Math 6510 • Differentiable Manifolds

Let \(M_0\) and \(M_1\) be orientable closed \(n\)-manifolds and \(f:M_0\to M_1\) a map
such that \(f^*:H^n(M_1;\mathbb{R})\to H^n(M_0;\mathbb{R})\) is an isomorphism.
\begin{enumerate}
\item Assume that \(H^i(M_0;\mathbb{R})\cong H^i(M_1;\mathbb{R})\) for all \(i\). Show
that \(f^*\) is an isomorphism on cohomology with \(\mathbb{R}\) coefficients in all
dimensions.
\item Assume that there exists a map \(g:M_1\to M_0\) such that
\(g^*:H^n(M_0;\mathbb{R})\to H^n(M_1;\mathbb{R})\) is an isomorphism. Show that \(f^*\) and
\(g^*\) are isomorphisms on cohomology with \(\mathbb{R}\) coefficients in all dimensions.
\end{enumerate}