DEPARTMENT OF MATHEMATICS University of Utah Ph.D. PRELIMINARY EXAMINATION IN GEOMETRY/TOPOLOGY August 2015

1. State the definition of a regular value and state the pre-image theorem.
2. Let \(F:\mathbb {R}^n\to \mathbb {R}^m\) be a smooth function with \(y\in \mathbb {R}^m\) a regular value and let \(M=F^{-1}(y)\). If \(f:\mathbb {R}^n\to \mathbb {R}^k\) is a smooth function and \(x\in M\) is critical point for \(f|_M\) what are the possible values for the dimension of \(\ker F_*(x)\cap \ker f_*(x)\)? (Here \(F_*(x)\) is the tangent map from \(T_x\mathbb {R}^n\to T_y\mathbb {R}^m\) and \(f_*(x)\) is also a tangent map.)

3. Define \(G:\mathbb {R}^n\to \mathbb {R}^{m+k}\) by \(G(x)=(F(x),f(x))\).

   1. If \(z\in \mathbb {R}^k\) is regular value of \(f|_M\) is \((y,z)\in \mathbb {R}^{m+k}\) always a regular value of \(G\)? Give a proof or find a counterexample.
   2. If \(z\in \mathbb {R}^k\) is a regular of \(f\) (as a function on \(\mathbb {R}^n\)) is \((y,z)\in \mathbb {R}^{m+k}\) always a regular value of \(G\). Give a proof or find a counterexample.

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

Let \(V\) be a smooth vector field on \(\mathbb {R}^2\) and assume that outside of a compact set \(V=\frac {\partial }{\partial x}\). Show that the flow for \(V\) is defined for all time.

1. State Stokes theorem.
2. If \(\omega \in \Omega ^n(\mathbb {R}^n)\) has compact support and \(\int _{\mathbb {R}^n}\omega \ne 0\) show that there does not exist an \(\alpha \in \Omega ^{n-1}(\mathbb {R}^n)\) with compact support and \(d\alpha =\omega \).
3. Now assume that \(n=1\) and that \(\int _{\mathbb {R}}\omega =0\). Find an \(\alpha \in \Omega ^0(\mathbb {R})\) with compact support and \(d\alpha =\omega \).

Let \(S^2=\{(x,y,z)\in \mathbb {R}^3\mid x^2+y^2+z^2\}\). Define \(\pi :\mathbb {R}^3\backslash \{0\}\to S^2\) by \[ \pi (x,y,z)= \left ( \frac {x}{\sqrt {x^2+y^2+z^2}}, \frac {y}{\sqrt {x^2+y^2+z^2}}, \frac {z}{\sqrt {x^2+y^2+z^2}} \right ) \] and \(T_\epsilon (x,y,z)=(x,y,z+\epsilon )\). For \(\epsilon \in (0,1)\) the map \(f_\epsilon =\pi \circ T_\epsilon \) is a Lefschetz map from \(S^2\) to itself. Calculate its Lefschetz number and conclude every map of \(S^2\) to itself that is homotopic to the identity has a fixed point. (Hint: It will be easier to calculate the derivative of \(\pi \) and \(T_\epsilon \) separately and use the chain rule to find the derivative of \(f_\epsilon \).)

Let \(\omega \) and \(\eta \) be closed forms on a manifold \(M\). Show that the de Rham cohomology class of \(\omega \wedge \eta \) only depends on the cohomology classes of \(\omega \) and \(\eta \).