ref{main} and Remark~\ref{half}, the considerations above should
be compared with the fact that an immersion of $\cee P^n$ in $\erre^N$
yields an immersion of $L^{2n+1}(2^{e+1})$ in $\erre^{N+1}$; and the latter, in turn, 
induces a corresponding immersion for $L^{2n+1}(2^{e})$ 
(cf.~\cite{nonimm}). 
}\end{nota}

Corollary~\ref{casi} below gives, in a certain range, the 
part of Theorem~\ref{main} not included in Remark~\ref{half}. The proof requires the next 
auxiliary result whose verification, although elementary, is deferred until Section~\ref{sec:4}
(Proposition~\ref{caracterizaciones}).
In the rest of the section we take $1\leq e\leq \infty$.

\begin{lema}\label{luissuggested}
There is an $e$-axial map $a\colon S^{2n+1}\times_e P^{2m-1}\rightarrow P^{N}$ 
if and only if there is an $e$-skew map $A\colon S^{2n+1}\times S^{2m-1}\rightarrow S^{N}$, that is, 
a map satisfying $A(x,-y)=-A(x,y)$ and $A(w\cdot x,y)=A(x,w\cdot y)$
for all $(x,y)\in S^{2n+1}\times S^{2m-1}$ and $w\in\zet/2^e$.
\end{lema}

\begin{corolario}\label{linealizar}
Assume $N-2m\geq n$. If there is an $e$-axial map $S^{2n+1}\times_e P^{2m-1}\rightarrow P^{N}$,
then there exists an {\rm ($N-2m+1$)}-dimensional vector bundle
$\nu$ over $L^{2n+1}(2^e)$ such that $m\xi_{n,e}\oplus\nu$ is trivial.
\end{corolario}

\dem Let $\sigma\colon S^{2n+1}\times_e \cee^{m}\rightarrow L^{2n+1}(2^e)\times\erre^{N+1}$ be 
obtained as the 
radial extension of a map $S^{2n+1}\times_e S^{2m-1}\rightarrow L^{2n+1}(2^e)\times S^{N}$ whose first 
component is the bundle projection 
$S(m\xi_{n,e})\rightarrow L^{2n+1}(2^e)$ and whose second 
component is induced by an $e$-skew map as in Lemma~\ref{luissuggested}. Then $\sigma$ is a skew bundle map 
(in the usual sense) which, in view 
of~\cite[Theorem~1.2]{HH}, can be deformed into a monomorphism of vector bundles. The result follows.
\hfill\cajita

\begin{corolario}\label{casi}
If there is an $e$-axial map $S^{2n+1}\times_e P^{2n+1}\rightarrow P^{2n+k+1}$ when
$n<k$, then $L^{2n+1}(2^e)$ admits a Euclidean immersion in codimension $k$.
\end{corolario}

\dem Take $N=2n+k+1$ and $m=n+1$, then
Corollary~\ref{linealizar} shows that $(n+1)\xi_{n,e}$ admits a $k$-dimensional complement over 
$L^{2n+1}(2^e)$ which, in view of~(\ref{iso}) and Hirsch's basic immersion theorem, corresponds to
the normal bundle of a codimension-$k$ Euclidean immersion of $L^{2n+1}(2^e)$.
\hfill\cajita

\section{Low dimensional immersion problem}
\label{sec:3}

Remark~\ref{inducir} and the following result, which is a particular case 
of~\cite[Lemma~2.1]{AGJ}, implies that the only cases 
of Theorem~\ref{main} not considered already in Section~\ref{sec:2} are those with $n\leq 7$. 

\begin{proposicion}\label{3.1}
Suppose there is a $1$-axial map $P^{2n+1}\times P^{2n+1}
\rightarrow P^{2n+k+1}$, where $n\geq k$.
Then $n\leq 7$.
\end{proposicion}

We analyze each of these low dimensional cases. The following portion of table~(\ref{tabla1}) 
consisting of known optimal 
immersions~(\cite{crabb, table, MR56:3839}) will be helpful. 
\begin{equation}\label{tabla2}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}\hline
 & $n=0$ & $n=1$ & $n=2$ & $n=3$ & $n=4$ & $n=5$ & $n=6$ & $n=7$ & $n=8$ \\\hline
 $\erre P^{2n+1}$ & $2$ & $4$ & $7$ & $8$ & $15$ & $16$ & $22$ & $22$ & $31$ \\\hline
 $\cee P^{n}$ &  & $3$ & $7$ & $9$ & $15$ & $17$ & $22$ & $22$ & $31$ \\\hline
\end{tabular}
\end{equation}
In particular, as observed in Remark~\ref{inducir}, this yields immersions of lens spaces
$L^{2n+1}(2^e)$.
Part of the work below is to improve such an immersion (whenever it is possible) and 
analyze the optimality of the corresponding $e$-axial map.
We only need to consider the case $e>1$ in view of~\cite{AGJ}.

\medskip
\noindent{\it Case $n=2^m$: } For $e\geq 2$, the immersion 
$L^{2n+1}(2^e)\subseteq\erre^{4n}$ in the introduction is optimal in view of~\cite[Corollary 1.1]{junod}.
If moreover $m\geq 2$, the nonimmersion $L^{2n+1}(2^e)\not\subseteq\erre^{4n-1}$
lies within the range of Corollary~\ref{casi}, verifying in this case the second part of Theorem~\ref{main}.

