Ph.D. PRELIMINARY EXAMINATION IN GEOMETRY/TOPOLOGY

Let \(A\) and \(B\) be disjoint closed subsets of a topological space \(X\). Then there exists a continuous function \(f:X\to [0,1]\) so that \(f(A)=0\) and \(f(B)=1\).

Let \(K\subset S^3\) be a knot (a homeomorph of \(S^1\)). Then \(\pi _1(S^3-K,x_0)=\mathbb {Z}\).

\[ \pi _k\left (\underbrace {S^1\times \cdots \times S^1}_{n\ \mathrm {times}},x_0\right ) = \begin {cases} \mathbb {Z}^n, & \text {if } k=1,\\ 0, & \text {if } k>1. \end {cases} \]

Let \(K\) denote a finite simplicial complex. The Euler characteristic is given by \[ \chi (K)=\sum _{j\ge 0}(-1)^j\alpha _j \] where \(\alpha _j\) is the number of \(j\)-simplices in \(K\). If \(\mathrm {Sd}\,K\) denotes barycentric subdivision then \(\chi (K)=\chi (\mathrm {Sd}\,K)\).

If \(X\) and \(Y\) are compact connected smooth \(n\)-manifolds then for \(0<k<n\), the singular homology satisfies \[ H_k(X\times Y,\mathbb {Z})=H_k(X,\mathbb {Z})\oplus H_k(Y,\mathbb {Z}). \]

\(\mathrm {U}(n)\), the space of complex \(n\times n\) unitary matrices, has the structure of a smooth real manifold.

Let \(X\) be a finite connected graph. Then the fundamental group \(\pi _1(X,x_0)\) is a free group with \(1-\chi (X)\) generators. (\(\chi \) is the Euler characteristic.)

Let \(\{C_n,d_n\}\) and \(\{C'_n,d'_n\}\) be chain complexes of abelian groups and \(f_n:C_n\to C'_n\) be a chain homotopy equivalence. Then the induced map \(f_{n*}:H_n(C)\to H_n(C')\) is isomorphism in homology.

Suppose a smooth oriented manifold \(M^n\) has a Riemannian metric which is given in any local coordinate system by \[ ds^2=\sum _{i,j=1}^{n}g_{ij}(x)\,dx^i\otimes dx^j. \] Then the form given locally in the same coordinate systems by \[ \omega =\det (g_{ij}(x))\,dx^1\cdots dx^n \] represents a globally defined nowhere vanishing \(n\)-form on \(M\).

Let \(f\in C^\infty (S^n)\) be a smooth function and let \(x_1,x_2\in S^n\) be two points such that \(f(x_1)<0<f(x_2)\). Let \(\omega \) be a smooth nonvanishing \(n\)-form. Then there is a diffeomorphism \(\varphi :S^n\to S^n\) such that \[ \int _{S^n}(\varphi ^*f)\,\omega =0. \]

II. Ten point questions.

Let \(\Sigma \) denote the surface obtained from an annulus by identifying antipodal points of the inner circle and by identifying antipodal points of the outer circle. Compute the fundamental group of \(\Sigma \). Can you identify \(\Sigma \) as a connected sum of tori and projective planes?

Let \(X\) be a connected, locally arcwise connected, semilocally simply connected, compact metric space. Show that \(\pi _1(X,x_0)\) is finitely generated.

Let \(S^n\) denote the unit sphere of \(\mathbb {R}^{n+1}\) and let \(a:S^n\to S^n\) denote the antipodal map \(a(x)=-x\). Show that \(S^n\) admits a nonvanishing tangent vector field if and only if \(a\) is homotopic to the identity. For which values of \(n\) does this happen?

Find the singular homology \(H_*(S^5-(A\cup B),\mathbb {Z})\) where \(A\) is a homeomorph of \(S^2\), \(B\) a homeomorph of \(S^3\), and \(A\cap B=\{z_0\}\) is a single point.

Suppose that maps between three spheres forms a fibration. Show that \(k=n-1\).

Diagram description: \(S^k\) maps to \(S^{n+k}\) by \(\iota \), and \(S^{n+k}\) maps down to \(S^n\) by \(p\).

By giving examples, show that such a fibration is indeed possible for some \(n\).

Find the homology, cohomology and homotopy groups

a. \(H_*(\mathbb {R}P^3,\mathbb {Z})\);

b. \(H_*(\mathbb {R}P^3,\mathbb {Z}/2\mathbb {Z})\);

c. \(H^*(\mathbb {R}P^3,\mathbb {Z})\);

d. \(\pi _k(\mathbb {R}P^3,x_0)\), for \(k=1,2,3\).

Compute the deRham cohomology of \(S^2\) in four distinct ways. You may assume the singular cohomology groups \(H^*(S^2,\mathbb {R})\). Hint: one way could be from first principles using the Poincaré Lemma.

The surface parameterized by \(X:(u,v)\mapsto (\cos u,\sin u,2\cos v,2\sin v)\) is a torus in \(\mathbb {R}^4\) with the induced metric \(ds^2=X^*ds^2_{\mathbb {R}^4}\). Show that this torus cannot be isometrically immersed into \(\mathbb {R}^3\).

Let \(M^2\) be a smooth closed connected orientable surface with Gauss curvature \(K(x)<0\) and genus \(g\). Let \(\{\gamma _1,\ldots ,\gamma _\ell \}\) be a system of pairwise nonintersecting closed geodesics of \(M\). (A closed geodesic is a smoothly embedded loop \(\gamma _i:S^1\to M\) whose geodesic curvature vanishes identically \(\kappa _i=0\).) Show that \(\ell \le 3g-3\). (Hint: can two nonintersecting closed geodesics be freely homotopic?)

Let \(f\) be a smooth function on a smooth manifold \(M^n\). A critical point \(x_0\in M\) is nondegenerate if in some coordinate system the Hessian matrix of second partials \[ \frac {\partial ^2 f}{\partial x^i\partial x^j}(x_0) \] is nonsingular at \(x_0\). A function all of whose critical points are nondegenenerate is called a Morse function. Show that the critical points of a Morse function are isolated. Given any smooth function \(f\in C^\infty (M)\), show that for every \(\epsilon >0\) there is a Morse function \(g\in C^\infty (M)\) with \[ \sup _{x\in M}|f(x)-g(x)|<\epsilon . \]

Let \(M\subset \mathbb {R}^{n+1}\) be the graph \(x^{n+1}=f(x^1,\ldots ,x^n)\) where \(f\in C^\infty (\Omega )\) and \(\Omega \subset \mathbb {R}^n\) is a convex open set. Suppose the Hessian of \(f\) has constant rank i.e. for some \(k\) such that \(0\le k<n\) and every \(x\in \Omega \), \[ \operatorname {rank}\left (\frac {\partial ^2 f(x)}{\partial x^i\partial x^j}\right )=k. \] Show that \(M\) is foliated by \((n-k)\) dimensional manifolds whose tangent vectors lie in \(\ker (dG)\) where \(G:M\to \mathbb {R}^n\) is given by \[ G(x)=\left (\frac {\partial f}{\partial x^1},\ldots , \frac {\partial f}{\partial x^n}\right ). \] Show that a connected component \(L\) of such a “leaf” manifold locally coincides with points where \(G\) is constant. In fact, the graph of \(L\) is an affine subspace of \(\mathbb {R}^{n+1}\) restricted to \(\Omega \times \mathbb {R}\).