DEPARTMENT OF MATHEMATICS University of Utah Ph.D. PRELIMINARY EXAMINATION IN GEOMETRY/TOPOLOGY May 12, 2016

Give a definition of the flag manifold (or the flag variety) and describe it as a homogeneous space.

Prove that the homogeneous space \(\mathrm {SL}_2(\mathbb {R})/\mathrm {SL}_2(\mathbb {Z})\) is not compact.

Give a sketch of the calculation that the Lie bracket on the Lie algebra of \(\mathrm {GL}_n(\mathbb {R})\), identified with the space of all \(n\times n\) matrices, is \([A,B]=AB-BA\).

Let \(M,N,P\) be smooth manifolds without boundary and \(F:M\times P\to N\) a smooth map which is transverse to a submanifold \(Q\subset N\). Prove the Transversality Theorem: for almost every \(p\in P\) the map \(F_p:M\to N\) defined by \(F_p(x)=F(x,p)\) is transverse to \(Q\). If you’d like, you can consider the special case when \(M,Q\) are submanifolds of \(P=N=\mathbb {R}^n\) and \(F(x,p)=x+p\).

Let \(a:S^2\to S^2\) be the antipodal map on the 2-sphere: \(a(x)=-x\), and let \[ p:S^2\to \mathbb {R}\mathrm {P}^2=S^2/x\sim -x,\qquad p(x)=[x] \] be the projection to the projective plane. Let \(\omega \) be a 2-form on \(\mathbb {R}\mathrm {P}^2\).

1. Prove that \(\int _{S^2}p^*\omega =0\). Hint: \(p^*\omega \) is \(a\)-invariant.
2. Prove that there is a 1-form \(\eta \) on \(\mathbb {R}\mathrm {P}^2\) such that \(\omega =d\eta \). You are allowed to use the fact that if \(\zeta \) is a 2-form on \(S^2\) and \(\int _{S^2}\zeta =0\) then \(\zeta \) is exact, but you are not allowed to use de Rham’s theorem.

Let \(\omega =xdy+dz\) be a 1-form on \(\mathbb {R}^3\). Is the 2-plane field \(\operatorname {Ker}(\omega )\) integrable?

Let \(X=S^1\vee S^1\) and \(x_0\in X\) be the attaching point.

1. Carefully state van Kampen’s Theorem and use it to compute \(\pi _1(X,x_0)\).
2. Find two covering spaces of \(X\) with three sheets; one of which is regular (determining a normal subgroup of \(\pi _1(X,x_0)\)) and the other of which is irregular.

What is the universal cover of \(\mathbb {R}\mathrm {P}^2\vee \mathbb {R}\mathrm {P}^2\)?

Define the singular homology groups \(H_i(X;\mathbb {Z})\) of a topological space \(X\) and compute the singular homology groups of the \(n\)-sphere \(S^n\), justifying all your steps.

Describe a cell structure on real projective space \(\mathbb {R}\mathrm {P}^n\) and use it to compute all the cohomology groups \(H^i(\mathbb {R}\mathrm {P}^n,\mathbb {Z})\) and \(H^i(\mathbb {R}\mathrm {P}^n,\mathbb {Z}/2)\).

Describe the cohomology ring \(H^*(\Sigma _g,\mathbb {Z})\) (i.e. the groups together with the cup product) of a compact oriented surface \(\Sigma _g\) of genus \(g\).

1. Prove that if \(g<h\), then every map \(f:\Sigma _g\to \Sigma _h\) has degree zero.
2. Show that for any \(g\ge h\), there is a map \(f:\Sigma _g\to \Sigma _h\) of degree one.

Find all cohomology and homology groups (with \(\mathbb {Z}\) coefficients) of a closed connected orientable 3-manifold \(M\) with \(\pi _1(M,m_0)=\mathbb {Z}^r\times G\) where \(G\) is a finite group.