\medskip
\noindent{\it Cases $n=6$ and $n=7$: } 
Table~(\ref{tabla2}) yields immersions $L^{2n+1}(2^e)\subseteq \erre^{23}$. We show the corresponding
$e$-axial maps are optimal. In view of the considerations on Remark~\ref{inducir}, 
it suffices to show the non-existence of a $2$-axial map
$S^{2n+1}\times_2 P^{2n+1}\rightarrow P^{22}$. Moreover, we only need to consider the case $n=6$
(take $f$ to be the inclusion $L^{13}(4)\hookrightarrow L^{15}(4)$, $\alpha=8\xi_{7,2}$ and
$\beta=7\xi_{6,2}$ in the considerations immediate to Remark~\ref{pullback}) which, again, lies
within the range of Corollary~\ref{casi}. Then the second part of Theorem~\ref{main}, as well as the 
immersion problem in this case, follows from the nonimmersion $L^{13}(4)\not\subseteq \erre^{22}$ 
in the recent Ph.~D.~Thesis~\cite{shimkus}.

\smallskip
This completes the proof of Theorem~\ref{main} 
as well as the details in columns $n=4,6,7,8$ of~(\ref{tabla1})
(the case $n=1$ follows immediately from table~(\ref{tabla2}) and Remark~\ref{inducir}, 
whereas the case $n=0$ is elementary). Table~(\ref{tabla2}) and the next result
complete the details in columns $n=2,3,5$ of~(\ref{tabla1}).

\begin{proposicion}\label{low}
\begin{itemize}
\item[{\rm(a)}] $L^{11}(4)$ immerses in $\erre^{17}$, while $L^{11}(8)$ does not.
\item[{\rm(b)}] $L^7(4)$ immerses in $\erre^9$, while $L^7(8)$ does not.
\item[{\rm(c)}] $L^5(4)$ does not immerse in $\erre^7$. $L^7(4)$ does not immerse in $\erre^8$. 
$L^{11}(4)$ does not immerse in $\erre^{16}$.
\end{itemize}
\end{proposicion}

With respect to the missing cases in Theorem~\ref{main}, 
except for the nonimmersion in (a), which lies in the range of Proposition~\ref{casi}, the authors do not know
whether there exist axial maps corresponding to the nonimmersions in Proposition~\ref{low}. 

\medskip
The last two nonimmersions in part (c) follow from~\cite[Theorem~1.1]{nonimm}, whereas the proof of the 
first nonimmersion in part (c) is similar to, and simpler than (thus omitted), that of part (b). 

\medskip
\noindent{\it Proof of Proposition~\ref{low} (b): } 
The normal bundle $\nu$ of a possible immersion $L^{7}(8)\subseteq \erre^9$ would be two dimensional 
and orientable and, therefore, of the form $\nu=r\eta^k$ ($0\leq k\leq 7$), 
where $\eta=\eta_{3,3}$ and $r$ stands for realification. 
In these conditions Hirsch's theorem and~(\ref{iso}) would yield 
an equation of the form
\begin{equation}\label{eq1}
10=r(\eta^k)\oplus 4r\eta.
\end{equation}
Since $r\eta=r(\eta^{-1})$, we only need to show~(\ref{eq1}) is impossible for $0\leq k\leq 4$. 
Recall~(\cite{jap8})
\begin{equation}\label{des1}
\widetilde{KO}(L^{7}(8))=\zet/16\oplus\zet/2
\end{equation}
where the first summand is generated by $r\sigma$ and the second by $\kappa$ ($\sigma=\eta-1_\ceei$ 
and $\kappa=\rho-1_\errei$, with $\rho$ the canonical real line bundle over $L^{7}(8)$).
Note $k$ must be positive 
in~(\ref{eq1}), otherwise we would have $8=4r\eta$ so that $0=4r\sigma$, in contradiction 
to~(\ref{des1}). Case $k=1$ is also inconsistent with~(\ref{eq1}) as it yields $0=5r\sigma$. 
To dispose of
case $k=4$ we recall $\eta^4=c\rho$~(\cite[Proposition.3]{jap4}), the complexification of $\rho$; 
thus, with 
$k=4$,~(\ref{eq1}) yields $10=rc\rho\oplus 4r\eta=2\rho\oplus 4r\eta$ and therefore 
$0=2\kappa+4r\sigma$, 
in contradiction to~(\ref{des1}). Cases $k=2$ and $k=3$ are easier to consider after complexification 
(which, according to~\cite[Corollary~4.4]{jap8}, is monic under the present conditions). 
For instance, with $k=2$, the
complexification of~(\ref{eq1}) produces
\begin{eqnarray*}
0 & = & (\eta^2-1 + \eta^6-1) + 4(\eta-1 + \eta^7-1)\\
 & = & (\sigma+1)^2-1 + (\sigma+1)^6-1 +4(\sigma + (\sigma+1)^7-1)\\
 & = & 160\sigma^3 +100\sigma^2+40\sigma 
\end{eqnarray*}
(recall the well known relation 
$\sigma^4=0$) which is incompatible with the fact~(\cite[Proposition~3.7]{jap8}) that
$\widetilde{K}(L^7(8)=\zet/32\oplus\zet/4\oplus\zet/4$ with 
respective generators $\sigma$, $\sigma^2-2\sigma$ and 
$\sigma^3+2\sigma^2$. In an entirely similar way we rule out $k=3$ proving, therefore,  the nonimmersion
$L^7(8)\not\subseteq\erre^9$.

