Ph. D. preliminary examination in Geometry/Topology

Show that \(\pi _1(\mathbb {R}P^2\#\mathbb {R}P^2\#\mathbb {R}P^2)\) is infinite.

Using homology prove Brouwer’s fixed-point theorem: If \(f:I^n\to I^n\) is a continuous map then there exists \(x\in I^n\) such that \(f(x)=x\). (Here and below \(I\) denotes the closed unit interval.)

Prove the fixed-point theorem for contracting maps. Let \(X\) be a complete metric space and \(f:X\to X\) be a map such that there exists \(\lambda <1\) so that \[ d(f(x),f(y))\le \lambda d(x,y) \] for all \(x,y\in X\). Then \(f\) has a fixed point.

Prove that \(\mathbb {R}^n\) is not homeomorphic to \(\mathbb {R}^k\) for \(k\ne n\). (Do not appeal to Brouwer’s invariance of domain theorem.)

True or false: There is no orientable smooth 3-dimensional manifold \(M\) and a smooth embedding \(\mathbb {R}P^2\to M\). Justify your answer.

Let \(p(z)=a_nz^n+\cdots +a_0\) be a non-constant complex polynomial. Prove \(p:\mathbb {C}\to \mathbb {C}\) is proper, has only a finite number of singular values, that the inverse images of regular values have constant cardinality. Deduce the Fundamental Theorem of Algebra: \(p\) has a root.

Give a proof or a counterexample to the following assertion: If \(H_n(X;\mathbb {Z})=0\) then \(H_n(X;G)=0\) for any abelian group \(G\).

Let \(M^n\subset \mathbb {R}^{n+1}\) be an embedded differentiable submanifold. Prove that for each \(P\in M\) there is a neighborhood \(U\subset \mathbb {R}^{n+1}\) of \(P\) so that \(U\cap M\) is expressible as the graph “above” at least one of the \(n+1\) coordinate hyperplanes of \(\mathbb {R}^{n+1}\).

Let \(f,g:K\to L\) be two chain maps (\(f_n,g_n:K_n\to L_n\)). Define what it means for \(f\) and \(g\) to be chain homotopic. Prove that if \(f\) and \(g\) are chain homotopic, then they induce the same maps in homology, \(f_*=g_*:H_n(K)\to H_n(L)\).

Identify \(\mathbb {R}^4\) with \(\mathbb {C}^2\) so that \(z=x_1+iy_1\), \(w=x_2+iy_2\). Let \(p=(1,0)\in \mathbb {C}^2\). Consider the following smooth submanifolds of \(\mathbb {C}^2-\{(0,0),(0,0.5)\}\): \[ M_1:=\{(z,w)\mid z^2+w^2=1\}, \] \[ M_2:=\{(z,w)\mid z^2-w^2+w=1\}, \] \[ M_3:=\{(x_1,y_1,x_2,y_2)\mid 1-x_1+x_2^2+y_1^2+y_2^2=0\}. \] Which of the pairs of manifolds \((M_1,M_2)\), \((M_2,M_3)\), \((M_1,M_3)\) are transverse at \(p\)?

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

Let \(X\) be the topological space described on the Figure below: it is obtained from the 2-dimensional square \(I^2\) by identifying segments with arrows (the identification respects the orientation of edges determined by the arrows). Compute \(\pi _1(X)\).

Diagram description: the square \(I^2\) has arrows on the left and right vertical sides pointing upward, on the top and bottom horizontal sides pointing to the right, and on the diagonal from the lower-left corner to the upper-right corner pointing toward the upper-right corner. The arrowed segments are identified respecting these orientations.

True or false: Any closed differential form of degree \(\ge 1\) on \(\mathbb {R}P^2\times \mathbb {R}P^2\) is exact. Justify your answer.

Let \(S^4\) be the 4-dimensional sphere and \(F\subset S^4\) be a smoothly embedded surface of genus \(g\) (this surface is orientable, compact and has no boundary). Compute the homology groups \[ H_k(S^4-F,\mathbb {R}) \] for \(k=2,3\). (You may assume that \(F\) is contained in an open subset \(U\) of \(\mathbb {R}^4\), which is homeomorphic to \(F\times \mathbb {R}^2\) so that \(F\subset U\) corresponds to \(F\times \{0\}\).)

Consider the set of matrices \[ O(n)=\{A\in \mathcal {M}_n:AA^T=1\} \] where \(\mathcal {M}_n\cong \mathbb {R}^{n^2}\) is the collection of all \(n\times n\) matrices and \(A^T\) means the transposition of \(A\). Show that \(O(n)\) is a smooth submanifold of \(\mathcal {M}_n\). Describe the set of matrices which form the tangent space to \(O(n)\) at the identity matrix.

a. Let \(S^3\) be the 3-sphere, and let \(T^2=S^1\times S^1\) be the 2-torus. Compute the singular homology of \(S^3\times T^2\) using its CW-complex structure.

b. Recompute the singular homology of \(S^3\times T^2\) using the Eilenberg-Zilber and Künneth Theorems.

Define what a Riemannian manifold is. Define the natural distance function which one obtains from the (infinitesimal) Riemannian metric, and prove that this distance function gives a metric space structure to each Riemannian manifold.

Let \(X=\{(x,y,z)\in \mathbb {R}^3\mid xyz=0\}\), i.e. \(X\) is the union of the three coordinate planes in \(\mathbb {R}^3\). Let \(0\in X\) be the origin. Prove that any homeomorphism of \(X\) must fix 0.

a. Let \(P:M\to N\) be a regular covering map between connected differentiable manifolds \(M,N\) and let \(M\) be orientable. Prove that \(N\) is orientable if and only if every deck transformation (covering space automorphism) preserves the orientation of \(M\).

b. Prove that \(\mathbb {R}P^n\) is orientable if and only if \(n\) is odd.