Ph. D. PRELIMINARY EXAMINATION IN TOPOLOGY/GEOMETRY

Let \(S_1\) and \(S_2\) be two (2-dimensional) surfaces. Derive the relationship between the Euler characteristic of the connected sum \(S_1\#S_2\) and the Euler characteristics of \(S_1\) and \(S_2\).

Suppose that \(X\) is a topological space, \(A\subset X\) is a deformation retract which is path connected. Prove that \(X\) is path-connected as well.

Let \(G\) be a subgroup of index 3 in the free group of rank 2. Prove that \(G\) is free and find its rank.

Suppose \(M\) is a connected manifold. Calculate \(H_1(M\times S^1)\) in terms of the homology groups of \(M\).

Find a cell-complex structure for the Klein bottle.

Suppose that \(p:\widetilde {X}\to X\) is a covering map between connected manifolds. Prove that if \(p\) is null-homotopic, then \(\widetilde {X}\) is contractible.

Show that every continuous map \(f:S^n\to S^n\) (\(n\ge 1\)) of degree 2 has a fixed point.

Let \(\Delta \) be the 2-dimensional distribution in \(\mathbb {R}^3\) defined by \[ \Delta =\operatorname {span}\left \{ \frac {\partial }{\partial x^1}+x^2\frac {\partial }{\partial x^3}, \frac {\partial }{\partial x^3}+x^3\frac {\partial }{\partial x^2} \right \}. \] Determine whether this distribution is tangent to a family of two-dimensional surfaces.

Let \((M^n,g)\) be a Riemannian manifold. Show that the form given locally by \[ \omega =\sqrt {g}\,dx^1\wedge dx^2\wedge \cdots \wedge dx^n \] represents a globally-defined, nowhere vanishing \(n\)-form. (Here \(\sqrt {g}\) is the standard shorthand for the square root of the determinant of the matrix \([g_{ij}]=[\langle \frac {\partial }{\partial x^i},\frac {\partial }{\partial x^j}\rangle ]\) which gives the metric \(g\) in local coordinates.)

Calculate \(\pi _2(\mathbb {R}P^3)\).

II. Pick four of the following seven questions to answer

Let \(M_2(\mathbb {R})\) be the set of all \(2\times 2\) matrices with real coefficients, and identify it with \(\mathbb {R}^4\) in the usual way. Let \(SL(2,\mathbb {R})\) be the subset of \(M_2(\mathbb {R})\) consisting of those matrices with determinant equal to one.

(a) Show \(SL(2,\mathbb {R})\) is a submanifold of \(M_2(\mathbb {R})\).

(b) Exhibit the tangent space to \(SL(2,\mathbb {R})\) at the identity matrix.

Let \(f:M\to N\) and \(g:N\to M\) be smooth maps, where \(M\) and \(N\) are compact connected smooth oriented manifolds of the same dimension. If \(g\circ f\) is a diffeomorphism, either prove that both \(f\) and \(g\) are diffeomorphisms, or find a counterexample.

Let the curve \((f(s),g(s))\) be parameterized by arclength in \(\mathbb {R}^2\) (\(f>0\)). Consider the surface of revolution \(S\) in \(\mathbb {R}^3\) given parametrically by \[ X(s,\theta )=(f(s)\cos \theta ,f(s)\sin \theta ,g(s)). \]

(a) Compute the mean curvature and Gauss curvature for points on \(S\).

(b) Find necessary and sufficient conditions for the “latitudes” obtained when \(s\) is constant to be geodesics.

Consider the domain \(D\) in \(\mathbb {R}^3\) which is the complement to the union of two unit balls with centers at the points \((0,0,2)\) and \((0,2,0)\). Calculate \(H_2(D,\mathbb {Z})\).

In the product \(S^1\times S^1\) consider the subset \[ A:=S^1\times \{p\}\cup \{p\}\times S^1 \] where \(p\) is a point on \(S^1\). Calculate \(H_j(A,\mathbb {Z})\), \(j=0,1,2\).

Prove that the Riemannian connection on a Riemannian manifold \((M^n,g)\) is uniquely determined. (Recall that a Riemannian connection is an affine connection which has zero torsion and is metric compatible.)

Let \(M\) be a differentiable manifold, \(\omega \) a closed \(p\)-form satisfying \[ \int _c \omega =0 \] for all smooth \(p\)-cycles \(c\). Prove \(\omega \) is exact.