As for the second claim, according to~\cite{jap4}, $\widetilde{KO}(L^7(4))=\zet/8\oplus\zet/2$, 
with respective generators
$r\sigma$ and $\kappa+2r\sigma$; moreover there hold the relations $0=(r\sigma)^2=-4r\sigma+2\kappa$, 
$\kappa^2=-2\kappa$ and $\kappa^3=0$. In particular, $0=2\kappa^2$ and $2\kappa=-2\kappa$. Thus, 
using~(\ref{iso}), the 
stable normal bundle of $L^7(4)$ is $-4r\sigma=2\kappa$ which has geometric dimension at most 2. 
The required
immersion follows from Hirsch's Theorem.\hfill\cajita

\medskip
The remainder of the section is devoted to the proof of Proposition~\ref{low}~(a) which is, 
by far, the hardest part in Proposition~\ref{low}.
In view of the considerations in Remark~\ref{inducir}, what we want is the critical 2-torsion
required for the nonimmersions $L^{11}(2^e)\not\subseteq\erre^{17}$ (proved 
in~\cite{Junod} to hold for $e\geq 7$).
Consider the maps
\begin{equation}\label{egeneral}
L^{11}(4)\mapright{p_1}L^{11}(8)\mapright{p_2}
\cee P^5\mapright{f}\bspin,
\end{equation}
where $p_i$ is the canonical projection map, while $f$ classifies
the stable normal bundle $-6\eta$ of $\cee P^5$, with $\eta=\eta_{5,\infty}$.
By Hirsch's theorem, it suffices to prove:
\begin{equation}\label{lift} 
fp_2p_1 \mbox{ \ factors through \ } \bspin(6)\to\bspin,
\end{equation} 
while  
\begin{equation}\label{DNL}
fp_2  \mbox{ \ does not. \ }
\end{equation}
%One bit of input that we will use is the following result of Junod,
%which we interpret as saying that, for $e\ge 7$, $fp_3$ does not factor through
%$\bspin(6)$.  (A reader who refers to \cite{Junod} should be aware that Junod's $L^n$
%is our $L^{2n+1}$).
%
%\begin{proposicion}\label{jun} {\em\cite[last lines]{Junod}} If $e\ge 7$, then
%$L^{11}(2^e)$ does not
%immerse in $\br^{17}$.
%\end{proposicion}

The method of proof is the obstruction-theoretic method of modified
Postnikov towers (MPTs). We will use the following MPT, through dimension 11, where the left hand 
side of the bottom row is as in~(\ref{egeneral}).

\begin{picture}(0,98)(115,0)
\put(107,0){\makebox(115,15)[br]{$L(4)\mapright{p_1}L(8)\mapright{p_2}
\cee P^5\mapright{f}$}}
\put(223,2){$\bspin$}
\put(240,2){$-\!\!\!-\!\!\!\longrightarrow K(\bz,7)\times K_8$.}
\put(230,25){\vector(0,-1){15}}
\put(233,18){$\scriptstyle q_1$}
\put(227,30){$E_1 \stackrel{k_8^1,k_{10}^1}{-\!\!\!-\!\!\!\longrightarrow}  K_8\times K_{10}$}
\put(204,43){\vector(2,-1){20}}
\put(142,41){\makebox(60,15)[br]{$K(\bz,6)\times K_7$}}
\put(230,55){\vector(0,-1){15}}
\put(233,48){$\scriptstyle q_2$}
\put(227,60){$E_2 \stackrel{k_{10}^2}{-\!\!\!-\!\!\!\longrightarrow}  K_{10}$}
\put(204,73){\vector(2,-1){20}}
\put(106,71){\makebox(96,15)[br]{$K_7\times K_9$}}
\put(230,85){\vector(0,-1){15}}
\put(221,90){$\bspin(6)$}
\end{picture}

\vspace{-3.9cm}
\begin{equation}\label{MPT}
\end{equation}

\vspace{3cm}
Here and in what follows we abbreviate $K(\bz/2,m)$ by $K_m$ and $L^{11}(2^e)$ by $L(2^e)$ ($2\leq e\leq 3$).
In this section all cohomology groups have coefficients in $\bz/2$.

MPTs were introduced in \cite{Mah64} and \cite{GM}.
Each vertical map in the above diagram is
part of a fiber sequence preceded by the map from the fiber represented by a
diagonal arrow, and followed by the classifying map represented by a horizontal
arrow. The information of the diagram can be obtained from the Adams
spectral sequence (ASS) of the
stunted real projective space $P_6$---the quotient of
$P^\infty$ by pinching $P^5$ to a point---which is, through dimension 12,
the fiber of $\bspin(6)\to \bspin$. This ASS can be found in Table 8.7 on
page 59 of \cite{MahMem}. We re-create it in the next chart.

