DEPARTMENT OF MATHEMATICS University of Utah Ph.D. PRELIMINARY EXAMINATION IN GEOMETRY/TOPOLOGY August 2013

Let \(S^1\) denote the unit circle in \(\mathbb {C}\) and define \(f:S^1\times S^1\to \mathbb {R}\) by \(f(z,w)=\operatorname {Re}(z)+\operatorname {Re}(w)\). Show that \(f\) is a Morse function and compute the indices of the critical points.

Give an example of a 1-form on \(\mathbb {R}^2-\{(0,0),(1,0)\}\) which is closed but not exact and prove both properties.

Let \(S\subset \mathbb {R}^3\) be a (nonempty) smooth surface. Prove that there exists a plane \(ax+by+cz=d\) in \(\mathbb {R}^3\) whose intersection with \(S\) is a collection of circles and lines.

Let \(\Delta \) be a plane field on a manifold \(M\) and let \(X,Y\) be two vector fields tangent to \(\Delta \). Show that if for some \(p\in M\) we have \(Y_p=0\) then \([X,Y]_p\in \Delta \).

Prove that every manifold admits a Riemannian metric. Use partitions of unity and not embedding into \(\mathbb {R}^N\).

Regard \(\SO (n)\) as a subset of the space of \(n\times n\) matrices, which in turn is identified with \(\mathbb {R}^{n^2}\). Show that \(\SO (n)\) is a submanifold and identify the tangent space to \(\SO (n)\) at the identity \(I\) (i.e. the Lie algebra of \(\SO (n)\)).

Let \(S^3=\{(z,w)\in \mathbb {C}^2\mid |z|^2+|w|^2=1\}\) and \(S^1\subset S^3\) is given by \(w=0\). Let \(X\) be the quotient space of \(S^3\) obtained by collapsing \(S^1\) to a point. Use any method to compute \(H_k(X;\mathbb {Z})\) for all \(k\in \mathbb {Z}\).

Let \(S_g\) denote the closed oriented surface of genus \(g\). Show that if \(h<g\) then every map \(S_h\to S_g\) has degree 0. Does every such map have to be null-homotopic?

Define the concepts of chain maps and chain homotopies, and prove that chain homotopy is an equivalence relation.

Let \(X\) be a path connected, locally path connected, semi-locally simply connected Hausdorff space. One of the main theorems of covering space theory asserts that there is a simply connected space \(\widetilde {X}\) and a covering map \(p:\widetilde {X}\to X\). Outline the construction of \(\widetilde {X}\) and \(p\) as follows:

1. Describe \(\widetilde {X}\) as a set.
2. Describe the topology on \(\widetilde {X}\).
3. Describe \(p\). You do not need to prove that \(p\) is a covering map.
4. Outline the proof that \(\widetilde {X}\) is simply connected.

Let \(p:\widetilde {X}\to X\) be a covering map between cell complexes. Assume that \(\widetilde {X}\) is connected and that \(p\) is null-homotopic. Prove that \(\widetilde {X}\) is contractible.

Give an example of a space \(X\) such that \(H_i(X;\mathbb {Z}/2)=0\) for \(i>0\) but for some \(i>0\) \(H_i(X;\mathbb {Z})\ne 0\). Is it possible to have a space \(X\) with \(H_i(X;\mathbb {Z})=0\) for \(i>0\) but for some \(i>0\) \(H_i(X;\mathbb {Z}/2)\ne 0\)? Prove it or give an example.