UNIVERSITY OF UTAH DEPARTMENT OF MATHEMATICS Ph. D. Preliminary Examination in Geometry / Topology August 13, 2019.

Let \(M^m\subset \mathbb {R}^n\) be a smooth embedded submanifold. Show that the unit tangent bundle \[ \UM =\{(p,V):p\in M\text { and }V\in T_pM\text { such that }|V|=1\} \] is a smooth embedded submanifold of \(T\mathbb {R}^n\approx \mathbb {R}^n\times \mathbb {R}^n\). What is its dimension?

Consider the Lie Group \(G=\Ogrp (n)=\{A\in \mathcal {M}_n:A^TA=I\}\) where \(\mathcal {M}_n\cong \mathbb {R}^{n^2}\) is the collection of real \(n\times n\) matrices and \(A^T\) is the transpose of \(A\).

1. Describe the set of matrices which form the tangent space \(T_PG\) at the point \(P\in G\).
2. Given \(V\in T_IG\) and \(P\in G\), find \(X_V(P)\in T_PG\) where \(X_V\) is the left invariant vector field on \(G\) such that \(X_V(I)=V\). Show that your \(X_V\) is left invariant.
3. Let \(\theta _t^X\) be the flow for the left invariant vector field \(X=X_V\) for some \(V\in T_IG\). Find and verify a formula for \(\theta _t^X(P)\) in terms of \(P\), \(t\) and \(V\). Compute \(\theta _t^X(I)\) in case \(G=O(2)\) and \[ V=\begin {pmatrix}0&1\\-1&0\end {pmatrix}. \]

Let \(M\) and \(N\) be smooth manifolds. Without using the de Rham Theorem, show that if the de Rham cohomology of \(M\) doesn’t vanish \(H^k_{\mathrm {dR}}(M)\ne \{0\}\) then \(H^k_{\mathrm {dR}}(M\times N)\ne \{0\}\).

Let \(M^m\) be a smooth compact orientable manifold and \(\omega \in \Omega ^m(M)\) a smooth, nonvanishing \(m\)-form that determines the orientation of \(M\). Show that \(\int _M\omega >0\).

Suppose the three sphere \(S^3=\{(x,y,z,w)\in \mathbb {R}^4:x^2+y^2+z^2+w^2=1\}\) is embedded into four space the standard way. Two independent vector fields are given by \(U=(y,-x,w,-z)\) and \(V=(w,z,-y,-x)\). Determine whether it is possible to find a smooth embedded two dimensional submanifold \(N\subset S^3\) such that the vector fields \(U\) and \(V\) are tangent to \(N\) at every point of \(N\), i.e., \(U(p),V(p)\in T_pN\) for every \(p\in N\)?

Let \(X\) be the wedge product of a circle and a projective plane. Find all connected three sheeted covers of \(X\).

Let \(n_1\), \(n_2\) and \(n_3\) be integers. Given an example of a topological space \(X\) with \[ \pi _1(X,x_0)=\langle a,b\mid a^{n_1}b^{n_2}a^{n_3}b\rangle . \] What is \(H_1(X)\)? Under what conditions is \(H_1(X)\) torsion free?

Give an example of a chain complex whose homology and cohomology are different.

For \(n\ge 3\) let \(X=S^{n-1}\times S^1\) be the product of an \((n-1)\)-sphere and a circle and let \(Y\) be obtained by attaching an \(n\)-cell to \(X\) where the attaching map is a degree \(k\) from the boundary of the cell to the \((n-1)\)-sphere \(S^{n-1}\times \{p\}\subset X\) via a degree \(k\) map. Calculate the homology of \(Y\).

Let \(X\) be obtained from attaching two tori along a circle in their product structure. In particular each individual torus is \(S^1\times S^1\) and we attach the two tori along circles \(S^1\times \{p\}\) in the first torus and \(S^1\times \{q\}\) in the second torus. Calculate the cohomology and cup product structure on \(X\) with \(\mathbb {Z}\) coefficients. You can use the cup product structure of a torus as a given. (Hint: \(X\) retracts onto each of the individual tori.)

1. Let \(M\) be a closed, connected, orientable 3-manifold such that \[ H_1(M;\mathbb {Z})=\mathbb {Z}\oplus \mathbb {Z}_2. \] Calculate the remaining homology and cohomology groups with \(\mathbb {Z}\) coefficients.
2. Show that the Euler characteristic of any closed 3-manifold is zero.
3. Now assume that \(M\) is a closed, non-orientable 3-manifold. Show that \(H_1(M;\mathbb {Z})\) is infinite.