\vspace{.8cm}
\begin{equation}\label{htpycht}
\end{equation}

\vspace{-1.2cm}
\begin{picture}(0,45)(-30,0)
\setlength{\unitlength}{.015in}
\def\elt{\circle*{3}}
\def\mp{\multiput}
\put(10,45){$\pi_*(P_{6}),\ *\le10$}
\put(138,5){$6$}
\put(167,5){$8$}
\put(197,5){$10$}
\put(130,20){\line(1,0){75}}
\put(140,20){\line(1,1){15}}
\mp(140,20)(0,15){4}{\elt}
\put(140,20){\vector(0,1){70}}
\mp(155,20)(0,15){2}{\elt}
\mp(155,20)(30,15){2}{\line(0,1){15}}
\mp(185,35)(0,15){2}{\elt}
\end{picture}
\nopagebreak

The generalized Eilenberg-MacLane spaces 
on the right side of (\ref{MPT}) give rise to the $k$-invariants for the
MPT; these classes have dimensions 1 greater than those of the corresponding
elements of $\pi_*(P_{6})$. We will prove that 
\begin{equation}\label{lift2}
f \mbox{ lifts to a map }
\cee P^5\mapright{\ell_2} E_2.
\end{equation}
Note that the nonimmersion $\cee P^5\not\subseteq\erre^{16}$ (\cite{MR80g:57042})
implies that 
$\ell_2^*(k^2_{10})\ne0$. We will analyze the indeterminacy 
of this lifting and obtain
the following result, which implies (\ref{lift}) and (\ref{DNL}).

\begin{proposicion}\label{indet} There is a lifting of $fp_2p_1$ to $E_2$
which sends $k_{10}^2$
trivially, but every lifting of $fp_2$ to $E_2$ sends $k_{10}^2$
nontrivially.\end{proposicion}

As a first step toward proving (\ref{lift2}), it is useful to factor $f$ as
$$\cee P^5\mapright{h}\ache P^2\mapright{f'}\bspin,$$
where $h$ is the canonical map to the quaternionic projective space,
and $f'$ classifies the stable bundle $-3H$, with $H$ the quaternionic Hopf
bundle. Since $\left({{-3}\atop{2}}\right)=6$ is even, $f'$ lifts to a map $\ell_1:\ache P^2\to 
E_1$. Since $H^{10}(\ache P^2)=0$, $\ell_1h$ sends $k_{10}^1$ to $0\in 
H^{10}(\cee P^5)$.

\smallskip
In order to determine the indeterminacy for lifting $\cee P^5$ and $L(2^e)$ 
in this MPT,
we must know the relations which give rise to the $k$-invariants. These are
computed by the method initiated in \cite{GM} and utilized in many subsequent
papers by the second-named author (e.g. \cite{DM}) and also in papers of K.~Y.~Lam and/or D.~Randall.
It is a matter of building a minimal resolution using Massey-Peterson algebras.
The relations for the MPT in~(\ref{MPT}) are given in the table below,
where $\beta$ is the integral Bockstein whose mod-2 reduction equals Sq$^1$.

\vspace{-.2cm}
\begin{center}
\begin{tabular}{|rl|}
\hline
&\\[-.1cm]
$\b w_{6}$&\\[.15cm]
$w_{8}$&\\ 
&\\[-.1cm]
\hline
&\\[-.1cm]
$k^1_{8}:$ & $\ \sq^2\b $$w_{6}+\sq^1$$w_8$ \\
&\\[-.1cm]
$\ k^1_{10}:$ & $\ (\sq^4+$$w_4)\b $$w_{6}+\sq^3$$w_8\ $ \\
&\\[-.1cm]
\hline
&\\[-.1cm]
$k^2_{10}:$&$\ \sq^2\sq^1$$k^1_{8}+\sq^1$$k^1_{10}$\\
&\\[-.1cm]\hline
\end{tabular}
\end{center}

If $\ell_1^*(k^{1}_{8})\ne0$, we vary the map $\ell_1h:\cee P^5\to E_1$ through the first factor of the 
fiber $K(\bz,6)\times K_7$.
The primary indeterminacy
is computed using the above relations.
The relation for $k^{1}_{10}$ means that the action map
$\mu:(K(\bz,6)\times K_7)\times E_1\to E_1$ sends $k^{1}_{10}$ to
$$1\otimes k^{1}_{10}+\mbox{Sq}^4\iota_{6}\otimes 1+\iota_{6}\otimes w_4
+\sq^3\io_{7}\ot1.$$
With $y$ denoting the generator of $H^2(\cee P^5)$, the new lifting $\ell_1'$
$$%\begin{equation}\label{comp}
\cee P^{5} \stackrel{y^{3}\times \ell_1 h}
{-\!\!\!-\!\!\!-\!\!\!\longrightarrow}
K(\bz,6)\times K_7\times E_1\stackrel{\mu}{\longrightarrow} E_1
$$%\end{equation}
sends $k^{1}_{10}$ to
$$h^*\ell_1^*(k^{1}_{10})+\mbox{Sq}^4(y^{3})+y^{3}\cdot w_4(-6\eta)=0+y^5+y^5=0.$$
Thus $h^*\ell_{1}^*(k_{10}^1)$ is unchanged.
A similar computation shows $${\ell_1'}^*(k^1_8)=h^*\ell_1^*(k_1^8)+\sq^2(y^3)=
y^4+y^4=0.$$
Since ${\ell'_1}^*$ sends both $k$-invariants to 0, it lifts to a map
$\ell_2:\cee P^5\to E_2$, yielding (\ref{lift2}).

\bigskip
Let $\ell^2_8=\ell_2p_2$, 
 $\ell^2_4=\ell_2p_2p_1$ and $\ell^1_{2^e}=q_2\ell^2_{2^e}$.
Since $\sq^1=0$ in $H^*(L(2^e))$, the primary indeterminacy for 
$\ell^2_4$ and $\ell^2_8$ is 0. This means that varying through either factor of 
$K_7\times K_9$ will not change the image of $k_{10}^2$ in $H^{10}(L(2^e))$.

Secondary indeterminacy requires checking whether varying $\ell^1_{2^e}$
through $K(\bz,6)\times K_7$ and then lifting the altered map might change
the image of $k_{10}^2$. First note that if we vary through $K(\bz,6)$,
the image of $k^1_8$ is changed, and so the altered map no longer lifts
to $E_2$.

