DEPARTMENT OF MATHEMATICS University of Utah Ph.D. PRELIMINARY EXAMINATION IN GEOMETRY/TOPOLOGY August 2007

Let \(S^1\) denote the unit circle in \(\mathbb {C}\) and define \(f:S^1\times S^1\to \mathbb {R}\) by \(f(z,w)=\operatorname {Re}(z)+\operatorname {Re}(w)\). Show that \(f\) is a Morse function and compute the indices of the critical points of \(f\).

Let \(f:\mathbb {R}P^3\to \mathbb {R}P^3\) be defined by \[ f([x_0:x_1:x_2:x_3])=[x_0^2:x_1^2:x_2^2:x_3^2]. \] Compute the degree and the Lefschetz number of \(f\).

Let \(\omega \) be an \(n\)-form on \(\mathbb {R}^n\) with compact support. Show that there exists a compactly supported \((n-1)\) form \(\eta \) with \(\omega =d\eta \) if and only if \[ \int _{\mathbb {R}^n}\omega =0. \] You are allowed to use the standard results about deRham cohomology of \(\mathbb {R}^n\) and spheres.

Let \(f:X\to Y\) be a smooth map between smooth manifolds that is one-to-one on a compact submanifold \(Z\) of \(X\). Suppose that for all \(x\in Z\) \[ df_x:T_x(X)\to T_{f(x)}(Y) \] is an isomorphism. Prove that there is a neighborhood \(U\) of \(Z\) such that \[ f|_U:U\to f(U) \] is a diffeomorphism.

Let \(X\) be any subset of \(\mathbb {R}^n\) and \(f:X\to \mathbb {R}\) a function with the following property. For every \(x\in X\) there is a neighborhood \(U_x\) of \(x\) in \(\mathbb {R}^n\) and a smooth function \(f_x:U_x\to \mathbb {R}\) such that \(f=f_x\) on \(X\cap U_x\). Show that there is an open set \(U\) in \(\mathbb {R}^n\) containing \(X\) and a smooth function \(g:U\to \mathbb {R}\) such that \(g|_X=f\).

Identify the set of all \(n\times n\) matrices with \(\mathbb {R}^{n^2}\) and let \(O(n)\) be the subset consisting of orthogonal matrices (i.e. those that satisfy \(AA^T=I\)). Show that \(O(n)\) is a submanifold of \(\mathbb {R}^{n^2}\) and compute the tangent space at \(I\).

Prove that for any \(x,y\in \mathbb {R}^n\) there a compactly supported isotopy taking \(x\) to \(y\) (i.e. there is an isotopy \(f_t:\mathbb {R}^n\to \mathbb {R}^n\) such that \(f_0=\id \), \(f_1(x)=y\) and \(f_t=\id \) outside some ball).

Suppose \(X\) is a smooth submanifold of \(\mathbb {R}^3\) and a nonempty closed surface. Prove that there is a plane \(P\subset \mathbb {R}^3\) such that \(X\cap P\) is a nonempty collection of circles.

Give an example of an irregular (i.e. not normal) covering space of the Klein bottle (with a proof).

Let \(f:M\to N\) be a covering map between closed connected nonempty manifolds. Suppose \(f\) is homotopic to a constant map. Prove that \(N\) is a point.

Give an example of a space \(X\) such that \(H_1(X;\mathbb {Z})\cong \mathbb {Z}/2\) and \(H_2(X;\mathbb {Z})\cong \mathbb {Z}\).

Suppose \(X\) is a connected 2-dimensional CW complex such that \(H_1(X;\mathbb {Z})\cong \mathbb {Z}/2\) and \(H_2(X;\mathbb {Z})\cong \mathbb {Z}\). Compute \(H_i(X;\mathbb {Z}_2)\), \(H^i(X;\mathbb {Z})\) and \(H^i(X;\mathbb {Z}_2)\) for all \(i\in \mathbb {Z}\).

Prove or disprove:

1. For every \(d\in \mathbb {Z}\) there is a degree \(d\) map \(f_d:T^2\to S^2\).
2. For every \(d\in \mathbb {Z}\) there is a degree \(d\) map \(g_d:S^2\to T^2\).

Let \(f,g:S^2\to S^2\) be two maps. Show that at least one of the following holds:

• there exists \(x\in S^2\) such that \(f(x)=g(x)\),
• there exists \(x\in S^2\) such that \(f(x)=-g(x)\), or
• both \(f\) and \(g\) are homotopic to constant maps.

Prove that \(\mathbb {C}P^3\) and \(S^2\times S^4\) have cell complex structures with isomorphic cellular chain complexes, and therefore \(H^i(\mathbb {C}P^3;\mathbb {Z})\cong H^i(S^2\times S^4;\mathbb {Z})\) for all \(i\). Also prove that cohomology rings \(H^*(\mathbb {C}P^3;\mathbb {Z})\) and \(H^*(S^2\times S^4;\mathbb {Z})\) are not isomorphic. Are \(\mathbb {C}P^3\) and \(S^2\times S^4\) homotopy equivalent? You are allowed to use the standard cell complex structure of spheres and complex projective spaces.

Compute \(\pi _2(X)\) where \(X\) is the wedge sum of \(S^1\) and \(S^2\).