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

What is the definition of a foliation on a manifold? Give an example of a foliation on a manifold.

Find the Lie bracket \[ \left [2\frac {\partial }{\partial x}+z\frac {\partial }{\partial y}, x\frac {\partial }{\partial x}+y^2\frac {\partial }{\partial z}\right ]. \] Is the plane field \[ \Delta _{(x,y,z)} =\operatorname {Span}\left \{2\frac {\partial }{\partial x}+z\frac {\partial }{\partial y}, x\frac {\partial }{\partial x}+y^2\frac {\partial }{\partial z}\right \} \] integrable?

Find \(a,b\in C^\infty (\mathbb {R}^3)\) such that \[ d(xyz\,dx\wedge dy+3x^3\,dy\wedge dz)=a\,dx\wedge dy\wedge dz \] and \[ dx\wedge (14x\,dz)\wedge dy=b\,dx\wedge dy\wedge dz. \]

Let \(f:T^2\to \mathbb {R}^3\) be an embedding. Find \(\displaystyle \int _{T^2} dx\wedge dy\).

Let \(G\) be a Lie group with a discrete subgroup \(\Gamma \leq G\). Prove that \(\Gamma \) acts freely and properly discontinuously on \(G\). (It will follow that \(\Gamma \backslash G\) is a smooth manifold, but you don’t have to show this.)

Let \(f:S^2\to S^2\) be smooth and surjective. Prove that there is an open set \(U\subseteq S^2\) such that \(f:U\to f(U)\) is a diffeomorphism.

Draw a nowhere vanishing vector field on \(S^1\). Explain why there is a nowhere vanishing vector field on \(S^3\).

Find the Lie derivative \(L_{y\frac {\partial }{\partial x}+x\frac {\partial }{\partial y}}(xy)\).

Find \[ \exp \begin {pmatrix} 0&1&1\\ 0&0&1\\ 0&0&0 \end {pmatrix} \] and \[ \exp \begin {pmatrix} 2&0&0\\ 0&5&0\\ 0&0&-3 \end {pmatrix}. \]

Let \(G\cong \SO (2)\ltimes \mathbb {R}^4\) be the Lie group given by matrices \[ \begin {pmatrix} \cos (\theta )&-\sin (\theta )&x&y\\ \sin (\theta )&\cos (\theta )&z&w\\ 0&0&\cos (\theta )&-\sin (\theta )\\ 0&0&\sin (\theta )&\cos (\theta ) \end {pmatrix} \] where \(\theta ,x,y,z,w\in \mathbb {R}\). Find the linear subspace of \(\mathfrak {gl}_4(\mathbb {R})\) that is the Lie algebra of \(G\).

What is the Lie algebra of \(\mathrm {SO}(n)\)?

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 \(S^3=\{(z,w)\in \mathbb {C}^2\mid |z|^2+|w|^2=1\}\) 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 \(k\in \mathbb {Z}\).

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

Let \(G\) be a group of homeomorphisms acting freely on the \(n\)-sphere \(S^n\) with \(n\) even. Prove that \(G\) has at most two elements.

Give an example of a connected cell complex \(X\) such that \(H_k(X;\mathbb {Z})=0\) for all \(k>0\) but \(\pi _1(X)\ne \{1\}\). You do not have to verify the second claim, but you should argue the first.

Prove that a closed orientable surface \(S_g\) of genus \(g\ge 1\) is not homotopy equivalent to the wedge \(X\vee Y\) of two finite cell complexes both of which have nontrivial \(H_1(\cdot ;\mathbb {Z})\).

Let \(M\) be a connected topological \(n\)-manifold. Define what it means for \(M\) to be orientable. Then state the Poincaré duality theorem (including a careful definition of the map that is claimed to be an isomorphism).

Let \(X\) be a cell complex obtained from the circle by attaching a 2-cell with a degree 3 attaching map. Prove that \(X\) cannot be embedded into \(\mathbb {R}^3\). You may assume that the embedding is such that there exists a “regular neighborhood”, i.e. a smooth compact manifold \(N\subset \mathbb {R}^3\) such that \(X\hookrightarrow N\) is a homotopy equivalence.