DEPARTMENT OF MATHEMATICS University of Utah Ph.D. PRELIMINARY EXAMINATION IN GEOMETRY/TOPOLOGY January 2016

1. Define \(F:\mathbb {R}^4\to \mathbb {R}^2\) by \[ F(x_1,x_2,x_3,x_4) =(x_1^2+x_2^2+x_3^2+x_4^2,\ x_1^2+x_2^2-x_3^2-x_4^2). \] Show that \(M=F^{-1}(1,0)\) is a smooth manifold.
2. For each \(x=(x_1,x_2,x_3,x_4)\in M\) show that the tangent space \(T_xM\) is spanned by \((x_2,-x_1,0,0)\) and \((0,0,x_4,-x_3)\).
3. Let \(G:\mathbb {R}^4\to \mathbb {R}\) be a smooth map and \(g=G|_M\) the restriction of \(G\) to \(M\). Show that \(x\) is a critical point of \(g\) if and only if \(\ker F\subset \ker G\).
4. If \(G(x_1,x_2,x_3,x_4)=x_1+x_3\) find the critical points of \(g\).

1. Define \(f:\mathbb {R}\to \mathbb {R}^2\) by \(f(x)=(x^2,x^3)\). Find \(f^*dx_1\).
2. Assume \(g:\mathbb {R}^2\to \mathbb {R}^2\) is a smooth map and for each \((x_1,x_2)\in \mathbb {R}^2\) the derivative is given by the matrix \[ \begin {pmatrix} 2x_1+x_2 & x_1+3\\ \dfrac {x_1}{(x_1^2+x_2^2)^2} & \dfrac {x_2}{(x_1^2+x_2^2)^2} \end {pmatrix}. \] Find \(g^*dx_1\).

Let \(M\) be a smooth manifold. Show that its tangent bundle \(TM\) is orientable.

Let \(U\subset \mathbb {R}^n\) be a connected, open subset. For all \(p,q\in U\) show that there exists a diffeomorphism \(\phi :\mathbb {R}^n\to \mathbb {R}^n\) with \(\phi (p)=q\) and \(\phi \) the identity outside of \(U\). (Hint: First assume that \(U\) is convex with compact closure and find a vector field with support in \(U\) whose flow takes \(p\) to \(q\).)

1. Define the DeRham cohomology groups \(H^k(M)\) for a smooth manifold \(M\).
2. Let \(N\) be a compact, smooth, \(k\)-dimensional manifold without boundary and \(f:N\to M\) a smooth map. Let \(\alpha \in \Omega ^k(M)\) be a closed \(k\)-form on \(M\) with \(\int _N f^*\alpha \ne 0\). Either show that \(\alpha \) represents a non-zero element of \(H^k(M)\) or find a counterexample.
3. The wedge product defines a map \(\Omega ^k(M)\times \Omega ^l(M):\Omega ^{k+l}(M)\) by \((\alpha ,\beta )\mapsto \alpha \wedge \beta \). Show that this descends to a map from \(H^k(M)\times H^l(M)\) to \(H^{k+l}(M)\).

Let \(G\) be an \(n\)-dimensional Lie group. Show that the tangent bundle \(TG\) is diffeomorphic to \(G\times \mathbb {R}^n\).

Show that \(\pi _1(S^1)=\mathbb {Z}\); that \(\pi _1(S^n)=1\) if \(n\ge 2\); that \(\pi _1(\Sigma _1-\{x\})=F_2\) (where \(\Sigma _1-\{x\}\) is a closed orientable genus 1 surface with a single point removed and \(F_2\) is a free group on two generators); and find \(\pi _1(\Sigma _2)\) (where \(\Sigma _2\) is a closed orientable surface of genus 2).

What is the universal cover of \(S^1\vee S^1\)?

Let \(X\) be a path connected space that is locally simply connected. Give the construction of \(p:\widetilde {X}\to X\), the universal cover of \(X\). You don’t have to define what a universal cover is, or show that your construction is a universal cover. Just state the description of the universal cover for \(X\) used in the standard proof that the universal cover for \(X\) exists.

Let \(\{B_n\}_{n\in \mathbb {Z}}\) and \(\{C_n\}_{n\in \mathbb {Z}}\) be chain complexes. What is the definition of a chain map from \(\{B_n\}_{n\in \mathbb {Z}}\) to \(\{C_n\}_{n\in \mathbb {Z}}\)?

Let \(SX\) be the suspension of \(X\). Show that \(\operatorname {H}_n(SX)\cong \operatorname {H}_{n-1}(X)\) for all \(n\).

Prove that if \(U\subseteq \mathbb {R}^n\) and \(V\subseteq \mathbb {R}^m\) are open, nonempty, and homeomorphic, then \(n=m\).

Let \(A\) be a retract of \(X\), and let \(i:A\to X\) be the inclusion. Prove that \(i_*:H_n(A)\to H_n(X)\) is injective for all \(n\).

Find all homology and cohomology groups (with integer coefficients) for the following three spaces: \(\mathbb {R}\mathrm {P}^3\), \(\mathbb {C}\mathrm {P}^3\), and \(S^1\vee S^3\).

Prove that \(S^1\vee S^3\) is not homotopy equivalent to a compact, orientable, 3-manifold without boundary.