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

Let \(S^n\) be the unit sphere in \(\mathbb {R}^{n+1}\) and \(h:\mathbb {R}^{n+1}\to \mathbb {R}\) the projection to the last coordinate. Prove that the restriction of \(h\) to \(S^n\) is a Morse function and find all critical points and their indices.

Find a perturbation of the identity map \(\mathbb {R}P^3\to \mathbb {R}P^3\) which is a Lefschetz map and compute its Lefschetz number.

Give a proof that \(\mathbb {R}P^2\) is nonorientable.

Compute de Rham cohomology of the 2-torus. You may use the Poincare lemma and the Mayer-Vietoris sequence. For example, you may want to compute \(H^*(S^1)\) first.

Let \(f:\mathbb {R}^n\to \mathbb {R}\) be a smooth function. Show that for every \(\epsilon >0\) there is \(v\in \mathbb {R}^n\) with \(|v|<\epsilon \) such that the function \(g_v:\mathbb {R}^n\to \mathbb {R}\) given by \[ g_v(x)=f(x)+v\cdot x \] is Morse.

The set \(SL_2(\mathbb {R})\) of \(2\times 2\) matrices with determinant 1 can be viewed as a subset of \(\mathbb {R}^4\) by choosing an ordering of the matrix entries. Show that this set is a submanifold of \(\mathbb {R}^4\) and compute the tangent space at the identity matrix.

Consider the vector fields in \(\mathbb {R}^3\): \[ X=x\frac {\partial }{\partial x}+\frac {\partial }{\partial y} \qquad \text {and}\qquad Y=\frac {\partial }{\partial x}-\frac {\partial }{\partial z}. \] Show that there is no nonempty surface \(S\subset \mathbb {R}^3\) that is tangent to both vector fields at each of its points.

Show that for any two points \(x,y\in \mathbb {R}^n\) there is a compactly supported isotopy \(\phi _t\) (i.e. it is identity outside a compact set) such that \(\phi _0=id\) and \(\phi _1(x)=y\).

Define a \(\Delta \)-complex structure on the Klein bottle and compute its homology with \(\mathbb {Z}\) and \(\mathbb {Z}/2\) coefficients.

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

Let \(f:M\to N\) be a map of degree 1 between two smooth, closed, connected, oriented \(n\)-manifolds. Prove that \(f_*:\pi _1(M)\to \pi _1(N)\) is surjective.

Let \(M\) be a closed connected 5-manifold such that \(\pi _1(M)\cong \mathbb {Z}/7\). If \(H_2(M;\mathbb {Z})\cong \mathbb {Z}\), compute all other homology and cohomology groups of \(M\) with integral coefficients.

Let \(F_n\) be the free group of rank \(n\) and let \(G\subset F_n\) be a subgroup of index \(m\). Prove that \(G\) is a free group and compute its rank.

Define carefully a CW structure on \(\mathbb {R}P^n\) (you don’t have to prove it here), and use it to compute \(H_*(\mathbb {R}P^n;\mathbb {Q})\) and \(H_*(\mathbb {R}P^n;\mathbb {Z}/2)\).

Prove that the map \(h:S^3\to \mathbb {C}P^1\) given by \(h(x,y)=[x:y]\) is a fiber bundle. Here, \(S^3\) is the unit sphere \(|x|^2+|y|^2=1\) in \(\mathbb {C}^2\).

Prove that \(\mathbb {R}P^3\) is not homotopy equivalent to \(\mathbb {R}P^2\vee S^3\).