Let $F_{0,2}$ denote the fiber of $E_2\to\bspin$. There is a fibration
$$K_7\times K_9\to F_{0,2}\to K(\bz,6)\times K_7$$
and compatible actions

\centerline{\xymatrix{
{ F_{0,2}\times E_2 } \ar[d] \ar[r] & E_2 \ar[d] \\
{ K(\bz,6)\times K_7\times E_1 } \ar[r] & { E_1. } 
}}

\medskip\noindent 
Because of the above remark about $K(\bz,6)$, we consider just $K_7$, and
define $X_1$ by the pullback diagram

\centerline{\xymatrix{
{ X_1 } \ar[d] \ar[r] & F_{0,2} \ar[d] \\
{ K_7 } \ar^{i_2\hspace{1cm}}[r] & { K(\bz,6)\times K_7. } 
}}

\smallskip\noindent 
There is a fiber sequence
\begin{equation}\label{2op} X_1\to K_7
\stackrel{\mbox{\scriptsize Sq}^1,\mbox{\scriptsize Sq}^3}{-\!\!\!-\!\!\!-\!\!\!\longrightarrow}
K_8\times K_{10}
\end{equation}
and a commutative diagram of actions

\centerline{\xymatrix{
{ X_1\times E_2 } \ar[d] \ar^{\ \ \ \ \mu'}[r] & E_2 \ar[d] \\
{ K_7\times E_1 } \ar[r] & {  E_1. } 
}}

\medskip
Let $x_7\colon L(2^e)\to K_7$ be the nontrivial map. Since $\sq^1=0$
in $H^*(L(2^e))$, $x_7$ lifts to a map ${x'}\colon L(2^e)\to X_1$, and the
new lifting ${\ell_2'}\colon L(2^e)\to E_2$ is given by 
$\mu'(x'\times\ell^2_{2^e})$.
It satisfies
$$(\ell_2')^*(k_{10}^2)=(\ell^2_{2^e})^*(k_{10}^2)+(x')^*j^*(k_{10}^2),$$
where $j:X_1\to E_2$ is the inclusion.

\smallskip
The fibration (\ref{2op}) and class $j^*(k_{10}^2)\in H^{10}(X_1)$
comprise a secondary cohomology operation, associated to the Adem
relation(s) $(\sq^2\sq^1)\sq^1+\sq^1\cdot\sq^3$. Note that each term
is an Adem relation by itself, and, since the mod-$4$ Bockstein $\b_2$ 
(that is, with respect to the nontrivial extension $0\rightarrow\zet/2\rightarrow\zet/8\rightarrow\zet/4
\rightarrow 0$, the boundary operation defined on $\zet/2$-classes with a $\zet/4$-representative)
is the secondary
operation associated with the Adem relation $\sq^1\cdot\sq^1$, our
secondary operation is $\sq^2\b_2+\b_2\sq^2$. Thus, since 
$(\ell^2_{2^e})^*(k_{10}^2)$
is the nonzero class $x_{10}$,
$$(\ell_2')^*(k_{10}^2)=x_{10}+\sq^2(\b_2x_7)+\b_2(\sq^2x_7).$$
Since $\sq^2(H^8(L(2^e)))=0$ and $\sq^2(H^7(L(2^e)))\ne0$, and $\b_2=0$
in $H^{*}(L(8))$ but $\b_2(x_{2i-1})\ne0\in H^*(L(4))$,
we obtain 
$$(\ell_2')^*(k_{10}^2)\ 
\left\{\begin{array}{c l}
 {=0}  & \mbox{for $L(4)$,} \\
 {\neq 0} & \mbox{for $L(8)$,}
\end{array}\right.$$
which implies Proposition \ref{indet}.\hfill\cajita

%\smallskip
%\centerline{\bf The following parragraph might be removed}
%\hrule\smallskip 
%\indent Section~\ref{postnikov} constructs
%$e$-axial maps which would correspond to the (nonexistent) immersions
%$L^{5}(2^e)\subseteq\erre^7$ and $\cee P^2\subseteq \erre^6$. 
%\hrule\smallskip

\section{Characterizations for axial maps}\label{sec:4}
%From now on we use the alternative notation $L^{2n+1}(2^\infty)$ for $\cee P^n$.
As in the previous section, we only consider cohomology groups with coefficients in $\zet/2$.
Let $x\in H^1(L^{2n+1}(2^e))$ (when $e<\infty$) and $y\in H^2(L^{2n+1}(2^e))$
denote the nontrivial elements 
(so that $y=x^2$ for $e=1$, but $x^2=0$ for $1<e<\infty$). 
Let $z\in H^1(S^{2n+1}\times_e P^{2m-1})$ 
be the Stiefel-Whitney
class of the canonical real line bundle over $S^{2n+1}\times_e P^{2m-1}$
(recall $S^{2n+1}\times_e P^{2m-1}$ is the projectivization of $m\xi_{n,e}$). 
By the Leray-Hirsch Theorem, the projection
$S^{2n+1}\times_e P^{2m-1}\rightarrow L^{2n+1}(2^e)$ injects
$H^*(L^{2n+1}(2^e))$ into $H^*(S^{2n+1}\times_e P^{2m-1})$ and, as an algebra over
the former ring, the latter is generated by $z$ subject to the single relation
\begin{equation}\label{relation}
0=\sum_{i=0}^{m} \left({m\atop i}\right)z^{2m-2i}y^i=(z^2+y)^{m}.
\end{equation}

