DEPARTMENT OF MATHEMATICS University of Utah Ph.D. PRELIMINARY EXAMINATION IN GEOMETRY/TOPOLOGY Autumn 1998

Let \(X\) be the set of points \((x,y,z)\in \mathbb {R}^3\) satisfying the equation \[ (1+z)x^2-(1-z)y^2=2z(1-z^2). \] Show that \(X\) is a smooth oriented surface embedded in \(\mathbb {R}^3\).

For \((x,y,z)\in \mathbb {R}^3\) let \(\eta =dx\otimes dx+dy\otimes dy-dz\otimes dz\) denote a symmetric 2-form on \(\mathbb {R}^3\). Let \[ M=\{(x,y,z)\in \mathbb {R}^3:1+x^2+y^2=z^2,\ z>0\} \] be one nappe of the hyperboloid.

1. Compute \[ \int _\Omega dx\wedge dy+(4-2z)dx\wedge dz+(z-2)dy\wedge dz \] where \(\Omega \subset M\) is the region inside \(x^2+y^2\le 3\).
2. Show that the restriction \(\eta |_M\) is a Riemannian metric for \(M\).
3. What is the length of the curve \(t\mapsto (\sinh t,0,\cosh t)\) for \(a\le t\le b\) in this metric?

Suppose the three sphere \[ S^3=\{(x,y,z,w)\in \mathbb {R}^4:x^2+y^2+z^2+w^2=1\} \] is embedded into four space the standard way. Two independent vector fields on \(S^3\) are given by \(V=(y,-x,w,-z)\), \(W=(w,z,-y,-x)\). Determine whether it is possible to embed a closed smooth two dimensional surface \(N\subset S^3\) so that the vector fields are tangent to \(N\) at every point of \(N\), i.e. \(V(p),W(p)\in T_pN\) for every \(p\in N\).

1. Show that \(\Phi :\mathbb {R}^2\to \mathbb {R}^2\) given by \[ (p,q)=\Phi (x,y)=(x+y^2+2y,x+y^2-2y) \] is a diffeomorphism. Compute \(d\Phi (\partial /\partial y)\) and the pullback \(\Phi ^*(p^2\,dp\wedge dq)\).
2. Let \(X\) be a smooth vector field and \(\omega \) a smooth \(n\)-form on the smooth compact boundaryless manifold \(M^n\). Let \(M\times \mathbb {R}\ni (x,t)\mapsto \phi _t(x)\in M\) be the flow generated by \(X\). Let \[ A(t)=\int _M\phi _t^*\omega \] be the integral of the pulled back \(n\)-form. Find \(dA/dt\).

Let \(M^m,N^n\subset P^p\) be smooth embedded submanifolds of a smooth manifold. \(M^m\) and \(N^n\) are said to intersect transversally if \(T_xM+T_xN=T_xP\) for every \(x\in M\cap N\). Prove that if \(M^m\) and \(N^n\) intersect transversally then \(M^m\cap N^n\) is a smooth submanifold of \(P^p\). What is its dimension?

Let \(X=S^1\times \mathbb {R}\). Find the de Rham cohomology of \(X\) in two ways.

1. Using de Rham’s Theorem.
2. Directly from the definition.

Let \(f:S^n\to S^n\) be a continuous map of spheres whose degree \(D(f)\ne 1\). Show that there is a point \(x\in S^n\) such that \(f(x)=a(x)\) where \(a\) is the antipodal map.

Suppose that \(K_1\supset K_2\supset K_3\supset \cdots \) is a nested sequence of compact connected subsets of \(S^2\) such that \(\pi _1(K_j)=\mathbb {Z}\) for all \(j\). Prove or provide a counterexample for each of the following statements:

1. \(\bigcap _{j=1}^{\infty }K_j\) is connected.
2. \(\pi _1\left (\bigcap _{j=1}^{\infty }K_j\right )\cong \mathbb {Z}\).

Let \(X\) be the space obtained from a disk by identifying the boundary according to the following diagram. Find the fundamental group \(\pi _1(X)\). Is \(X\) homeomorphic to a closed surface? Explain.

Diagram description: a hexagonal disk with paired boundary labels. Reading clockwise around the boundary, the labeled sides are \(a^{-1},c,b,a,c^{-1},b^{-1}\); paired sides have matching labels with inverse orientations.

Determine all covering spaces of the two-torus \(T^2\) up to equivalence. Explain.

Let \(t^2\subset S^5\) be a homeomorph of the two torus \(T^2\). Find the singular homology \(H_*(S^5-t^2;\mathbb {Z})\).

Let \(X_p=D^4\cup _f S^3\) be the space obtained by attaching a 4-cell to the 3-sphere by an attaching map \(f:\partial D^4\to S^3\) of degree \(p\).

1. Compute the singular homology of \(H_*(X_p;\mathbb {Z})\) and \(H_*(X_p\times X_q;\mathbb {Z})\) for any \((p,q)\) using the CW-complex cell structure.
2. Find \(H_*(X_p\times X_q;\mathbb {Z})\) using the Künneth Theorem.

Find the singular homology groups \(H_*(\mathbb {R}P^5;\mathbb {Z})\).

Explicitly describe a nontrivial homotopy class in \(\pi _3(S^2)\). Show that it is nonzero.

Suppose \((C_k,\partial _k)\) is a free chain complex. Suppose that it is acyclic, i.e. the homology groups \(H_k(C)=0\) for all \(k\). Show that \(C\) is contractible, that is, the identity map and the zero map \(\iota _k,\zeta _k:C_k\to C_k\) are chain homotopic.

Prove that \(\mathbb {C}P^2\) cannot be decomposed as a union \(M_1\cup M_2\) where \(M_1\) and \(M_2\) are homeomorphic compact submanifolds with boundary and \(M_1\cap M_2\) is the common boundary of \(M_1\) and \(M_2\).

Let \(X=S^1\vee S^2\vee T^2\) be the one point union. Compute \(\pi _1(X)\) and show that \(\pi _2(X)\) is infinitely generated.

Let \(X=\Delta -\bigcup _{j=1}^{\infty }D_j^\circ \) where \(\Delta \) is the closed unit disk and \(D_j^\circ \subset \Delta \) are open round disks with pairwise disjoint closures. Prove that the homology of \(X\) is infinitely generated.