DEPARTMENT OF MATHEMATICS University of Utah Ph.D. PRELIMINARY EXAMINATION IN GEOMETRY/TOPOLOGY August 2008

Let \(\mathrm {SL}(2,\mathbb {R})\) denote the subset of \(M^2\) (\(2\times 2\) real matrices), with determinant equal to one. Prove that \(\mathrm {SL}(2,\mathbb {R})\) is a regular submanifold of \(M^2=\mathbb {R}^4\), and find it’s dimension.

Let \(G\) be a finite group that acts smoothly, and freely (without fixed points) on a manifold \(M\). Prove that the orbit space \(M/G\) admits a smooth structure (that is, has the structure of a smooth manifold).

Let \(M\) and \(N\) be compact orientable \(n\)-dimensional manifolds and \(f:M\to N\) a smooth map. Suppose that \[ \int _M f^*\Omega \ne 0 \] for some \(n\)-form \(\Omega \). Prove that \(f\) is surjective.

Let \(M\) be a compact manifold. Prove that there does not exist a smooth submersion \(f:M\to \mathbb {R}^n\) for any \(n>0\).

Let \(M\) be a compact oriented Riemannian manifold with boundary. Suppose that \(u\) and \(v\) are harmonic functions that are equal on \(\partial M\). Show that \(u=v\) on \(M\). Recall that \(u\) is harmonic on \(M\) if \(\Delta u=\operatorname {div}(\nabla u)=0\) on \(M\).

Show that the group of diffeomorphisms of a compact manifold \(M\) acts transitively on \(M\). That is, if \(p,q\in M\) there exists a diffeomorphism \(\phi :M\to M\) such that \(\phi (p)=q\). Hint: First assume \(p\) and \(q\) are in the same coordinate neighborhood \(U\) and construct \(\phi \) as the flow of a vector field supported in \(U\).

Let \(S^2\) be the 2-sphere and \(T\) the torus.

1. Show that every map from \(S^2\) to \(T\) is null-homotopic.
2. Show that there are infinitely many homotopy classes of maps from \(T\) to \(S^2\).

Let \(X\) be a compact topological space and \(\widetilde {X}\) its universal cover. Show that \(\pi _1(X)\) is finite if and only if \(\widetilde {X}\) is compact.

Let \(X_n\) be the space obtained by attaching a disk \(D\) to the torus \(T=S^1\times S^1\) where the attaching map is a degree \(n\) map from \(\partial D\) to the circle \(S^1\times \{p\}\) in \(T\).

1. Calculate \(\pi _1(X_n)\).
2. Calculate the homology and cohomology of \(X_n\) with \(\mathbb {Z}\)-coefficients and with \(\mathbb {Z}_m\)-coefficients where \(\mathbb {Z}_m\) is the cyclic group of order \(m\).

Show that \(S^2\vee S^4\) and \(\mathbb {C}P^2\) are not homotopy equivalent.

Construct a \(\Delta \)-complex structure for the torus and use it to calculate the homology, cohomology and cup product structure for the torus.

Let \(M\) be a closed, orientable manifold of dimension \(2k\) and assume that \(H_{k-1}(M)\) is torsion free. Show that \(H_k(M)\) is torsion free.