Consider the maps
$\iota\colon P^{2m-1}\rightarrow S^{2n+1}\times_e P^{2m-1}$
and $\Delta\colon L^{2d+1}(2^e)\rightarrow S^{2n+1}\times_e P^{2m-1}$,
where $d=\min\{n,m-1\}$, defined as follows: (a)
$\iota$ is the composite
$P^{2m-1}{\hookrightarrow} S^{2n+1}\times P^{2m-1}\rightarrow S^{2n+1}\times_e P^{2m-1}$,
where the first map is inclusion into the second factor; (b)
$\Delta$ is induced by the composite $S^{2d+1}{\rightarrow} S^{2n+1}\times S^{2m-1}
\rightarrow S^{2n+1}\times_e P^{2m-1}$, where the first map takes $x$ into the pair
$(x,\overline{x})$. Here $\overline{x}$ means
conjugating all complex coordinates in $x\in S^{2d+1}\subset \cee^{d+1}$.

\begin{proposicion}\label{hotaxial}
Assume $N\geq 2m$. A map $a\colon S^{2n+1}\times_e P^{2m-1}\rightarrow P^N$ is $e$-axial 
if and only if
 $a\Delta$ is null-homotopic but $a\iota$ is essential.
\end{proposicion}

\dem The inclusion $P^N\hookrightarrow P^\infty$ has $S^N$ as homotopy  
fiber and, therefore,
induces a one-to-one %n injective (in fact bijective) 
map $[X,P^N]\rightarrow[X,P^\infty]$ for any CW complex $X$
of dimension less than $N$. Thus, in this situation, the (non)triviality of a given homotopy class
$X\rightarrow P^N$ can be decided by its effect on $H^1(\;\;)$. Let $\omega\in H^1(P^N)$ be
the generator and, for $a\colon S^{2n+1}\times_e P^{2m-1}\rightarrow P^N$ set $a^*(\omega)=rx+sz$, where
$r,s\in\zet/2$ and $x$ and $z$ are as established at the beginning of the section. By definition
$a$ is $e$-axial precisely when $r=0$ and $s=1$. Hence, by the
dimensional restriction, it suffices to verify the relations $\iota^*(x)=0$, $\Delta^*(x)=x$,
$\iota^*(z)\neq 0$ and $\Delta^*(z)=0$ (only the last two when $e=\infty$).
As $x$ lies in the image of $\pi^*$,
the first one follows at once from the fibration
$$P^{2m-1}\stackrel{\iota}{\rightarrow} S^{2n+1}\times_e P^{2m-1} 
\stackrel{\pi}{\rightarrow} L^{2n+1}(2^e).$$
The second one holds since
$\pi\Delta$ is the inclusion $L^{2d+1}(2^e)\hookrightarrow
L^{2n+1}(2^e)$. The third one is a consequence of the general fact that, when restricted to the fiber 
of $\pi$, the canonical line bundle over $S^{2n+1}\times_e P^{2m-1}$ agrees with the universal bundle over 
$P^{2m-1}$.
The last relation follows from dimensional reasons for $e=\infty$, and by naturality (Remark~\ref{pullback})
of the following diagram when $e<\infty$.

\vspace{.2cm}
{\ }\hspace{2cm}\xymatrix{
{ L^{2d+1}(2^e) } \ar[d] \ar^{\Delta\hspace{.6cm}}[r] & 
{ S^{2n+1}\times_e P^{2m-1} } \ar[d] \ar@{=}[r] & { P(m\xi_{n,e}) }  \ar[d]  & & & \\
{\cee P^d}  \ar^{\Delta\hspace{.9cm}}[r] & {S^{2n+1}\times_\infty P^{2m-1}} \ar@{=}[r] & 
{P(m\xi_{n,\infty}).} & & & {\qquad\hspace{.03cm}\cajita}
}

\begin{nota}\label{converse}{\em
In view of Remark~\ref{accionlibre}, the last result implies that
a $1$-axial map is just an axial map in the usual sense. 
Moreover, a $1$-axial map
$P^{2n+1}\times P^{2m-1} \rightarrow P^N$ factoring 
through the maps in~(\ref{comparar}) induces a map 
$S^{2n+1}\times_{e+1} P^{2m-1} \rightarrow P^N$ which is ($e+1$)-axial
if and only if the 
first condition
in Proposition~\ref{hotaxial} holds (note that, for $e=\infty$,
the first condition is always automatic).
At any rate, since the projection $L^{2d+1}(2^e)\rightarrow L^{2d+1}(2^{e+1})$ induces the 
trivial morphism
in mod 2 cohomology of dimension 1, we see that the composite $S^{2n+1}\times_e P^{2m-1}\rightarrow
S^{2n+1}\times_{e+1} P^{2m-1}{\rightarrow} P^N$ would indeed be %an
$e$-axial.% map.
}\end{nota}

\begin{proposicion}\label{caracterizaciones}
The following conditions are equivalent:
\begin{itemize}
\item[{\rm (a)}] There is an $e$-axial map $S^{2n+1}\times_eP^{2m-1}\to P^N$.
\item[{\rm (b)}] There is an $e$-skew map $S^{2n+1}\times S^{2m-1}\to S^N$.
\item[{\rm (c)}] The $N+1$ iterated Whitney sum of the canonical line bundle over $S^{2n+1}\times_eP^{2m-1}$
admits an everywhere nontrivial section.
\end{itemize}
\end{proposicion}

