DEPARTMENT OF MATHEMATICS University of Utah Ph.D. PRELIMINARY EXAMINATION IN GEOMETRY/TOPOLOGY August 2012

\(13x\,dx+y^2\,dy+xyz\,dz\) is a form on \(\mathbb {R}^3\). What’s its exterior derivative?

Let \(f:\mathbb {R}^3\to \mathbb {R}^2\) be the function \(f(x,y,z)=(xy,z)\). Find the pullback form \(f^*(dy\wedge dz+x^2\,dx\wedge dy)\).

Let \(g:\mathbb {R}^3\to \mathbb {R}^3\) be the function \(g(x,y,z)=(x^2y,3xz,y+z)\). Find \(D_{(2,3,5)}g:\mathbb {R}^3\to \mathbb {R}^3\), the derivative of \(g\) at the point \((2,3,5)\).

Let \(h:\mathbb {R}^3\to \mathbb {R}\) be the function \(h(x,y,z)=xyz\). Let \(s\) be the vector field on \(\mathbb {R}^3\) given by \[ s(x,y,z)=xy\frac {\partial }{\partial x} +(y-z^3)\frac {\partial }{\partial y} +3\frac {\partial }{\partial z}. \] Find \(L_s(f)(1,1,2)\), the Lie derivative of \(f\) in the direction of \(s\) at the point \((1,1,2)\).

Find the following bracket of two vector fields in \(\mathbb {R}^3\) \[ \left [ (x+y)\frac {\partial }{\partial x}+z\frac {\partial }{\partial y}, yz\frac {\partial }{\partial y}+x^2\frac {\partial }{\partial z} \right ]. \]

Let \(S^2\) be the vectors of length 1 in \(\mathbb {R}^3\). Find \(\int _{S^2}dx\wedge dy\).

Let \(\Gamma \) be a group of diffeomorphisms acting on a smooth manifold \(M\). When is \(\Gamma \backslash M\) a manifold?

Let \(f:M\to N\) be a smooth map of smooth manifolds. If \(Q\subseteq N\) is an embedded submanifold, and \(f\) is transverse to \(Q\), prove that \(f^{-1}(Q)\) is a manifold.

Let \(M\) be a compact, connected smooth manifold, and suppose that \(N\) is a connected smooth manifold. If \(F:M\times [0,1]\to N\) is smooth and \(m\mapsto F(m,0)\) is an immersion of \(M\) into \(N\), then show there is some \(\varepsilon >0\) such that \(m\mapsto F(m,\delta )\) is an immersion of \(M\) into \(N\) for any fixed \(\delta <\varepsilon \).

Let \(M\) be a smooth compact manifold and let \(X\) be a smooth vector field with a corresponding 1-parameter flow group \(\{\theta _t^X:M\to M\}_{t\in \mathbb {R}}\). Let \(f\) be a diffeomorphism of \(M\) and let \(f_*X\) be its pushforward. Show that the flow group for \(f_*X\) is \(f\circ \theta _t^X\circ f^{-1}\).

Let \(f\) be the homeomorphism of the annulus \(S^1\times [0,1]\) given by \[ f(z,s)=(ze^{2\pi is},s) \] where we view \(S^1\) as the set of unit norm complex numbers.

1. Construct an explicit isotopy (i.e. homotopy through homeomorphisms) between \(f\) and the identity that does not move the points of \(S^1\times \{0\}\).
2. Prove that there is no homotopy between \(f\) and the identity that does not move points on both \(S^1\times \{0\}\) and \(S^1\times \{1\}\).

Consider the disk with 2 holes \[ P=\{(x,y)\in \mathbb {R}^2\mid \lVert (x,y)\rVert \le 4,\, \lVert (x,y)-(2,0)\rVert \ge 1,\, \lVert (x,y)+(2,0)\rVert \ge 1\}. \] Orient each boundary component counterclockwise. Let \(X\) be the space obtained from \(P\) by identifying all boundary components via orientation preserving homeomorphisms. Find a presentation of \(\pi _1(X)\).

Give a definition of the chain homotopy between chain maps and prove that chain homotopic chain maps induce the same homomorphism in homology.

A map \(f:S^n\to S^n\) is said to be even if \(f(-x)=f(x)\) for every \(x\in S^n\). Show that even maps have even degree, and in fact that the degree must be 0 when \(n\) is even.

1. Describe the simplest cell structures on \(\mathbb {C}\mathrm {P}^2\) (including attaching maps).
2. Prove that \(H_i(\mathbb {C}\mathrm {P}^2;G)\) is isomorphic to \(H_i(S^2\vee S^4;G)\) for every abelian group \(G\) and all \(i\), and the same is true for cohomology.
3. Prove that \(\mathbb {C}\mathrm {P}^2\) is not homotopy equivalent to \(S^2\vee S^4\) by considering ring structures in cohomology.

Prove that when \(g<h\) every map \(S_g\to S_h\) between orientable closed surfaces of genus \(g\) and \(h\) respectively is equal to 0.