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

Let \(\mathbf {P}^n(\mathbb {R})\) be \(n\)-dimensional real projective space. Use an explicit collection of charts to prove that \(\mathbf {P}^n(\mathbb {R})\) is a smooth manifold.

Prove that there is no immersion of the \(n\)-sphere into \(\mathbb {R}^n\).

Let \(M\) be the smooth submanifold of \(\mathbb {R}^3\) that is the solution set of the equation \(x^2+y^2-z^2=1\). Let \(N\) be the smooth submanifold of \(\mathbb {R}^3\) that is the solution set of the equation \(x^2+y^2+z^2=1\). Prove that \(M\) and \(N\) do not intersect transversally in \(\mathbb {R}^3\).

Let \(M\) and \(N\) be smooth manifolds, and let \(M\) be compact. Suppose \(F:M\times [0,1]\to N\) is a smooth function, and for any \(t\in [0,1]\) let \(F_t:M\to N\) be the function defined by \(F_t(p)=F(p,t)\). If \(F_0\) is an immersion, prove that there is some \(\varepsilon \in (0,1]\) such that \(F_t\) is an immersion for all \(t\in [0,\varepsilon )\).

Let \(G\) be a Lie group, and let \(H\) be the connected component of \(G\) that contains the identity. Prove that \(H\) is a Lie subgroup of \(G\), and that \(H\) is a normal subgroup of \(G\). (Do not use the theorem that closed subgroups of Lie groups are Lie groups.)

Let \(\Sigma \) be a closed, smooth surface. Show that if \(\Sigma \) admits a nonvanishing vector field, then the Euler characteristic of \(\Sigma \) equals 0.

Prove that if \(X\) and \(Y\) are path connected spaces, then \(\pi _1(X\times Y)=\pi _1(X)\times \pi _1(Y)\).

Let \(\Sigma _g\) be a closed, orientable surface of genus \(g\ge 1\). Prove that \[ \pi _1(\Sigma _g)= \left \langle a_1,b_1,a_2,b_2,\ldots ,a_g,b_g \,\middle |\, \prod _{i=1}^{g}[a_i,b_i] \right \rangle . \]

Show that \(\pi _1(\Sigma _2)\) contains \(\pi _1(\Sigma _3)\) as a normal subgroup.

Let \(D^k\) be the closed \(k\)-dimensional disk. Use that \(D^k/\partial D^k=S^k\) to find \(H_n(S^k;\mathbb {Z})\) for all \(n\).

Suppose \(f:S^k\to S^k\) is continuous and not surjective. Prove that \(f_*:H_k(S^k;\mathbb {Z})\to H_k(S^k;\mathbb {Z})\) is the zero homomorphism.

Suppose \(M\) is a closed, orientable, simply connected, smooth manifold of dimension 3. Let \(S_3\) be the symmetric group on 3 letters, and suppose \(S_3\) acts on \(M\) freely, by orientation preserving diffeomorphisms. Let \(N=S_3\backslash M\), and find \(H_n(N;\mathbb {Z})\) and \(H^n(N;\mathbb {Z})\) for all \(n\).