\dem Note that the principal $\zet/2$-bundle associated to the canonical line bundle
over $S^{2n+1}\times_eP^{2m-1}$ is given by the obvious projection
$S^{2n+1}\times_e S^{2m-1}\to S^{2n+1}\times_e P^{2m-1}$. 
In these terms, any $e$-axial
map $a\colon S^{2n+1}\times_eP^{2m-1}\to P^N$ can be covered by a $\zet/2$-equivariant map 
$\alpha\colon S^{2n+1}\times_eS^{2m-1}\to S^N$. Conversely, any such $\alpha$
covers, by definition, some $e$-axial map. This proves the equivalence between~(a) and~(b).
The equivalence between~(a) and~(c) follows from~\cite[Proposition~1.3]{sh}.\hfill\cajita

\medskip
We close the section by discussing the relationship of both the immersion problem for 2-torsion 
lens spaces and axial maps to the existence of equivariant maps into Stiefel 
type manifolds (in the case $e=1$, this
was first considered in~\cite{CF} and studied in more detail in~\cite{BR}).

Let Mono$(\erre^{2m},\erre^{N+1})$ be the Stiefel manifold
of orthogonal monomorphisms, and let Skew$(S^{2m-1},S^N)$ be the space of (regular) skew maps,
that is, continuous maps $q\colon S^{2m-1}\rightarrow S^N$ satisfying
$q(-x)=-q(x)$. In~\cite{HH} (see also~\cite{metaJames}) 
it is proved that the inclusion
\begin{equation}\label{incl}
{\rm{Mono}}(\erre^{2m},\erre^{N+1})\hookrightarrow {\rm{Skew}}(S^{2m-1},S^N)
\end{equation}
is a ($2N-4m+1$)-homotopy equivalence. Note that~(\ref{incl}) is a $\zet/2^e$-equivariant map, 
where $w\in\zet/2^e$ acts by precomposing with $x\mapsto wx$. 
Now, while the adjoint of a Gauss map $m\xi_{n,e}\to\erre^{N+1}$ (as the one given in Corollary~\ref{linealizar})
can be interpreted as a {\it twisted} $\zet/2$-equivariant map 
$g\colon S^{2n+1}\to \mbox{Mono}(\erre^{2m},\erre^{N+1})$ ---that is, $g(wx)=w^{-1}g(x)$, for $x\in S^{2n+1}$
and $w\in\zet/2^e$---, the adjoint of a $e$-skew map as in~(b) of Proposition~\ref{caracterizaciones} gives a 
corresponding twisted $\zet/2$-equivariant map $g\colon S^{2n+1}\to \mbox{Skew}(S^{2m-1},S^{N})$. In particular,
this yields an indirect approach for Theorem~\ref{main} as well as for the fact that, in the metastable range, any 
twisted equivariant map from $S^{2n+1}$ into $\mbox{Skew}(S^{2m-1},S^N)$ can {\it equivariantly} be deformed into 
Mono$(\erre^{2m},\erre^{N+1})$. For the sake of completeness, we provide a direct proof for this last fact.

\begin{proposicion}\label{deformar}
Under the conditions above, and if $N-2m\geq n$, any twisted $\zet/2$-equivariant map 
$S^{2n+1}\to \mbox{\rm Skew}(S^{2m-1},S^{N})$ can equivariantly be deformed into a twisted 
$\zet/2$-equivariant map 
$S^{2n+1}\to \mbox{\rm Mono}(\erre^{2m},\erre^{N+1})$.
\end{proposicion}

\dem The map in the hypothesis can be interpreted as a section $s$ for the bundle
on the right hand side in the diagram below.

\vspace{.2cm}
\centerline{
\xymatrix{
{ {\rm{Mono}}(\erre^{2m},\erre^{N+1}) } \ar[r] \ar[d] & {\rm{Skew}}(S^{2m-1},S^N) \ar[d] \\
{ S^{2n+1}\times_{\zeti/2^e}{\rm{Mono}}(\erre^{2m},\erre^{N+1}) } \ar[r] \ar^{\pi_1}[d] &
{ S^{2n+1}\times_{\zeti/2^e}{\rm{Skew}}(S^{2m-1},S^N) } \ar_{\pi_2}[d] \\
{ L^{2n+1}(2^e)} \ar@{=}[r] & { L^{2n+1}(2^e) } \ar@/_1pc/[u]_s
}}

\medskip\noindent
Now,~(\ref{incl}) and the long exact homotopy sequences associated to these fibrations
show that the middle horizontal map in the diagram is a 
($2N-4m+1$)-homotopy equivalence. Therefore, under the present hypothesis,
$s$ can be deformed into a map
$$\sigma\colon L^{2n+1}(2^e)\rightarrow S^{2n+1}\times_{\zeti/2^e}{\rm{Mono}}(\erre^{2m},\erre^{N+1})$$
such that $\pi_1\circ\sigma$ is homotopic to the identity.
However, as $\pi_1$ is a fibration, the homotopy lifting property assures that
$\sigma$ can further be deformed into an actual section for $\pi_1$. \hfill\cajita

\medskip
As with the technique of bilinear maps, it would be interesting to
explore the extent to which the above ideas can push known immersions of real projective spaces to 
immersions of 2-torsion lens spaces and complex projective spaces.

