Ph. D. preliminary examination in Geometry/Topology

Let \(M\) be the Moebius band which has Riemannian metric with totally-geodesic boundary and curvature \(K(p)\le 0\) in all points \(p\in M\). Show that the curvature of this metric is identically zero.

Consider the space \(\mathcal {L}\) of all straight lines \(ax+by+c=0\) in the Euclidean plane \(\mathbb {R}^2\). Topologize \(\mathcal {L}\) so that a sequence of lines \(L_i\) is convergent to a line \(L=\{ax+by+c=0\}\) if and only if \(L_i\) can be represented by equations \(a_ix+b_iy+c_i=0\) so that \(\lim _i a_i=a\), \(\lim _i b_i=b\), and \(\lim _i c_i=c\). Prove that the topological space \(\mathcal {L}\) is homeomorphic to the Moebius band without boundary.

Let \(F:\mathbb {R}^2\to \mathbb {R}^3\) be a mapping given by the formula \[ F(u,v)=(e^u\cos (v),e^u\sin (v),0). \] Compute the pull-back \(F^*(\omega )\) of the differential form \[ \omega =xy\,dx\wedge dy+x\,dy\wedge dz\in \Omega ^*(\mathbb {R}^3). \]

Construct a nontrivial circle bundle over the Klein bottle.

Consider the complex plane \(\mathbb {C}\) with the Euclidean metric. Let \(f:\mathbb {C}\to \mathbb {C}\) be a complex-analytic mapping and \(z\in \mathbb {C}\) be a point such that \(f'(z)\ne 0\). Show that the derivative \(Df_z:T_z\mathbb {C}\to T_{f(z)}\mathbb {C}\) preserves angles.

Take the 2-dimensional triangle \(T\) and identify all its sides as indicated below. What is the universal covering of the resulting topological space \(X\)?

Diagram description: the triangle \(T\) has all three sides identified. The left and right sides have arrows pointing toward the top vertex, and the bottom side has an arrow pointing from left to right.

Let \(n>2\). By \(\mathbb {Z}/n\) denote the cyclic group of order \(n\) with discrete topology and by \(S^2\) the 2-sphere with standard topology. Let \(p\) and \(q\) be two distinct points of \(S^2\). Compute the homology groups of the \(n\)-link sausage space \(X\) which is the space that can be represented as the identification space \((S^2\times \mathbb {Z}/n)/{\sim }\) with the equivalence relation induced by \((p,i)\sim (q,i+1)\).

Diagram description: the space \(X\) is drawn as a cyclic chain of lobe-shaped copies of \(S^2\). Each copy is attached to the next one around the cycle, with the \(p\)-point of one copy identified with the \(q\)-point of the next.

Denote by \([X,Y]\) the set of free homotopy classes of maps \(X\to Y\). Show that \([S^2,\mathbb {R}P^2]\) is infinite. In other words, construct a sequence of maps \(f_i:S^2\to \mathbb {R}P^2\), \(i=1,2,3,\ldots \), such that no two are homotopic to each other.

Give an explicit construction of the Hopf fibration \(h:S^3\to S^2\) and argue that \(h\) is not null-homotopic.

Section II. Pick four of the following eight questions to answer:

Let \(f:S^n\to S^n\) be a continuous map from the \(n\)-sphere to itself and assume that \(n\) is even. Show that there is a point \(x\in S^n\) such that \(f(x)\in \{x,-x\}\). Does the same assertion hold if \(n\) is odd? If it does, prove it, and if it does not, find a counterexample.

Consider the “tripod” \(T\) with the central point \(O\) (i.e. the union of three segments in the plane which intersect only at the origin \(O\)). Let \(P=(O,O)\) be the point on \(X=T\times T\) which projects to the point \(O\) in the both factors. Compute \(H_1(X,X-P;\mathbb {Z})\).

Diagram description: the tripod \(T\) consists of three line segments meeting at the central point \(O\): one vertical segment above \(O\), one diagonal segment down and to the left, and one diagonal segment down and to the right.

True or false: Suppose that \(X\) is a finite CW-complex and \(A\subset X\) is a subcomplex. Then the relative homotopy groups \(\pi _j(X,A)\) are isomorphic to the homotopy groups \(\pi _j(X/A)\), where \(X/A\) is the CW-complex obtained from \(X\) by collapsing \(A\) to a point. Justify your answer.

Suppose that the curvature and torsion of a regular curve in \(\mathbb {R}^3\) are non-zero constants \(a,b\). Find a parameterization of the curve by arclength.

Let \[ X(s,\theta )=(\varphi (s)\cos (\theta ),\varphi (s)\sin (\theta ),\gamma (s)) \] be a surface of revolution such that \(\varphi '(s)^2+\gamma '(s)^2=1\). Prove that the meridians \(\theta =\mathrm {const}\) are geodesics. What condition on the lattitude \(s=\mathrm {const}\) is necessary and sufficient to made it geodesic?

Prove that torus can not be immersed in \(\mathbb {R}^3\) in such a way that the Gauss curvature of the induced metric is everywhere non-negative.