DEPARTMENT OF MATHEMATICS University of Utah Ph.D. PRELIMINARY EXAMINATION IN GEOMETRY/TOPOLOGY August 17, 1999, 9:00--12:00, JWB 333

Let \(f:M\to N\) be a submersion between two smooth manifolds (without boundary).

1. Show that \(f\) is an open map, i.e., if \(U\subseteq M\) is open, then \(f(U)\subseteq N\) is open.
2. Show that if \(M\) is closed (i.e. compact without boundary) and \(N\) is connected, then \(f\) is onto.
3. Show that there is no submersion of a closed manifold to \(\mathbb {R}^n\).

Let \(M\) be a smooth manifold and \(p:E\to M\) a smooth vector bundle on \(M\) (with fibers finite-dimensional \(\mathbb {R}\)-vector spaces).

1. Define the notion of a smooth (Euclidean, also called Riemannian) metric on \(E\).
2. Using partitions of unity, show that \(E\) admits a smooth metric.

Prove that the \(n\)-sphere admits a nowhere vanishing vector field if and only if \(n\) is odd.

Show that \(SL_2(\mathbb {R})\), viewed as the subset \[ \{(a,b,c,d)\mid ad-bc=1\} \] of \(\mathbb {R}^4\), is a manifold.

Consider the two vector fields in \(\mathbb {R}^3\) given by \(V(x,y,z)=(x^2+y^2,0,-y)\) and \(W(x,y,z)=(0,x^2+y^2,x)\). Find a two dimensional surface \(M^2\subset \mathbb {R}^3\) containing the point \(p=(1,0,0)\) which is tangent to the vector fields \(V\) and \(W\) in a neighborhood of \(p\), or show that such a surface \(M\) doesn’t exist.

A symmetric quadratic form on \(\mathbb {R}^2\) is given by \[ ds^2=\frac {dx^2+dy^2}{(1+x^2+y^2)^2}. \]

1. Show that \(ds^2\) is a Riemannian metric for \(\mathbb {R}^2\).
2. Compute the length of the curve \(t\mapsto (0,t)\) for \(a\le t\le b\) in this metric.
3. Compute the area of \(\mathbb {R}^2\) in this metric.
4. Show that \((x,y)\mapsto \frac {(x,-y)}{x^2+y^2}\) is an isometry on \((\mathbb {R}^2-\{(0,0)\},ds^2)\).

Let \(\Sigma ^2\) be a (nonempty) smooth closed (compact without boundary) surface smoothly embedded in \(\mathbb {R}^3\). Show that there is a two-plane \(P\) in \(\mathbb {R}^3\) which intersects \(\Sigma \) such that \(P\cap \Sigma \) consists of finitely many disjoint smooth closed loops.

True or false (give a short explanation or a counterexample): Let \(\Sigma \) be a smooth, closed, oriented surface. Assume that \(\Sigma \) has a Riemannian metric of constant curvature \(K\). Suppose that \(\gamma \subset \Sigma \) is a closed embedded geodesic loop which separates \(\Sigma \). (That is, \(\Sigma -\gamma \) has two components.)

1. Then \(K\le 0\).
2. Then \(K\ne 0\).

Let \(M^n\) be a smooth oriented manifold. Suppose \(f:M\to \mathbb {R}\) is a smooth function such that \(df\ne 0\) at each point of \(M\). Show that there is a smooth vector field \(V\) on \(M\) so that \(df(V)\le -1\). Show also, that if \(\phi _t\) is the flow generated by \(V\), then \(f(\phi _1(x))\le f(x)-1\) for all \(x\in M\) such that \(\phi _1(x)\) exists.

Let \(\omega \) be a smooth closed \(p\)-form on \(S^n\) with \(p>0\). Show that there is a smooth \((p-1)\)-form \(\alpha \) so that \(d\alpha =\omega \) if \(p<n\) but that this need not be the case when \(p=n\). You are allowed to quote standard calculations in deRham cohomology.

Let \(T^2=S^1\times S^1\) be the standard 2-torus and \(p,q\in T^2\) be two distinct points. Let \(X=T^2/{\sim }\) be the identification space where the two points are identified \(p\sim q\). Find \(\pi _1(X)\).

Let \(U(n)\) denote the set of unitary matrices (the complex \(n\times n\) matrices \(A\) which satisfy \(A\overline {A}^{T}=I\)). Show that \(U(n-1)\to U(n)\to S^{2n-1}\) is a fiber bundle. (Hint 1: Find explicit local trivializations. Hint 2: State a general theorem whose conclusion is that a smooth map between closed manifolds is a submersion and verify the hypotheses.)

Deduce that \(\pi _r(U(n))\simeq \pi _r(U(n+1))\) for \(r<2n\). What are \(\pi _1(U(3))\), \(\pi _2(U(3))\), \(\pi _3(U(3))\)?

Find the Euler characteristic \(\chi (\mathbb {R}P^3\#(S^1\times S^2))\) of the connected sum.

In #14 and #15 by \(\Sigma _g\) we denote a closed oriented surface of genus \(g\).

Show that every map \(f:\Sigma _2\to \Sigma _3\) has degree 0.

Prove that \(\pi _2(\Sigma _2)=0\).

Prove that \(\mathbb {C}P^2\) and the wedge sum (one point union) \(S^2\vee S^4\) are not homotopy equivalent.

True or false (give a short explanation or a counterexample): The Klein bottle cannot be embedded in \(\mathbb {R}^3\) as a smooth submanifold.

Classify all connected covering spaces of \(\mathbb {R}P^2\vee \mathbb {R}P^2\) up to equivalence.

Describe all possible ways in which the homology of a space can change after attaching an \(n\)-cell.

Let \(M\) and \(N\) be two connected smooth manifolds. Show that \(M\times N\) is orientable if and only if both \(M\) and \(N\) are orientable.