DEPARTMENT OF MATHEMATICS University of Utah Ph.D. PRELIMINARY EXAMINATION IN GEOMETRY/TOPOLOGY August 17, 2005, 9:00--12:00, LCB 219

In this problem we identify \(\mathbb {C}\) with \(\mathbb {R}^2\) in the usual manner via \[ z\mapsto (\operatorname {Re}z,\operatorname {Im}z). \] Consider the following smooth submanifolds of \(\mathbb {C}^2-\{(0,0),(0,0.5)\}\): \[ M_1=\{(z,w)\in \mathbb {C}^2\mid z^2+w^2=1\} \] \[ M_2=\{(z,w)\in \mathbb {C}^2\mid z^2-w^2+w=1\} \] \[ M_3=\{(x_1,y_1,x_2,y_2)\in \mathbb {R}^4\mid 1-x_1+x_2^2+y_1^2+y_2^2=0\}. \] Which of the pairs \(M_i,M_j\) (\(i,j\in \{1,2,3\}\)) are transverse at \(p=(1,0)\in \mathbb {C}^2\)?

Let \(V,W:\mathbb {R}^4\to \mathbb {R}^4\) be vector fields on \(\mathbb {R}^4\) defined by \[ V(x,y,z,w)=(y,-x,w,-z) \] and \[ W(x,y,z,w)=(w,z,-y,-x). \] Is there a nonempty surface \(\Sigma \subset \mathbb {R}^4\) such that for every \(p\in \Sigma \), we have \(V(p),W(p)\in T_p\Sigma \)? Find such a surface or prove that it does not exist.

Let \(M\) be a nonempty compact manifold with empty boundary. Prove that there is no compact manifold \(W\subset M\times M\) whose boundary \(\partial W\) is the diagonal \(\Delta =\{(x,x)\in M\times M\}\).

Is there a smooth embedding of \(\mathbb {R}P^2\) into an orientable 3-manifold? Provide a proof or an example.

Let \(f:\mathbb {R}^2\to \mathbb {R}^3\) be the function given by \(f(x,y)=(x^2-y,x+y^2,\sin (x))\). Let \(\omega \) be the 2-form on \(\mathbb {R}^3\) given by \(\omega (u,v,w)=u\,dv\wedge dw\). Compute the pull-back \(f^*\omega \).

Prove that for any two disjoint closed subsets \(A,B\subset \mathbb {R}^n\) there is a smooth function \(f:\mathbb {R}^n\to \mathbb {R}\) such that \(f=0\) on \(A\) and \(f=1\) on \(B\).

Show that the total space of the tangent bundle \(T(S^2)\) of the 2-sphere is not diffeomorphic to \(S^2\times \mathbb {R}^2\). Hint: One strategy is to find a 2-sphere in \(T(S^2)\) whose self-intersection number is nonzero and argue that there are no such 2-spheres in \(S^2\times \mathbb {R}^2\).

Compute the curvature of the hyperboloid \[ x^2+y^2-z^2=1 \] at the point \((1,0,0)\).

Suppose \(p:\widetilde {X}\to X\) is a covering map. Prove that \(p\) is null homotopic if and only if \(\widetilde {X}\) is contractible.

Let \(T^2=S^1\times S^1\) be the standard 2-torus and \(p,q\in T^2\) be two distinct points. Let \(X=T^2/{\sim }\) be the identification space where the two points are identified \(p\sim q\). Find \(\pi _1(X)\).

Carefully construct a \(\Delta \)-complex structure on \(\mathbb {R}P^2\) and use it to compute the homology groups of \(\mathbb {R}P^2\) with \(\mathbb {Z}\) and \(\mathbb {Z}_2\) coefficients.

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 (for every odd \(n\)).

1. Describe the (simplest) cell structure of \(\mathbb {R}P^n\) (give the attaching maps of all the cells – you don’t have to give a proof here).
2. Write down the cellular chain complex associated with this cell decomposition.
3. Use this chain complex to compute the homology and cohomology groups of \(\mathbb {R}P^n\) with coefficients in \(\mathbb {Z}\) and \(\mathbb {Z}_2\).

Prove that \(\mathbb {R}P^3\) and \(\mathbb {R}P^2\vee S^3\) are not homotopy equivalent. (Hint: Cup products.)

Let \(\Sigma \) be a simply connected compact 3-manifold. Prove that \(H_k(\Sigma )\cong H_k(S^3)\) for any integer \(k\).

Let \(p:S^3\to \mathbb {C}P^1=S^2\) be the map that sends \((z,w)\in S^3\subset \mathbb {C}^2\) to its equivalence class \([z:w]\in \mathbb {C}P^1\).

1. Prove that \(p\) is a fiber bundle by explicitly writing down local trivializations.
2. Write down the associated long exact sequence of homotopy groups.
3. Compute \(\pi _3(S^2)\). Justify all auxiliary results you are using.