DEPARTMENT OF MATHEMATICS University of Utah Ph.D. PRELIMINARY EXAMINATION IN GEOMETRY/TOPOLOGY January 2011

1. Let \(M\) be a smooth manifold. What’s the definition of an orientation on \(M\)?
2. Suppose \(M\) is a smooth oriented manifold with boundary. What is the induced orientation on \(\partial M\)?
3. State Stokes’ Theorem.
4. Let \(M\) be a smooth manifold without boundary that is oriented by the form \(\omega \in \Omega ^{\dim (M)}(M)\). Suppose that \([0,1]=\{x\in \mathbb {R}\mid 0\le x\le 1\}\) is oriented by the form \(dx\), and that \(M\times [0,1]\) has the orientation defined by \(\omega \wedge dx\). Endow \(\partial (M\times [0,1])=(M\times \{0\})\cup (M\times \{1\})\) with the orientation induced by \(M\times [0,1]\) and let \(i_0:M\to M\times \{0\}\) and \(i_1:M\to M\times \{1\}\) be the obvious diffeomorphisms. Prove that \(i_0\) preserves orientation if and only if \(i_1\) does not.
5. Let \(f:M\to N\) and \(g:M\to N\) be smooth maps between smooth manifolds without boundaries. What is the definition of \(f\) and \(g\) being smoothly homotopic to each other?
6. Suppose \(M\) is a smooth, compact, oriented manifold without boundary. Suppose \(N\) is a smooth manifold and that \(f:M\to N\) and \(g:M\to N\) are smoothly homotopic smooth maps. Prove that if \(\theta \in \Omega ^k(N)\) for some \(k>\dim (M)\), then \[ \int _M f^*\theta =\int _M g^*\theta . \]

1. What is a subalgebra of a Lie algebra?
2. Let \(G\) be a Lie group, with Lie algebra \(\mathfrak {g}\). Prove that any Lie subgroup \(H\) of \(G\) corresponds to a subalgebra of \(\mathfrak {g}\).
3. What is a smooth plane field on \(G\)?
4. Give an example of a foliation on a 2-torus that has compact leaves and noncompact leaves.
5. State Frobenius’ theorem.

6. Prove that any subalgebra of \(\mathfrak {g}\) corresponds to a foliation of \(G\).

7. Given a subalgebra of \(\mathfrak {g}\), and the corresponding foliation of \(G\), prove that the leaf containing the identity is a group, and thus that there is a correspondence between connected Lie subgroups of \(G\) and subalgebras of \(\mathfrak {g}\).

Define a deformation retract. Construct a 2-dimensional cell complex that contains both an annulus \(S^1\times I\) and a Möbius band as deformation retracts.

Let \(G\) be a Lie group and \(1\in G\) the identity element. Prove that \(\pi _1(G,1)\) is an abelian group.

Let \(f:S^1\to S^1\) be given by \(f(z)=z^2\), where we regard \(S^1\) as the unit circle in \(\mathbb {C}\). Find a presentation of the fundamental group of the mapping torus \[ M_f=S^1\times I/(z,1)\sim (f(z),0) \] of \(f\).

Suppose that \(X\) and \(Y\) are two connected manifolds or cell complexes homotopy equivalent to each other. Prove that their universal covers \(\widetilde {X}\) and \(\widetilde {Y}\) are homotopy equivalent to each other. Carefully state all theorems you are using.

An algebraic fact, often used in homology theory, is that a short exact sequence of chain complexes induces a long exact sequence in homology. State this fact carefully and give a definition of the “connecting homomorphism” (the one that lowers the degree of the homology group).

Let \(X\) denote the solid torus \(S^1\times D^2\). Using Mayer-Vietoris, compute the homology groups of the double of \(X\), obtained by taking two copies of \(X\) and gluing them along the boundary via the identity map. Also compute \(H_i(X,\partial X)\).

Let \(X\) be a finite connected cell complex such that \(H^2(X;\mathbb {Z})=\mathbb {Z}/2\mathbb {Z}\) and \(H^i(X;\mathbb {Z})=0\) for \(i=1\) and \(i>2\). Compute \(H_i(X;\mathbb {Z})\) for all \(i\). State all theorems you are using.

Let \(M\) be a closed orientable connected 4-manifold with \(H^1(M)=H^3(M)=0\) and \(H^2(M)=H^4(M)=\mathbb {Z}\). What are the possible cup product structures on \(H^*(M)\)? State all theorems you are using. All cohomology groups are with integral coefficients.