DEPARTMENT OF MATHEMATICS University of Utah Ph.D. PRELIMINARY EXAMINATION IN GEOMETRY/TOPOLOGY August 2003

Let \(X\) and \(Y\) be closed submanifolds of \(\mathbb {R}^n\). Show that for almost all \(a\in \mathbb {R}^n\) the translate \(X+a=\{x+a\mid x\in X\}\) intersects \(Y\) transversally.

Let \(f:X\to Y\) be a smooth map between smooth manifolds that is one-to-one on a compact submanifold \(Z\) of \(X\). Suppose that for all \(x\in Z\) \[ df_x:T_x(X)\to T_{f(x)}(Y) \] is an isomorphism. Prove that there is a neighborhood \(U\) of \(Z\) such that \[ f|_U:U\to f(U) \] is a diffeomorphism.

Identify the set of all \(n\times n\) matrices \(M(n)\) with real entries with \(\mathbb {R}^{n^2}\) and let \[ SL_n(\mathbb {R})=\{M\in M(n)\mid \det (M)=1\}. \]

1. Show that \(SL_n(\mathbb {R})\) is a submanifold of \(M(n)\).
2. Compute the tangent space \(T_I(SL_n(\mathbb {R}))\) at the identity matrix.

Let \(X\) be any subset of \(\mathbb {R}^n\) and \(f:X\to \mathbb {R}\) a function with the following property. For every \(x\in X\) there is a neighborhood \(U_x\) of \(x\) in \(\mathbb {R}^n\) and a smooth function \(f_x:U_x\to \mathbb {R}\) such that \(f=f_x\) on \(X\cap U_x\). Show that there is an open set \(U\) in \(\mathbb {R}^n\) containing \(X\) and a smooth function \(g:U\to \mathbb {R}\) such that \(g|_X=f\).

Consider the 1-form on \(\mathbb {R}^2\setminus \{0\}\) \[ \omega =\frac {x\,dy-y\,dx}{x^2+y^2}. \]

1. Prove that \(\omega \) is closed, i.e. \(d\omega =0\).
2. Prove that \(\omega \) is not exact, i.e. there is no smooth function \(f:\mathbb {R}^2\setminus \{0\}\to \mathbb {R}\) such that \(\omega =df\).

Give a proof of the Boundary Theorem: If a smooth map \(f:X\to Y\) between two closed oriented \(n\)-manifolds extends to a smooth map \(F:W\to Y\) of a compact oriented \((n+1)\)-manifold \(W\) with \(\partial W=X\) then \(\deg f=0\).

Regard the real projective plane \(\mathbb {R}P^2\) as the space of triples \((x,y,z)\in \mathbb {R}^3\setminus \{0\}\) modulo the relation \((x,y,z)\sim (tx,ty,tz)\) for \(t\in \mathbb {R}\setminus \{0\}\). Define \[ f:\mathbb {R}P^2\to \mathbb {R} \] by \[ f(x,y,z)=\frac {x^2+2y^2}{x^2+y^2+z^2}. \] Compute all critical points of \(f\). Show that \(f\) is a Morse function and compute the Morse index of each critical point.

Consider the following vector fields in \(\mathbb {R}^3\): \[ X=\frac {\partial }{\partial x},\qquad Y=x\frac {\partial }{\partial y}+\frac {\partial }{\partial z}. \] Here \(x,y,z\) are the coordinates on \(\mathbb {R}^3\) and \(\frac {\partial }{\partial x}\), \(\frac {\partial }{\partial y}\), \(\frac {\partial }{\partial z}\), are the three standard coordinate vector fields. Show that there is no (nonempty) surface \(M\subset \mathbb {R}^3\) which is tangent to both vector fields at each of its points.

Show that every map \(f:S^2\to S^2\) from the 2-sphere to itself admits a point \(x\in S^2\) such that \(f(x)=\pm x\).

Let \(\Sigma \) be an oriented genus 2 surface smoothly embedded in \(\mathbb {R}^3\).

1. Describe the Gauss map \(g:\Sigma \to S^2\) and define the Gaussian curvature at a point of \(\Sigma \) in terms of \(g\).
2. State the Gauss-Bonnet theorem for this situation.
3. Prove that \(\Sigma \) has (many) saddle points, i.e. points of negative curvature.

Compute the fundamental group of the space obtained from two tori \(S^1\times S^1\) by identifying a circle \(S^1\times \{x_0\}\) in one torus with the corresponding circle \(S^1\times \{x_0\}\) in the other torus.

Give an explicit example of a covering space of \(X=S^1\vee S^1\) which is not normal.

Compute the homology groups of the space obtained from a torus \(S^1\times S^1\) by attaching a Möbius band via a homeomorphism from the boundary circle of the Möbius band to the circle \(S^1\times \{x_0\}\) in the torus.

Suppose \(M\) is a closed connected 5-manifold such that \(\pi _1(M)\cong \mathbb {Z}/3\mathbb {Z}\). If \(H_2(M;\mathbb {Z})\cong \mathbb {Z}\), compute all other homology and cohomology groups of \(M\) with integral coefficients.

1. Describe the (simplest) cell structure of \(\mathbb {R}P^n\) (give the attaching maps of all 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 homology and cohomology groups of \(\mathbb {R}P^n\) with coefficients in \(\mathbb {Z}\) and \(\mathbb {Z}/2\mathbb {Z}\).

Let \(p:\widetilde {X}\to X\) be a 2-sheeted covering map between cell complexes such that the cell structure on \(\widetilde {X}\) is obtained by lifting the cell structure on \(X\).

1. Construct a short exact sequence of cellular chain complexes with coefficients in \(\mathbb {Z}/2\mathbb {Z}\) \[ 0\to C_n(X;\mathbb {Z}/2\mathbb {Z})\to C_n(\widetilde {X};\mathbb {Z}/2\mathbb {Z}) \to C_n(X;\mathbb {Z}/2\mathbb {Z})\to 0 \] (i.e. give the maps explicitly and prove that the sequence is exact).
2. Write down the induced long exact sequence of homology groups (the transfer sequence).
3. Prove that if \(\widetilde {X}\) is acyclic over \(\mathbb {Z}/2\mathbb {Z}\) (i.e. \(H_i(\widetilde {X};\mathbb {Z}/2\mathbb {Z})=0\) for \(i>0\) and \(H_0(\widetilde {X};\mathbb {Z}/2\mathbb {Z})\cong \mathbb {Z}/2\mathbb {Z}\)) then \(H_i(X;\mathbb {Z}/2\mathbb {Z})\cong \mathbb {Z}/2\mathbb {Z}\) for all \(i\ge 0\).

Compute \(\pi _2(S^1\vee S^2)\) using the Hurewicz theorem.

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.

Let \(F_n\) be a free group of rank \(n\) and let \(G\subset F_n\) be a subgroup of index \(m\). Prove that \(G\) is a free group and compute its rank.

Let \(\Sigma \) be a closed oriented surface of genus \(g\) and suppose that \(X\subset \Sigma \) is a graph which is a retract of \(\Sigma \). Prove that \[ \operatorname {rank}H_1(X)\le g. \]