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

Let \(M\) be the smooth submanifold of \(\mathbb {R}^3\) consisting of points that satisfy \(x^2+y^2-z^2=1\). Let \(N\) be the smooth submanifold of \(\mathbb {R}^3\) consisting of points that satisfy \(x+y+z=0\). Prove that \(M\cap N\) is a smooth manifold.

Let \(M\) be a smooth manifold containing a point \(p\), and let \(C_p^\infty (M)\) be the germs of smooth real-valued functions on \(M\) at the point \(p\). Prove, using only the definition of a derivation, that \(\delta ([\lambda ])=0\) for any constant \(\lambda \in \mathbb {R}\) and any derivation \(\delta :C_p^\infty (M)\to \mathbb {R}\).

Let the inclusion of \(\SO (2)\) into \(\SO (3)\) be induced by the inclusion of \(\mathbb {R}^2\) into \(\mathbb {R}^3\). Find all connected, 2-dimensional Lie subgroups of \(\SO (3)\) that contain \(\SO (2)\).

Let \[ A= \begin {pmatrix} 0 & 1 & 2\\ 0 & 0 & 3\\ 0 & 0 & 0 \end {pmatrix} \] Find \(\exp (A)\).

Let \(M\) and \(N\) be smooth manifolds, and let \(f:M\to N\) be smooth. Prove that the set of all \(p\in M\) for which the differential \(D_pf:T_pM\to T_{f(p)}N\) is injective is an open set.

Let \(f:M\to N\) and \(g:N\to P\) be diffeomorphisms of smooth manifolds. Let \(\sigma (f)\) equal 1 if \(f\) preserves orientation and \(-1\) otherwise. Defining \(\sigma (g)\) and \(\sigma (g\circ f)\) similarly, prove that \(\sigma (g\circ f)=\sigma (g)\sigma (f)\).

Let \(X\) be a path-connected topological space.

1. Let \(x_0,x_1\) be points in \(X\), and let \(\alpha :I\to X\) be any path from \(x_0\) to \(x_1\). Explain how \(\alpha \) induces an isomorphism of groups \(\alpha _*:\pi _1(X,x_0)\xrightarrow {\sim }\pi _1(X,x_1)\).

2. Give an example where \(x_1=x_0\) (so \(\alpha \) is a loop at \(x_0\)), but the map \(\alpha _*:\pi _1(X,x_0)\to \pi _1(X,x_0)\) is not the identity map.
3. Give an example of a space \(X\) with \(H_1(X;\mathbb {Z})=0\) but \(\pi _1(X,x_0)\ne 0\). State carefully the results you are using.

Let \(n\ge 1\) be a positive integer.

1. Give an elementary description of the universal cover \(X\to \mathbb {R}\mathrm {P}^n\) of \(\mathbb {R}\mathrm {P}^n\).
2. Describe a CW structure on \(\mathbb {R}\mathrm {P}^n\).
3. Compute the cohomology ring \(H^*(\mathbb {R}\mathrm {P}^n;\mathbb {Z}/2)\) and the map (from (a)) \(H^*(\mathbb {R}\mathrm {P}^n;\mathbb {Z}/2)\to H^*(X;\mathbb {Z}/2)\).

1. State the van Kampen theorem for fundamental groups. Use it to compute, \(\pi _1(K,*)\), where \(K\) is the Klein bottle, and \(\pi _1(S_g,*)\), where \(S_g\) is the orientable surface of genus \(g\). (Here \(*\) is any base-point.)
2. Show that \(S_g\) is indeed orientable; carefully state the results you are using.

Let \(C_\bullet \) be a chain complex of finitely-generated \(\mathbb {Z}\)-modules such that \(C_i=0\) for all \(|i|\gg 0\) (i.e., the complex \(C_\bullet \) is bounded). Define the Euler characteristic of \(C\) to be \[ \chi (C)=\sum _{i\in \mathbb {Z}}(-1)^i\operatorname {rank}(C_i), \] where \(\operatorname {rank}(C_i)\) denotes the rank of the (finitely-generated) abelian group \(C_i\).

1. Let \(X\) be a finite CW complex, i.e. one having finitely many cells in total. Define \(\chi (X)\) to be \(\chi (C_\bullet ^{\mathrm {CW}}(X))\), where \(C_\bullet ^{\mathrm {CW}}(X)\) denotes the cellular chain complex of \(X\). Show that \[ \chi (X)=\sum _{i\in \mathbb {Z}}(-1)^i\operatorname {rank}(H_i(X)), \] and that this quantity equals \(\sum _{i\in \mathbb {Z}}(-1)^i\dim _F(H_i(X;F))\) for any field \(F\).
2. Let \(X\) be a connected compact manifold of odd dimension; you may assume \(X\) is homeomorphic to a finite CW complex. Show that \(\chi (X)=0\).