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

Let \(M=\{z\in \mathbb {C}^n=\mathbb {R}^{2n}:z_1^2+\cdots +z_n^2=1\}\). Prove that \(M\) is a regular submanifold of \(\mathbb {R}^{2n}\), and find it’s dimension.

Let \(G\) be an \(n\)-dimensional Lie group, and \(TG\) it’s tangent bundle. Prove that \(TG\) is diffeomorphic to \(G\times \mathbb {R}^n\).

Let \(M\) be a compact orientable manifold with boundary. Use differential forms to show that there is no smooth retraction \(r:M\to \partial M\).

Let \(M\) be a manifold. Prove that the tangent bundle \(TM\) and the cotangent bundle \(TM^*\) are isomorphic (as smooth vector bundles).

Let \(M\) be a Riemannian manifold. Define \(d:M\times M\to \mathbb {R}\) by \[ d(x,y)=\inf \int _0^1 |\gamma '(t)|\,dt \] where the infimum is taken over piecewise \(C^1\) paths \(\gamma \) in \(M\) with \(\gamma (0)=x\) and \(\gamma (1)=y\). Prove that \(d\) is a metric on \(M\).

Let \(M\) be an orientable manifold of dimension \(n\). Prove that the top dimensional de Rham cohomology with compact supports \(H_c^n(m)\) is nontrivial.

Let \(T=S^1\times S^1\subset \mathbb {C}\times \mathbb {C}\) be the torus. Let \(X\) be the vector field on \(T\) defined by \(X(z_1,z_2)=(iz_1,aiz_2)\), where \(a\in \mathbb {R}\). Compute the flow \(\Phi (t,p)\), \(p=(p_1,p_2)\) of \(X\). For \(a\) rational, show that \(t\to \Phi (t,p)\) is periodic.

Compute \(H_i(S^n-X)\) when \(X\) is a subspace of \(S^n\) homeomorphic to \(S^p\vee S^q\vee S^r\) with \(p,q,r\in \{1,2,\ldots ,n-1\}\) and \(p\ne q\ne r\ne p\).

Let \(S_g\) be the closed, orientable surface of genus \(g\). Calculate \(\pi _2(S_g)\) for all \(g\).

Prove that the map \(h:S^3\to \mathbb {C}P^1\) given by \(h(x,y)=[x:y]\) is a fiber bundle. Here, \(S^3\) is the unit sphere \(|x|^2+|y|^2=1\) in \(\mathbb {C}^2\).

Let \(M\) be an open 3-manifold such that every compact set of \(M\) is contained in a solid torus.

1. Show that \(\pi _1(M)\) is abelian.
2. Assume that \(\pi _1(M)\) is finitely generated. Show that \(\pi _1(M)\) is trivial or \(\mathbb {Z}\).

Define the higher homotopy groups, \(\pi _i\), for \(i\ge 2\) and prove that they are abelian.

Let \(p:\widetilde {X}\to X\) be a 2-sheeted covering map between cell complexes such that the cell structure on \(\widetilde {X}\) is obtained by lifting the cell structure on \(X\).

1. Construct a short exact sequence of cellular chain complexes with coefficients in \(\mathbb {Z}/2\mathbb {Z}\) \[ 0\to C_n(X;\mathbb {Z}/2\mathbb {Z})\to C_n(\widetilde {X},\mathbb {Z}/2\mathbb {Z}) \to C_n(X,\mathbb {Z}/2\mathbb {Z})\to 0 \] (i.e. give the maps explicitly and prove that the sequence is exact).
2. Write down the induced long exact sequence of homology groups (the transfer sequence).
3. Prove that if \(\widetilde {X}\) is acyclic over \(\mathbb {Z}/2\mathbb {Z}\) (i.e. \(H_i(\widetilde {X};\mathbb {Z}/2\mathbb {Z})=0\) for \(i>0\) and \(H_0(\widetilde {X};\mathbb {Z}/2\mathbb {Z})\cong \mathbb {Z}/2\mathbb {Z}\)) then \(H_i(X;\mathbb {Z}/2\mathbb {Z})\cong \mathbb {Z}/2\mathbb {Z}\) for all \(i\ge 0\).

Let \(\Sigma \) be a closed oriented surface of genus \(g\) and suppose that \(X\subset \Sigma \) is a graph which is a retract of \(\Sigma \). Prove that \[ \operatorname {rank} H_1(X)\le g \]

Let \(\Sigma _g\) denote the closed oriented surface of genus \(g\). Show that every map \(f:\Sigma _2\to \Sigma _3\) has degree 0.