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\bibitem{table}
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\end{thebibliography}

\small
\vglue .5cm
\noindent
\textsc{Departamento de Matem\'aticas, CINVESTAV--IPN, M\'exico}

\vspace{-.3cm}
\begin{verbatim}
lastey@math.cinvestav.mx
\end{verbatim}

\noindent
\textsc{Lehigh University, Bethlehem, PA 18015}

\vspace{-.3cm}
\begin{verbatim}
dmd1@lehigh.edu
\end{verbatim}

\noindent
\textsc{Departamento de Matem\'aticas, CINVESTAV--IPN, M\'exico}

\vspace{-.3cm}
\begin{verbatim}
jesus@math.cinvestav.mx
\end{verbatim}

\end{document}



\section{Low dimensional $e$-axial maps}
\label{postnikov}
As indicated in Section~\ref{sec:3}, there is no 7-dimensional immersion of $L^5(2^e)$; yet, 
in this section,
we construct $e$-axial maps of the form $S^5\times_e P^5\rightarrow P^7$. In view of 
Remark~\ref{inducir} we only need to consider the case $e=\infty$. Set $\eta=\eta_{2,\infty}$ 
and $\xi=\xi_{2,\infty}$ (so that
$S^5\times_\infty P^5$ is the projectivization of $3\xi$) and
recall from~(\ref{relation}) that $H^*(P(3\xi);\zet/2)$ is the algebra over 
$H^*(\cee P^2;\zet/2)$ generated by $z$ ---the first Stiefel-Whitney class of the canonical 
real line bundle $h$ over $P(3\xi)$--- subject to the single relation $(z^2+y)^3=0$,
where $y$ is the generator of $H^2(\cee P^2;\zet/2)$. Let $\cee P(3\eta)$ be the complex 
projectivization of 
$3\eta$ and consider the sphere bundle
\begin{equation}\label{??+1}
S^1\rightarrow P(3\xi)\rightarrow \cee P(3\eta)
\end{equation}
associated with the tensor square of the canonical complex line bundle $\overline{h}$ over 
$\cee P(3\eta)$. The integral cohomology ring $H^*(\cee P(3\xi);\zet)$
is the algebra over $H^*(\cee P^2;\zet)$ generated by the first Chern class
$\overline{z}$ of $h$, subject to the single relation
$(\overline{z}-\overline{y})^3=0$, where $\overline{y}$ is the generator of 
$H^2(\cee P^2;\zet)$. As the Euler class of~(\ref{??+1})
is $2\overline{z}$, we see from the corresponding Gysin sequence that the subalgebra of 
$H^*(P(3\xi);\zet)$ generated by (the image of) $\overline{z}$ is 
\begin{equation}\label{??+2}
H^*(\cee P^2;\zet)[\overline{z}] \left/ \rule{0cm}{.4cm}\left( (\overline{z}-\overline{y})^3, 
2\overline{z}  \right). \right. 
\end{equation}

We have $\beta(z)=\overline{z}$, where $\beta$ is the Bockstein operator, as well as 
$\rho(\overline{z})=z^2$ and $\rho(\overline{y})=y$, where $\rho$ is mod-2 reduction. Then
\begin{eqnarray}
\beta(y^2z^3+yz^5) & = & \beta(\rho(\overline{y}^2\overline{z}+\overline{y} \,\overline{z}^2)z) 
\nonumber\\
 & = & (\overline{y}^2\overline{z}+\overline{y} \,\overline{z}^2)\overline{z} \label{??+3}\\
 & = & \overline{y}^2\overline{z}^2+\overline{y}(\overline{y}^2\overline{z}+\overline{y} 
\,\overline{z}^2) \; = \; 0 \nonumber
\end{eqnarray}
by~(\ref{??+2}), so $y^2z^3+yz^5$ is the reduction of an integral class. Note also that
$\mbox{Sq}^2(y^2z^3)=y^2z^5$ and $\mbox{Sq}^2(yz^5)=0$.


\begin{eqnarray}\label{??+4}
\mbox{Sq}^2(y^2z^3+yz^5)=yz^7=yz(y^2z^2+yz^4)=y^2z^5
\end{eqnarray}
is the generator of $H^9(P(3\xi);\zet/2)\simeq\zet/2$. Now consider the bundle $8h$. 
It has a complex structure, namely that of $4(h\otimes\cee)$, and the total Chern class is
\begin{equation}
c(8h)=\left( 1+c_1(h\otimes\cee)\right)^4=(1+\overline{z})^4=1+\overline{z}^4 \nonumber
\end{equation}
because $c_1(h\otimes\cee)=\beta(w_1(h))=\beta(z)=\overline{z}$. Since 
$\overline{z}^4=(\overline{y}^2\overline{z}
+\overline{y}\,\overline{z}^2)\overline{z}=\overline{y}^2\overline{z}^2+
\overline{y}(\overline{y}^2\overline{z}+\overline{y}\,\overline{z}^2)=0$, again by~(\ref{??+2}),
we have that $8h$ has trivial Chern classes. In fact:

\begin{lema}\label{cxseccion}
$8h$ admits a nowhere vanishing (complex) section.
\end{lema}

Note the required axial map is an easy consequence of Lemma~\ref{cxseccion}; indeed, the diagram 

\vspace{.2cm}
\centerline{\xymatrix{
{ } & { P^7 } \ar[d] \ar[rr] & & { BSU(3) }  \ar[d]  \\
{ P(3\xi) } \ar^{h}[r] & {P^\infty} \ar^{4(\xi\otimes\ceei)}[rr] & & 
{BSU(4).} 
}}

\bigskip
has a pull back square, so that $h$ lifts to $P^7$.