DEPARTMENT OF MATHEMATICS University of Utah Ph.D. PRELIMINARY EXAMINATION IN GEOMETRY/TOPOLOGY May, 2017

Let \(A:\mathbb {R}^2\to \mathbb {R}^3\) be given by \[ A= \begin {pmatrix} 3 & 2\\ 1 & 4\\ 0 & -1 \end {pmatrix} \] Find \(\lambda \) such that \(A^*(3dy_1\wedge dy_2-2dy_2\wedge dy_3+7dy_1\wedge dy_3) =\lambda dx_1\wedge dx_2\).

Let \(\Sigma _g\) be a closed, orientable surface of genus \(g\), and let \(T\Sigma _g\) be its tangent bundle. Prove that \(T\Sigma _g\) is trivial if and only if \(g=1\).

Let \(M\) be a compact manifold with a vector field \(X\in \Gamma (TM)\) and resulting one-parameter group \(\{\theta _t^X\}_{t\in \mathbb {R}}\le \operatorname {Diff}(M)\). For any \(f\in \operatorname {Diff}(M)\), let \(f_*X\in \Gamma (TM)\) be the pushforward of \(X\). Show that the one-parameter group for \(f_*X\) is \(\{f\circ \theta _t^X\circ f^{-1}\}_{t\in \mathbb {R}}\).

Specify the leaves of a 2-dimensional foliation on \(S^2\times S^1\), such that not every leaf is compact.

Find all connected subgroups of \(\mathrm {SL}_2(\mathbb {R})\) that contain the group \[ V=\{g\in \mathrm {SL}_2(\mathbb {R})\mid g_{1,2}=0\text { and }g_{1,1}=g_{2,2}=1\}. \]

Let \(G\) be a Lie group. Prove that \(G\) is orientable, and that if \(L_g\in \operatorname {Diff}(G)\) is left multiplication by \(g\in G\), then \(L_g\) preserves orientation.

Let \(\Sigma _g\) be a closed, orientable surface of genus \(g\). Given \(p\ge 1\) distinct points \(\{x_1,\ldots ,x_p\}\subseteq \Sigma _g\), let \(\Sigma _{g,p}=\Sigma _g-\{x_1,\ldots ,x_p\}\). Find the cohomology of \(\Sigma _{g,p}\) with integer coefficients, including its cup product structure.

For \(k\in \N \) and a topological space \(X\), let \(C_kX\) be a collection of \(k\) cones on \(X\), with their bases – each homeomorphic to \(X\) – identified. So for example, \(C_2X\) is the suspension of \(X\). Express the homology groups (with integer coefficients) of \(C_kX\) in terms of the homology groups of \(X\).

Use covering spaces to write the generators of a normal rank 3 free subgroup of a rank 2 free group. Use covering spaces to write the generators of a non-normal rank 3 free subgroup of a rank 2 free group.

Let \(n,k\in \N \) with \(n\ge 2\). Suppose \(f:\mathbb {R}\mathrm {P}^n\to T^k\) is continuous. Determine \[ f_*:H_m(\mathbb {R}\mathrm {P}^n)\to H_m(T^k) \] for all \(m\).

Give the chain complex used to define cellular homology of a cell complex \(X\). Prove that it’s a chain complex.

Let \(D^n\) be a closed \(n\)-dimensional disk. Prove that if \(f\) is a homeomorphism of \(D^n\), then \(f\) fixes some \(x\in D^n\).