\documentclass{amsart}
\newtheorem{theorem}{Theorem}
\newcommand{\zz}{{\mathbb Z}}
\begin{document}
\title{Manifolds that have degree 1 maps from spheres}
\author{Kevin Iga}
\date{January 16, 2001}
\maketitle
In \cite{S}, Sol Schwartzman, while studying parallel tangent hyperplanes
to immersed hypersurfaces, considers a situation where an $n$-sphere maps
to an $n$-manifold in a way that induces an isomorphism on the top homology
group $H_n$. He shows in a few cases how this implies that the $n$-manifold
in question must be a sphere.
In this note, I will prove a more general theorem that gives a necessary and
sufficient condition for such a map to exist: the manifold must be a
homotopy sphere.
\begin{theorem}
Let $M^n$ be a connected manifold. Then $M^n$ is a homotopy
sphere if and only if there exists a continuous map $f:S^n\to M^n$
of degree 1 (meaning $f_*:H_n(S^n)\to H_n(M^n)$ is an isomorphism).
\end{theorem}
\begin{proof}
If $M^n$ is a homotopy sphere, then by the definition of homotopy equivalence,
the map $f$ exists and gives an isomorphism on all homology groups.
Conversely, suppose $f$ exists with the above property. We will prove
$M^n$ is a homotopy sphere.
Now $H_n(M^n)\cong \zz$ only when $M^n$ is a closed orientable manifold, so for
$f_*$ on $H_n$ to be an isomorphism, $M^n$ must be closed and orientable.
Now if $n=0$ or $n=1$, then by the classification of compact manifolds in these
dimensions, the theorem follows trivially. So we assume $n>1$.
If $n>1$, we show that $M^n$ must be simply-connected. Indeed, let
$\tilde{M}$ be the universal cover of $M$. Then since $n>1$, $f$ lifts to
$\tilde{f}:S^n\to \tilde{M}$, in such a way that if $\pi:\tilde{M}\to
M$ is the projection map for the covering space $\tilde{M}$, then
$f=\pi\circ \tilde{f}$. Therefore, $f_*=\pi_*\circ \tilde{f}_*:H_n(S^n)\to
H_n(M^n)$. The degree of $f_*$ is 1, and the degree is multiplicative
under composition, so the degree of $\pi_*$ is $\pm 1$. Therefore, $\pi$
is the identity covering map, and thus $\tilde{M}=M$. Therefore, $M$ is
simply connected.
Next we prove $M^n$ is a homology sphere. Indeed, if there were
non-zero homology in dimension $i$ for some $i$ between $1$ and $n-1$,
it would arise either in homology with rational coefficients (if it
were free) or homology with $\zz/p$ coefficients for some prime $p$
(if it were a torsion element). In any case, if we let $F$ be
whichever field has non-zero $H_i(M,F)$, then Poincar\`e duality
implies that there is an isomorphism $PD:H_i(M,F)\to H^{n-i}(M,F)$
such that $(\alpha\cup PD(c)) [M^n] = \alpha(c)$. Since $F$ is a
field, this means $H^i(M,F)$ is isomorphic to $\hom(H_i(M,F),F)$
through the evaluation map. So (assuming $H_i(M,F)$ is non-zero) there
exist $\alpha\in H^i(M,F)$ and $\beta=PD(c)\in H^{n-i}(M,F)$ so that
$(\alpha\cup\beta)[M^n]=1$. Applying $f$ we get
\begin{eqnarray*}
f^*(\alpha\cup \beta)(f_*[M^n])&=&1\\
(f^*(\alpha)\cup f^*(\beta))(f_*[M^n])&=&1.
\end{eqnarray*}
We note that here $[M^n]$ is a homology class in $H_n(M,F)$ not
$H_n(M)$ as before, but since $H_{n-1}(M)$ has no torsion,
$H_n(M)\otimes F$ and $H_n(M,F)$ are isomorphic through the usual
correspondence on chains. Furthermore, $f_*:H_n(S^n)\to H_n(M^n)$
being degree 1 implies that $f_*:H_n(S^n,F)\to H_n(M^n,F)$ is degree 1.
Therefore
\[(f^*(\alpha)\cup f^*(\beta))[S^n]=1.\]
On the other hand, $S^n$ has no cohomology classes except in dimension $0$ and
$n$, and $\alpha$ and $\beta$ are not in these dimensions, so $f^*(\alpha)$
and $f^*(\beta)$ are zero. This gives us a contradiction, unless $M^n$ has
no homology between dimensions $0$ and $n$. Therefore, $M^n$ is a homology
sphere.
Thus $f$ is a map between simply-connected spaces that induces an
isomorphism on homology. By Whitehead's theorem \cite{Sp}, $f$ is a
weak homotopy equivalence. Since manifolds are CW complexes, $f$ is a
homotopy equivalence \cite{Sp} and thus $M^n$ is a homotopy sphere.
\end{proof}
\begin{thebibliography}{99}
\bibitem{S}Sol Schwartzman, {\em Parallel Tangent Hyperplanes}, Proceedings
of the AMS, in preparation.
\bibitem{Sp}Edwin Spanier, {\em Algebraic Topology}, Springer-Verlag, New York,
1966, pp. 399, 405.
\end{thebibliography}
\end{document}