DEPARTMENT OF MATHEMATICS University of Utah Ph.D. PRELIMINARY EXAMINATION IN GEOMETRY/TOPOLOGY January, 2005, 9:00--12:00, JWB 333

Let \[ X=\{(x,y)\in \mathbb {R}^3\times \mathbb {R}^3 \mid x\cdot x=y\cdot y=1,\ x\cdot y=0\}. \] Show that \(X\) is a submanifold of \(\mathbb {R}^3\times \mathbb {R}^3\).

Prove that the tangent bundle \(TM\) of a smooth manifold \(M\), viewed as a smooth manifold, is orientable.

Let \(f:M\to N\) be a map between two closed connected oriented manifolds. If \(f\) has degree 1, show that \(f_*:\pi _1(M)\to \pi _1(N)\) is an epimorphism.

Let \(M\) be a manifold and \(\Delta \) a smooth \(k\)-plane field on \(M\). Suppose that \(X\) and \(Y\) are smooth vector fields on \(M\) with values in \(\Delta \). Show that if for some \(p\in M\) we have \(X_p=0\) then \([X,Y]_p\in \Delta \).

Let \[ X=y\frac {\partial }{\partial x}-(x+1)\frac {\partial }{\partial y} \] and \[ Y=y\frac {\partial }{\partial x}-(x-1)\frac {\partial }{\partial y} \] be two vector fields in \(\mathbb {R}^2\). Compute \([X,Y]\).

Let \(X\) be the standard “middle thirds” Cantor set in \(\mathbb {R}\), viewed as a subset of the plane \(\mathbb {R}^2\). Assume that \(f:X\to \mathbb {R}\) is a function with the property that for every \(x\in X\) there is an open set \(U_x\ni x\) in \(\mathbb {R}^2\) and a smooth function \(f_x:U_x\to \mathbb {R}\) such that \(f_x|_{U_x\cap X}=f|_{U_x\cap X}\). Show that there is an open set \(U\) in \(\mathbb {R}^2\), \(U\supset X\), and a smooth function \(g:U\to \mathbb {R}\) such that \(g|_X=f\).

Prove or give a counterexample (with proofs): If \(M\) is a closed submanifold of \(\mathbb {R}^n\) then its tangent bundle is trivial if and only if its normal bundle is trivial.

Let \(f:\mathbb {R}^n\to \mathbb {R}\) be a smooth function. Show that for almost every \(a\in \mathbb {R}^n\) the function \[ f_a:\mathbb {R}^n\to \mathbb {R} \] defined by \[ f_a(x)=f(x)+x\cdot a \] is Morse.

Let \[ \omega =(x^2+y^2+z^2)^\alpha (x\,dy\wedge dz+y\,dz\wedge dx+z\,dx\wedge dy) \] be a 2-form in \(\mathbb {R}^3\setminus \{0\}\).

1. Find \(\alpha \in \mathbb {R}\) such that \(\omega \) is closed.
2. Show that \(\omega \) is not exact.

Let \(X\) and \(Y\) be two closed oriented manifolds. Prove that Euler characteristic is multiplicative, i.e. that \[ \chi (X\times Y)=\chi (X)\times \chi (Y). \] You may not appeal to the Kunneth formula.

Find a non-regular cover of \(S^1\vee S^1\), the wedge of 2 circles. You should both explicitly describe the cover and the corresponding subgroup of \(\pi _1(S^1\vee S^1)\).

Let \(X\) be the space obtained from two tori, \(S^1\times S^1\), by gluing \(S^1\times \{pt\}\) in one torus to \(S^1\times \{pt\}\) in the other torus with a degree two map. Calculate \(\pi _1(X)\).

Let \(K\) be the \(\Delta \)-complex obtained by identifying all the 2-dimensional faces of a 3-simplex. Calculate the simplicial homology of \(K\).

Let \[ f:S^1\times S^1\to S^1\times S^1 \] be map of the torus to itself. Since \(H_1(S^1\times S^1)=\mathbb {Z}\oplus \mathbb {Z}\), after choosing generators for \(H_1(S^1\times S^1)\) we can represent \[ f_*:H_1(S^1\times S^1)\to H_1(S^1\times S^1) \] as a \(2\times 2\) matrix \(A\) with integer entries. If \(f\) does not have fixed points what are the possibilities for the trace and determinant of \(A\) and the degree of \(f\)? For each possible triple give an example of a fixed point free map.

Construct examples of CW-complexes \(X\) with each of the following properties.

1. For some \(n\), \(H_n(X)\not \cong H^n(X;\mathbb {Z})\).
2. For some \(n\), \(H_n(X;\mathbb {Z}_2)\ne 0\) and \(H_n(X;\mathbb {Z}_3)\ne 0\) but \(H_n(X;\mathbb {Z}_5)=0\).

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

Let \(M\) be a compact manifold with boundary. Show that \(M\) does not retract onto \(\partial M\).

Let \(M\) be a closed, orientable manifold \(M\) of dimension \(2k\). If \(H_{k-1}(M)\) is torsion free show that \(H_k(M)\) is torsion free.

1. Let \(X\) be a CW-complex and \(A\) a sub-complex. Assume that all cells of \(X-A\) have dimension \(>n\). If \(X\) retracts to \(A\) show that \(\pi _n(X)=\pi _n(A)\).
2. Show that \(\mathbb {R}P^n\) does not retract to \(\mathbb {R}P^k\) if \(k<n\).

Calculate \(\pi _3(S^2)\).