DEPARTMENT OF MATHEMATICS University of Utah Ph.D. PRELIMINARY EXAMINATION IN GEOMETRY/TOPOLOGY August 2010

Give a nowhere vanishing vector field on \(\mathrm {SL}_n(\mathbb {R})\).

Suppose \(M\) is a smooth manifold with a smooth plane field. If the plane field is integrable, then prove that the bracket of any pair of smooth vector fields on \(M\) that are tangent to the plane field is another smooth vector field on \(M\) that is tangent to the plane field.

Give an example of a smooth plane field on a manifold that is not integrable.

Given a Lie group \(G\) with Lie algebra \(\mathfrak {g}\) and some \(v\in \mathfrak {g}\), let \(\{\theta _t^v\}_{t\in \mathbb {R}}\) be the corresponding 1-paramater group of diffeomorphisms of \(G\). Prove that \(\theta _t^v(gh)=g\theta _t^v(h)\) for any \(t\in \mathbb {R}\) and \(g,h\in G\).

Prove that there are uncountably many 3-dimensional foliations on a 5-dimensional torus.

Let \(U\) be a connected Lie subgroup of \(\mathrm {GL}_n(\mathbb {R})\). Suppose that every element of \(U\) has all of its entries below the main diagonal equal to 0, and all of its entries on the main diagonal equal to 1. Why is \(U\) diffeomorphic to \(\mathbb {R}^k\) for some \(k\)?

Let \(G\) be a Lie group with Lie algebra \(\mathfrak {g}\). Given a subalgebra \(\mathfrak {h}\subseteq \mathfrak {g}\), prove there is a Lie subgroup \(H\le G\) whose Lie algebra is \(\mathfrak {h}\). (The entire proof with all details would be best. If not, then try to write the main ideas of the proof.)

Let \(X\) be a smooth vector field on \(S^2\). Prove that \(X(p)=0\) for some \(p\in S^2\).

Let \(X\) be a topological space and \(x_0\in X\) a basepoint.

1. Define \(\pi _1(X,x_0)\) (describe it as a set and define the group operation; you don’t have to prove that it is well defined or that it is a group).
2. If \(X\) is path-connected and \(x_1\in X\) prove that \(\pi _1(X,x_0)\cong \pi _1(X,x_1)\) (write down an explicit isomorphism and check that it works).

1. Define a covering space \(p:\widetilde {X}\to X\).
2. Let \(p:\widetilde {X}\to X\) be a covering space, \(\widetilde {x}_0\in \widetilde {X}\), \(x_0\in X\), and \(p(\widetilde {x}_0)=x_0\). Show that \(p_*:\pi _1(\widetilde {X},\widetilde {x}_0)\to \pi _1(X,x_0)\) is injective (carefully state the lifting property you are using).

Let \(X\) be the cell complex obtained from the circle \(S^1\) by attaching two 2-cells \(e^2_2\) and \(e^2_3\) with attaching maps of degrees 2 and 3 respectively. Compute the fundamental group of \(X\). Carefully state any theorems you are using.

1. Define the concepts of chain morphisms and chain homotopies.
2. Show that every map \(S^2\to T^2\) from the 2-sphere to the 2-torus is null-homotopic.
3. Write down explicit cellular chain complexes \(C(S^2)\) and \(C(T^2)\) for \(S^2\) and \(T^2\) and a chain morphism \(\Phi :C(S^2)\to C(T^2)\) which is not chain homotopic to a chain morphism representing a constant map.

Let \(X\) be the space obtained from two \(n\)-spheres by identifying them along their equatorial \((n-1)\)-spheres. Using any method you like compute \(H_i(X)\) for all \(i\). State any theorems you are using (e.g. Mayer-Vietoris).

1. State the universal coefficient theorem for cohomology.
2. Suppose \(X\) is a space such that \(H_0(X)=\mathbb {Z}\), \(H_1(X)=\mathbb {Z}/2\mathbb {Z}\) and \(H_i(X)=0\) for \(i>1\). Compute \(H^i(X;\mathbb {Z}/2\mathbb {Z})\) for all \(i\).