DEPARTMENT OF MATHEMATICS University of Utah Ph.D. PRELIMINARY EXAMINATION IN GEOMETRY/TOPOLOGY January 2013

Let \(A\) be an element of the Lie algebra of \(\mathrm {GL}_n(\mathbb {R})\). What is the left-invariant vector field on \(\mathrm {GL}_n(\mathbb {R})\) that is associated to \(A\)? Show that it is left-invariant.

For \(t\in \mathbb {R}\), find the matrix \[ \exp \begin {pmatrix}0&t\\-t&0\end {pmatrix}. \]

The cup product for the de Rham cohomology of a smooth manifold \(M\) is defined as \[ \cup :H^k(M)\times H^\ell (M)\to H^{k+\ell }(M) \] where \([\omega ]\cup [\theta ]=[\omega \wedge \theta ]\). Show that the cup product is well-defined.

For \((x,y)\in \mathbb {R}^2-\{(0,0)\}\), let \[ \omega (x,y)=\frac {-y}{x^2+y^2}\,dx+\frac {x}{x^2+y^2}\,dy. \] Find \(\int _{C(r)}\omega \) where \(C(r)\) is the circle of vectors in \(\mathbb {R}^2\) of norm equal to \(r>0\) oriented counterclockwise. Find a smooth function \(f:\mathbb {R}^2-\{(0,0)\}\to \mathbb {R}\) such that \(\omega =df\) or explain why there is no such \(f\).

Is the plane field on \(\mathbb {R}^3\) that is spanned at the point \((x,y,z)\) by \(2\frac {\partial }{\partial x}+y\frac {\partial }{\partial y}\) and \(\frac {\partial }{\partial y}+y\frac {\partial }{\partial z}\) integrable? Explain.

Let \(G\) be a Lie group, and let \(\widetilde {G}\) be the universal cover of \(G\). Endow \(\widetilde {G}\) with the structure of a Lie group.

Let \(G\) be a Lie group, and \(H\) a closed Lie subgroup. Sketch the proof that \(G/H\) is a manifold.

Let \(X=\mathbb {R}\mathrm {P}^2\vee \mathbb {R}\mathrm {P}^2\). Give an example of an irregular covering space \(\widetilde {X}\to X\). You should draw the picture of the 1-skeleton of \(\widetilde {X}\) (make sure 2-cells lift).

Compute the fundamental group of the space obtained from the disjoint union \(A\sqcup B\) of two 2-tori by identifying them along 3 points, i.e. \(a_i\sim b_i\) for \(i=1,2,3\) where \(\{a_1,a_2,a_3\}\subset A\) and \(\{b_1,b_2,b_3\}\subset B\).

Let \(M\) be a closed connected orientable 3-manifold with \(H_1(M;\mathbb {Z})=\mathbb {Z}\oplus \mathbb {Z}/2\mathbb {Z}\). Compute \(H_i(M;\mathbb {Z})\) and \(H^i(M;\mathbb {Z})\) for all \(i\).

Let \(S_g\) be the closed connected oriented surface of genus \(g\).

1. Prove that for \(g<h\) every map \(S_g\to S_h\) has degree 0.
2. Prove or disprove: for \(g<h\) every map \(S_g\to S_h\) is nullhomotopic?

Let \(X=S^1\times [0,1]\) and \(\partial X=S^1\times \{0,1\}\). Compute \(H_i(X,\partial X;\mathbb {Z})\) for all \(i\).

1. The complex projective space \(\mathbb {C}\mathrm {P}^n\) can be obtained from \(\mathbb {C}\mathrm {P}^{n-1}\) by attaching a single \(2n\)-cell. Describe the attaching map.
2. What is \(H^k(\mathbb {C}\mathrm {P}^n;\mathbb {Z})\)? Give a proof by induction on \(n\) using (a). State any theorems you are using.
3. What is the ring structure on \(H^*(\mathbb {C}\mathrm {P}^n;\mathbb {Z})\)? Give an inductive proof. You may use Poincaré duality.

Carefully state the van Kampen theorem.