DEPARTMENT OF MATHEMATICS University of Utah Ph.D. PRELIMINARY EXAMINATION IN GEOMETRY/TOPOLOGY August 18, 2004, 9:00--12:00, JWB 333

Let \(M\subset \mathbb {R}^n\) be a compact manifold without boundary. Prove that there is \(\epsilon >0\) such that for any manifold \(X\) the following holds: any two smooth maps \[ f,g:X\to M \] with \(|f(x)-g(x)|<\epsilon \) for every \(x\in X\) are homotopic. Find an explicit \(\epsilon >0\) for the case when \(M\) is the unit sphere \(S^m\).

Regard the real projective plane \(\mathbb {R}P^2\) as the space of triples \((x,y,z)\in \mathbb {R}^3\setminus \{0\}\) modulo the relation \((x,y,z)\sim (tx,ty,tz)\) for \(t\in \mathbb {R}\setminus \{0\}\). Define \[ f:\mathbb {R}P^2\to \mathbb {R} \] by \[ f(x,y,z)=\frac {x^2+2y^2}{x^2+y^2+z^2}. \] Compute all critical points of \(f\). Show that \(f\) is a Morse function and compute the Morse index of each critical point. (Hint: Work in local coordinates, in standard charts \(U_0,U_1,U_2\) where \(U_0=\{x\ne 0\}\) etc.)

Compute the Lefschetz number of the mapping \[ f_k:\mathbb {C}P^n\to \mathbb {C}P^n \] given by \[ [x_0:x_1:\cdots :x_n]\mapsto [x_0^k:x_1^k:\cdots :x_n^k] \] for \(k=1,2,\cdots \).

Let \(p\) be a homogeneous polynomial in \(k\) real variables. This means that \[ p(tx_1,tx_2,\cdots ,tx_k)=t^m p(x_1,x_2,\cdots ,x_k) \] for all \(t,x_1,x_2,\cdots ,x_k\in \mathbb {R}\). Prove that for \(a\ne 0\) the set \[ \{x\in \mathbb {R}^k\mid p(x)=a\} \] is a submanifold of \(\mathbb {R}^k\).

Let \(X\) and \(Y\) be two submanifolds of \(\mathbb {R}^n\). Show that for almost every \(a\in \mathbb {R}^n\) the translate \[ X+a=\{x+a\mid x\in X\} \] is transverse to \(Y\).

Prove that the Euler characteristic of the orthogonal group \(O(n)\) is 0 for \(n\ge 1\).

Let \(V\) be the vector field in the complex plane \(\mathbb {C}\) given by \[ V(z)=z^m \] for some \(m=1,2,\cdots \), where we use the standard identification \(T_z\mathbb {C}=\mathbb {C}\). Compute the index of \(V\) at the origin.

Let \(\omega \) be the 1-form on \(\mathbb {R}^2-\{0\}\) given by \[ \omega (x,y)=\frac {-y\,dx}{x^2+y^2}+\frac {x\,dy}{x^2+y^2}. \]

1. Calculate \(\int _C\omega \) for any circle \(C\) centered at the origin.
2. Prove that in the half-plane \(x>0\) there is a smooth function \(f\) such that \(\omega =df\).
3. Prove that there is no smooth function \(f\) defined on \(\mathbb {R}^2-\{0\}\) such that \(\omega =df\).

Give an example (with proof) of a nonintegrable plane field.

Compute de Rham cohomology of the 3-torus \(S^1\times S^1\times S^1\).

Let \(P\) be a discrete set of points in \(\mathbb {R}^n\). Show that \(\mathbb {R}^n\setminus P\) is simply connected if \(n\ge 3\).

Find all connected covering spaces of \(\mathbb {R}P^2\vee S^1\) of degree less than three. Which of these coverings are regular?

Construct a \(\Delta \)-complex for the Klein bottle and calculate its simplicial homology for both \(\mathbb {Z}\) and \(\mathbb {Z}_2\) coefficients.

Compute \(H_i(S^n-X)\) when \(X\) is a subspace of \(S^n\) homeomorphic to \(S^k\vee S^\ell \).

Construct a homeomorphism \(f:S^1\times S^1\to S^1\times S^1\) that is not homotopic to a map without fixed points.

Let \(X\) be the space obtained by attaching a disk \(D\) to \(S^1\times S^1\) where the attaching map is a degree \(d\) map from \(\partial D\) to \(S^1\times \{pt\}\). Calculate the homology and cohomology groups for \(X\) with \(\mathbb {Z}\) and \(\mathbb {Z}_p\) coefficients where \(p\) is prime.

Let \(S_g\) be the closed, orientable surface of genus \(g\). Show that there exists a map \(f:S_g\to S_h\) with \[ f_*:H_2(S_g)\to H_2(S_h) \] non-trivial if and only if \(g\ge h\).

Let \(X=S(S^1\times S^1)\) be the suspension of the torus and \(Y=SX\) the suspension of \(X\). Show that \(X\) and \(Y\) aren’t manifolds.

Let \(X\) be a connected CW-complex obtained by attaching an \((n+2)\)-dimensional cell to an \(n\)-dimensional CW-complex \(A\). What is the smallest possible dimension \(k\) where \(\pi _k(X)\) is not isomorphic to \(\pi _k(A)\)?

1. Let \(S_g\) be the closed, orientable surface of genus \(g\). Calculate \(\pi _2(S_g)\) for all \(g\).
2. Show that \(\pi _3(S_0)\) is non-trivial. (\(S_0\) is the 2-sphere.)