DEPARTMENT OF MATHEMATICS University of Utah Ph.D. PRELIMINARY EXAMINATION IN GEOMETRY/TOPOLOGY January 2004

Let \(f:\mathbb {R}^n\to \mathbb {R}\) be a smooth function. Show that there is \(v\in \mathbb {R}^n\) such that the function \(g_v:\mathbb {R}^n\to \mathbb {R}\) given by \[ g_v(x)=f(x)+v\cdot x \] is Morse (i.e. all critical points are nondegenerate).

Identify the set \(M(n)\) of all \(n\times n\) matrices with real entries with \(\mathbb {R}^{n^2}\) as usual. Show that the set \[ O(n)=\{M\in M(n)\mid MM^t=I\} \] is a submanifold of \(\mathbb {R}^{n^2}\) (here \(M^t\) denotes the transpose of \(M\) and \(I\) is the identity matrix).

Consider vector fields in \(\mathbb {R}^3\): \[ X=x\frac {\partial }{\partial x}+\frac {\partial }{\partial y} \qquad Y=\frac {\partial }{\partial x}-\frac {\partial }{\partial z}. \] Show that there is no nonempty surface \(S\subset \mathbb {R}^3\) that is tangent to both vector fields at each of its points.

Let \(\omega \) be a compactly supported smooth \(n\)-form on \(\mathbb {R}^n\). Show that there exists a compactly supported smooth \((n-1)\)-form \(\eta \) with \(\omega =d\eta \) if and only if \[ \int _{\mathbb {R}^n}\omega =0. \] You are allowed to quote facts about \(H^k(X)\) but not about \(H_c^k(X)\).

Let \(f:\mathbb {R}^2\to \mathbb {R}^2\) be given by \[ f(x,y)=(x^2+y,x-y^2). \] Explicitly compute \(f^*dx\), \(f^*dy\), \(f^*(dx\wedge dy)\).

Show that the product \(M\times N\) of two manifolds is orientable if and only if both \(M\) and \(N\) are orientable.

Using winding numbers, show that there is a complex number \(z\) such that \[ z^2=e^{-|z|^2}+i e^{-|z|}. \]

Let \(M\) be a manifold and \(f:M\to \mathbb {R}\) a continuous function. Show that there is a smooth function \(g:M\to \mathbb {R}\) such that \[ |f(x)-g(x)|<1 \] for every \(x\in M\).

Let \(f:\Sigma \to S^1\) be a smooth map from the surface of genus 2 to the circle. Suppose that \(y\in S^1\) is a regular value of \(f\) and that \(f^{-1}(y)\) is a nonseparating circle in \(\Sigma \) (i.e. \(\Sigma \setminus f^{-1}(y)\) is connected). Prove that \(f\) is not nullhomotopic.

Let \(Y\subset \mathbb {R}^n\) be a compact manifold without boundary. Prove that there is \(\epsilon >0\) with the following property. If \(f,g:X\to Y\) are smooth maps defined on a manifold \(X\) and if \(|f(x)-g(x)|<\epsilon \) for every \(x\in X\), then \(f\) and \(g\) are homotopic.

Give an example of a topological space that does not have a universal cover.

Let \(M\) be a 3-manifold such that for every compact set \(K\subset M\) there is a solid torus \(W\subset M\) with \(K\subset W\). Show that \(\pi _1(M)=\mathbb {Z}\) or \(0\).

1. Let \(M\) be a manifold with boundary. Show that if \(M\) is orientable then \(\partial M\) is orientable.
2. Let \(M\) be an \(n\)-dimensional manifold and \(N\subset M\) an \(n-1\) dimensional submanifold such that \(N\) has a neighborhood \(U\subset M\) with \(U\) homeomorphic to \(N\times (0,1)\). Show that if \(M\) is orientable then \(N\) is orientable.

Let \(S\) be a closed, orientable surface and \(\gamma \) a simple closed curve on \(S\). Show that \(S\) retracts onto \(\gamma \) if and only if \(\gamma \) represents a non-trivial element in homology.

Show that \(\mathbb {R}P^3\) is not homotopy equivalent to \(\mathbb {R}P^2\vee S^3\).

Let \(M\) be a closed 4-manifold, \(S\) a surface with \(\chi (S)\le 0\) and \(p:M\to S\) a fiber bundle. Show that there exists a closed surface \(F\) with \(\pi _n(M)=\pi _n(F)\) for \(n\ge 2\).

Let \(M\) be a Möbius strip and \(D\) a disk. Let \(f:\partial D\to \partial M\) have degree \(n\). Set \(X_n=M\coprod D/{\sim }\) where \(x\sim y\) if \(f(x)=y\). Calculate \(\pi _1(X_k)\) and construct all of its connected covering spaces.

1. Let \(M\) be a closed, orientable 3-manifold such that \[ H_1(M;\mathbb {Z})=\mathbb {Z}\oplus \mathbb {Z}/2\mathbb {Z}. \] Calculate the remaining homology and cohomology groups for \(M\) with \(\mathbb {Z}\) coefficients.
2. Now assume that \(M\) is a closed, non-orientable 3-manifold. Show that \(H_1(M;\mathbb {Z})\) is infinite.

Let \(X\) be a topological space obtained by adding a \(k\)-cell to \(\mathbb {C}P^n\). For what values \(i\) must \(H_i(X;\mathbb {Z})=H_i(\mathbb {C}P^n;\mathbb {Z})\)? What are the possibilities for \(H_i(X;\mathbb {Z})\) when the two groups are not equal?

Let \(f:S^n\to S^n\) be a continuous map from the \(n\)-sphere to itself and assume that \(n\) is even. Show that there is a point \(x\in S^n\) such that \(f(x)\in \{-x,x\}\). Does the same assertion hold if \(n\) is odd? If it does, prove it, if it does not construct a counterexample.