DEPARTMENT OF MATHEMATICS University of Utah Ph.D. PRELIMINARY EXAMINATION IN GEOMETRY/TOPOLOGY January, 2018

Let \(M\) and \(N\) be smooth manifolds. If \(f:M\to N\) is continuous, proper, and injective, prove that \(f^{-1}:f(M)\to M\) is continuous. Conclude that any proper, injective, immersion \(f:M\to N\) is an embedding.

Let \(M\) and \(N\) be smooth manifolds of the same dimension, and suppose \(M\) is compact. Prove that if \(f:M\to N\) is an immersion then \(f^{-1}(y)\) is finite for any \(y\in N\).

Give a foliation on \(S^3\) whose leaves are each diffeomorphic to \(S^1\). (To describe the foliation, you don’t need to give the charts. Just a description of the leaves will do.)

Let \(G\) be a Lie group with Lie algebra \(\mathfrak {g}\) and exponential map \(\exp :\mathfrak {g}\to G\). For a nonzero \(v\in \mathfrak {g}\), let \(L\) be the line spanned by \(v\). Prove that \(\exp |_L:L\to G\) is an immersion.

Prove that \(S^n\) is orientable.

Let \(M\) and \(N\) be smooth manifolds, and let \(f:M\to N\) be smooth. Prove that if \(\theta \) and \(\psi \) are closed differentiable \(n\)-forms on \(N\) (so that \([\theta ]\) and \([\psi ]\) are classes in de Rham cohomology) then \([\theta ]=[\psi ]\) implies \([f^*\theta ]=[f^*\psi ]\).

Suppose \(X\) and \(Y\) are cell complexes and that \(A\subseteq X\) is a subcomplex. Let \(f:A\to Y\) be continuous and let \(Y\sqcup _f X\) be \(Y\) with \(X\) attached along \(A\) via \(f\). Prove that \(Y\sqcup _f X\) is homotopy equivalent to a cell complex.

Prove \(\pi _1(S^n)=1\) if \(n\ge 2\).

Let \(p:\widetilde {X}\to X\) be a cover of connected spaces. Prove \(p_*:\pi _1\widetilde {X}\to \pi _1X\) is injective.

Let \(\Sigma _2\) be a closed, orientable surface of genus 2. Prove that \(\pi _1(\Sigma _2)\) has a normal subgroup \(G\) such that \(\pi _1(\Sigma _2)/G\cong \mathbb {Z}\).

Prove \(H_1(\Sigma _2)\cong \mathbb {Z}^4\).

Prove \(H_c^k(\mathbb {R}^n)\) equals \(\mathbb {Z}\) if \(k=n\) and \(0\) if \(k\ne n\).