DEPARTMENT OF MATHEMATICS University of Utah Ph.D. PRELIMINARY EXAMINATION IN GEOMETRY/TOPOLOGY January 4, 2006, 9:00--12:00, LCB 215

Let \(X\) be a manifold and \(f:X\to \mathbb {R}^k\) a continuous function. Show that for every \(\epsilon >0\) there is a smooth function \(g:X\to \mathbb {R}^k\) such that \[ \|g(x)-f(x)\|<\epsilon \] for every \(x\in X\).

Let \(\mathcal {M}_{n\times n}\) be the set of all real \(n\times n\) matrices. This set is naturally a manifold, since it can be identified with \(\mathbb {R}^{n^2}\) by choosing an ordering of entries. Let \(O(n)\subset \mathcal {M}_{n\times n}\) be the set of orthogonal matrices, i.e. matrices \(A\) with \(AA^T=I\). Prove that \(O(n)\) is a submanifold of \(\mathcal {M}_{n\times n}\). What is the dimension of \(O(n)\)?

1. Give a careful definition of the tangent bundle of a manifold \(X\).
2. Show that the total space of the tangent bundle of \(S^2\) is not diffeomorphic to \(S^2\times \mathbb {R}^2\).

1. State the Lefschetz fixed point theorem (for Lefschetz maps).
2. Compute the Lefschetz number of the identity map \(id:\mathbb {C}P^2\to \mathbb {C}P^2\) by first perturbing it to a Lefschetz map and then applying the Lefschetz fixed point theorem.

1. State the Frobenius integrability theorem.
2. Show that the plane field in \(\mathbb {R}^3\) spanned by the vector fields \[ X=\frac {\partial }{\partial x}+z\frac {\partial }{\partial y} \qquad \text {and}\qquad Y=\frac {\partial }{\partial z} \] is not integrable on any nonempty open subset of \(\mathbb {R}^3\).

Let \(\omega \) be a compactly supported smooth \(n\)-form on \(\mathbb {R}^n\). Prove that the following two statements are equivalent.

1. There exists a compactly supported smooth \((n-1)\)-form \(\eta \) such that \(\omega =d\eta \).
2. \(\displaystyle \int _{\mathbb {R}^n}\omega =0\).

Note: You are allowed to use the fact that \(H^n(S^n)=\mathbb {R}\), but you are not allowed to quote the fact that \(H_c^n(\mathbb {R}^n)=\mathbb {R}\).

1. Let \(a:S^n\to S^n\) be the antipodal map. Suppose that \(\omega \) is a smooth form on \(S^n\) such that \(a^*\omega =\omega \). Prove that if \(\omega \) is exact, then there is a smooth form \(\eta \) with \(\omega =d\eta \) and \(a^*\eta =\eta \).
2. Use (a) and the fact that \(H^k_{\mathrm {DR}}(S^n)=0\) for \(0<k<n\) to deduce that \(H^k_{\mathrm {DR}}(\mathbb {R}P^n)=0\) for \(0<k<n\).

1. State the tubular neighborhood theorem for a compact submanifold \(X\subset \mathbb {R}^n\).
2. Give an outline of the proof with details of the step that shows that the constructed map is a diffeomorphism.

Two covering spaces \(p_0:X_0\to X\) and \(p_1:X_1\to X\) of a space \(X\) are isomorphic if there is a homeomorphism \(h:X_0\to X_1\) such that \(p_0=p_1\circ h\). In this question all spaces are path-connected, locally path-connected and semilocally path-connected.

1. Prove the following piece of the classification theorem for covering spaces: Let \(x\in X\), \(x_0\in X_0\) and \(x_1\in X_1\) be basepoints such that \(p_0(x_0)=p_1(x_1)=x\). The covers are isomorphic if and only if \((p_0)_*(\pi _1(X_0,x_0))\) and \((p_1)_*(\pi _1(X_1,x_1))\) are conjugate subgroups of \(\pi _1(X,x)\).
2. Show that there are non-isomorphic covers of a genus two surface of every degree.

Let \(M\) be an open 3-manifold such that every compact set of \(M\) is contained in a solid torus.

1. Show that \(\pi _1(M)\) is abelian.
2. Assume that \(\pi _1(M)\) is finitely generated. Show that \(\pi _1(M)\) is trivial or \(\mathbb {Z}\).

Let \(c\) be a separating simple closed curve on a compact surface \(S\) without boundary. Show that \(S\) doesn’t retract onto \(c\).

Construct a \(\Delta \)-complex structure for a genus two surface. Use the \(\Delta \)-complex to calculate the cohomology and cup product structure for the surface.

Show that \(S^2\vee S^4\) and \(\mathbb {C}P^2\) are not homotopy equivalent.

Let \(X_d\) be the CW-complex obtained by attaching a disk with a degree \(d\) map to the circle \(S^1\times \{x_0\}\) on the torus \(S^1\times S^1\) (and \(x_0\in S^1\) is a basepoint). Calculate the homology and cohomology of \(X_d\) with \(\mathbb {Z}\), \(\mathbb {R}\) and \(\mathbb {Z}_p\) coefficients where \(p\) is prime.

Let \(M_0\) and \(M_1\) be orientable closed \(n\)-manifolds and \(f:M_0\to M_1\) a map such that \(f^*:H^n(M_1;\mathbb {R})\to H^n(M_0;\mathbb {R})\) is an isomorphism.

1. Assume that \(H^i(M_0;\mathbb {R})\cong H^i(M_1;\mathbb {R})\) for all \(i\). Show that \(f^*\) is an isomorphism on cohomology with \(\mathbb {R}\) coefficients in all dimensions.
2. Assume that there exists a map \(g:M_1\to M_0\) such that \(g^*:H^n(M_0;\mathbb {R})\to H^n(M_1;\mathbb {R})\) is an isomorphism. Show that \(f^*\) and \(g^*\) are isomorphisms on cohomology with \(\mathbb {R}\) coefficients in all dimensions.

Define the higher homotopy groups, \(\pi _i\), for \(i\ge 2\) and prove that they are abelian.