UNIVERSITY OF UTAH DEPARTMENT OF MATHEMATICS Ph. D. Preliminary Examination in Geometry / Topology May 27, 2020.

Let \(M\) be a smooth manifold and \(\gamma :S^1\to M\) be a smooth embedding. Show that there exists a smooth vector field \(X\) on \(M\) that does not vanish on \(\gamma (S^1)\) and which is tangent to \(\gamma (S^1)\) at each point. (The last condition means that for each point \(p=\gamma (t)\), the vector \(X_p\) is a multiple of \(\gamma '(t)\).)

Let \(M^2\) be a smooth two manifold without boundary and \(i:M^2\to \mathbb {R}^5\) be a smooth immersion. Show that there is a projection \(\pi :\mathbb {R}^5\to \mathbb {R}^4\) such that \(\pi \circ i\) is an immersion.

Let \(S^2=\{(x,y,z)\in \mathbb {R}^3:x^2+y^2+z^2=1\}\) be the two sphere in three space. Consider the vector fields \[ V=y\frac {\partial }{\partial z}-z\frac {\partial }{\partial y}, \qquad W=z\frac {\partial }{\partial x}-x\frac {\partial }{\partial z}. \]

1. Show \(V\) and \(W\) are tangent to \(S^2\), and hence by restriction define vector fields on \(S^2\).
2. Find the flow \(\Theta _t^V(x,y,z)\) on \(S^2\) generated by \(V\).
3. Compute \((\Theta _h^V)_*W\) on \(S^2\).
4. Compute the Lie Derivative \(\mathcal {L}_VW\) on \(S^2\) in two ways.

Let \(M\) and \(N\) be a smooth orientable manifolds. Show that the maps \(i_a:M\to \mathbb {R}\times M\) given by \(i_a(x)=(a,x)\) and \(\pi :\mathbb {R}\times M\to M\) given by \(\pi (t,x)=x\) induce inverse maps in de Rham Cohomology. Conclude that if smooth maps \(f:M\to N\) and \(g:M\to N\) are smoothly homotopic, then they induce equal maps in de Rham Cohomology.

Suppose that \(M\) and \(N\) are smooth surfaces in \(\mathbb {R}^3\) that intersect at a point \(p\) and that do not have the same tangent plane at that point. Show that \(p\) is not an isolated point of \(M\cap N\).

Let \(\alpha \) be a closed two form on \(S^4\). Prove that \(\alpha \wedge \alpha \) vanishes at some point.

State the homotopy lifting property of covering maps and use it to prove that the covering map \(S^n\to \mathbb {R}\mathrm {P}^n\) from the \(n\)-sphere to the real projective \(n\)-space is not nullhomotopic for any \(n\ge 1\).

Let \(V\) be a continuous vector field on the unit ball \(B^n\subset \mathbb {R}^n\) which is nowhere zero. Prove that there are points \(x,y\in S^{n-1}=\partial B^n\) and positive numbers \(a,b>0\) such that \(V(x)=ax\) and \(V(y)=-by\) (so the vector field points outward at \(x\) and inward at \(y\)).

Let \(X\) be a connected CW complex and \(A\subset X\) a connected subcomplex. Suppose that inclusion \(A\subset X\) is \(\pi _1\)-injective (for some, hence all, basepoints in \(A\)). Let \(p:\widetilde {X}\to X\) be the universal cover of \(X\). Prove that each component of \(p^{-1}(A)\) is simply connected.

Let \(m,n\) be positive integers and view \(S^1\) as the unit circle in the complex plane. Let \(X\) be the space obtained from the cylinder \(S^1\times [0,1]\) by identifying points \((z,0)\) and \((w,0)\) if \(z^n=w^n\), and identifying points \((z,1)\) and \((w,1)\) if \(z^m=w^m\).

1. Use the Mayer-Vietoris sequence to compute \(H_i(X)\) for all \(i\).
2. Use the Seifert-van Kampen theorem to find a presentation for \(\pi _1(X)\).

Suppose \(M\) is a closed connected orientable 5-manifold such that \(H_1(M)=\mathbb {Z}/2\mathbb {Z}\) and \(H_2(M)=\mathbb {Z}\oplus \mathbb {Z}/3\mathbb {Z}\). Compute \(H_i(M)\) and \(H^i(M)\) for all \(i\).