DEPARTMENT OF MATHEMATICS University of Utah Ph.D. PRELIMINARY EXAMINATION IN GEOMETRY/TOPOLOGY August 14, 2001, 9:00--12:00, JWB 333

Let \(X\) and \(Y\) be two submanifolds of \(\mathbb {R}^n\). Show that there is \(a\in \mathbb {R}^n\) such that \(X+a=\{x+a\mid x\in X\}\) and \(Y\) are transverse.

Recall that an \(n\times n\) real matrix \(M\) is orthogonal if \[ \langle Mv,Mw\rangle =\langle v,w\rangle \] for all \(v,w\in \mathbb {R}^n\) (and \(\langle \cdot ,\cdot \rangle \) is the inner product). Let \(O(n)\) denote the set of all orthogonal \(n\times n\) matrices, viewed as a subset of \(\mathbb {R}^{n^2}\). Show that \(O(n)\) is a manifold, and compute its dimension.

By \(a:S^n\to S^n\) denote the antipodal map. Then the space \(A^p(S^n)\) of smooth \(p\)-forms on \(S^n\) breaks up into the \(+1\) and the \(-1\) eigenspace of \(a^*\), i.e. \[ A^p(S^n)=A^p_+(S^n)\oplus A^p_-(S^n) \] where \[ A^p_\epsilon (S^n)=\{\omega \in A^p(S^n)\mid a^*(\omega )=\epsilon \omega \}. \] Show that if \(\omega \) is an exact \(p\)-form and belongs to the \(+1\)-eigenspace, then \(\omega =d\eta \) for some \((p-1)\)-form \(\eta \) in the \(+1\)-eigenspace. What can you conclude about DeRham cohomology \(H^i(\mathbb {R}P^n)\) from this?

1. Give a definition of the tangent bundle \(TM\) of a manifold \(M\).
2. Give a definition of what it means for a manifold to be orientable.
3. Show that \(TM\), viewed as a differentiable manifold, is always orientable.

Recall that a map \(f:X\to \mathbb {R}\) defined on a subset \(X\subset \mathbb {R}^n\) is said to be smooth if for every \(x\in X\) there is an open set \(U_x\ni x\) in \(\mathbb {R}^n\) and a smooth function \(f_x:U_x\to \mathbb {R}\) such that \(f=f_x\) on \(X\cap U_x\). Prove that if \(f:X\to \mathbb {R}\) is smooth then there is a neighborhood \(U\) of \(X\) and a smooth extension \(\tilde f:U\to \mathbb {R}\).

Define the index of a singularity of a vector field and state the Poincaré-Hopf index theorem.

Consider the vector fields \[ X=\frac {\partial }{\partial x}\qquad \text {and}\qquad Y=x\frac {\partial }{\partial y}+\frac {\partial }{\partial z} \] on \(\mathbb {R}^3\). Prove or disprove: There is a nonempty surface \(M^2\subset \mathbb {R}^3\) such that for every \(p\in M\) the tangent space \(T_pM\) is spanned by \(X_p\) and \(Y_p\).

Let \(X\) be a closed manifold, \(\Delta =\{(x,x)\in X\times X\}\) the diagonal, and \[ N(\Delta ,X\times X)=\{((x,x),(v,-v))\mid x\in X,\ v\in T_x(X)\} \] the normal bundle of \(\Delta \). Show that there is a diffeomorphism of a neighborhood of \(X\) (viewed as the zero section) in the tangent bundle \(T(X)\) with a neighborhood of \(\Delta \) in \(X\times X\), extending the usual diffeomorphism \(X\to \Delta \), \(x\mapsto (x,x)\).

Give a definition of (Gaussian) curvature for surfaces in \(\mathbb {R}^3\) and state the Gauss-Bonnet theorem.

Compute the pullback \(f^*\omega \) of the differential form \[ \omega =dx\wedge dy \] by the map \(f:\mathbb {R}^3\to \mathbb {R}^3\), \(f(x,y,z)=(xyz,y+x,z)\).

Let \(M\) be a nonempty compact manifold, smooth or triangulated (you can make a choice). Show that \(M\) does not admit a retraction to its boundary.

Let \(M\) be a closed connected 7-manifold such that \(H_1(M;\mathbb {Z})=0\), \(H_2(M;\mathbb {Z})=\mathbb {Z}\), \(H_3(M;\mathbb {Z})=\mathbb {Z}/2\). Compute \(H_i(M;\mathbb {Z})\) and \(H^i(M;\mathbb {Z})\) for all \(i\).

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

Let \(X\subset \mathbb {R}^3\) be the union of \(n\) lines through the origin. Compute \(\pi _1(\mathbb {R}^3\setminus X)\).

Let \(p:M\to N\) be a covering map between two closed connected manifolds of positive dimension. Show that \(p\) is not null-homotopic.

1. Define carefully the Hopf map \(h:S^3\to S^2\).
2. Define what it means for a map to be a fiber bundle and sketch a proof that \(h\) is a fiber bundle.
3. State the long exact sequence in homotopy groups associated to a fiber bundle and compute \(\pi _3(S^2)\).

Let \(X\) be the space obtained from the torus \(S^1\times S^1\) by attaching a Möbius band via a homeomorphism from the boundary circle in the Möbius band to the circle \(S^1\times \{x_0\}\) in the torus. Compute \(\pi _1(X)\) and describe the universal cover of \(X\).

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

The surface \(M_g\) of genus \(g\), embedded in \(\mathbb {R}^3\) in the standard way, bounds a compact region \(R\). Two copies of \(R\), glued together by the identity map between their boundary surfaces \(M_g\), form a closed 3-manifold \(X\). Compute the homology groups of \(X\) via the Mayer-Vietoris sequence for this decomposition of \(X\) into two copies of \(R\).

Diagram: a standard embedded genus \(g\) surface, drawn schematically as a handled surface bounding a compact region \(R\).

1. Show that the Klein bottle cannot be embedded (as a smooth submanifold) into \(\mathbb {R}^3\).
2. It is a fact that the Klein bottle can be embedded into \(\mathbb {R}^4\). Show that no such embedding \(K\subset \mathbb {R}^4\) can be given by a system of equations \[ f_1=0,\qquad f_2=0 \] with independent smooth functions \(f_1,f_2\) (i.e. where \(df_1,df_2\) are linearly independent at each point \(x\in K\)).