DEPARTMENT OF MATHEMATICS University of Utah Ph.D. PRELIMINARY EXAMINATION IN GEOMETRY/TOPOLOGY January 5, 2017

Let \(B\) be the open ball in \(\mathbb {R}^n\) that is centered at \(0\) and has radius \(1\). If \(y\in B\) and \(\lVert y\rVert <1/2\), define a vector field on \(B\) and use the resulting flow to find a diffeomorphism \(h:B\to B\) with \(h(0)=y\) such that \(h\) is homotopic to the identity map on \(B\). Conclude that if \(N\) is a smooth connected manifold and \(p,q\in N\), then there is a diffeomorphism \(H:N\to N\) such that \(H(p)=q\) and such that \(H\) is homotopic to the identity map on \(N\).

Let \(A:\mathbb {R}^n\to \mathbb {R}^n\) be linear and let \(\phi \in \Lambda ^n(\mathbb {R}^n)\). Prove \(A^*\phi =\det (A)\phi \).

Let \(\Phi :\mathfrak {sl}_2(\mathbb {R})\to \mathbb {R}\) be a Lie algebra homomorphism, where \(\mathfrak {sl}_2(\mathbb {R})\) is the Lie algebra of \(\mathrm {SL}_2(\mathbb {R})\) and \(\mathbb {R}\) is the Lie algebra of the Lie group \(\mathbb {R}\). Prove that \(\Phi =0\).

Let \(G\) be a connected Lie group with Lie algebra \(\mathfrak {g}\) such that \([X,Y]=0\) for all \(X,Y\in \mathfrak {g}\). Prove that \(G\) is abelian.

Prove that \(S^n\) is a smooth manifold for any \(n\).

Let \(M\) and \(N\) be smooth, compact, oriented manifolds without boundary, both of dimension \(n\). Suppose that \(f:M\to N\) and \(g:M\to N\) are smoothly homotopic. Prove that for any \(\omega \in \Omega ^n(N)\), we have \[ \int _M f^*\omega =\int _M g^*\omega . \]

Prove that the fundamental group of a finite connected graph is a free group.

Show that the union of circles in \(\mathbb {R}^2\) with centers at \((0,\frac {1}{2n})\) and radius \(\frac {1}{n}\) does not have a universal cover.

Show that \(S^n\) has a continuous field of nonzero tangent vectors iff \(n>0\) is odd. Conclude that the even-dimensional spheres are not Lie groups.

Construct a \(\Delta \)-complex structure on \(\mathbb {R}\mathrm {P}^3\) and use it to compute the cohomology rings of \(\mathbb {R}\mathrm {P}^3\) with \(\mathbb {Z}\) and \(\mathbb {Z}_2\) coefficients.

Prove that nonempty open subsets of \(\mathbb {R}^m\) and \(\mathbb {R}^n\) are not homeomorphic if \(m\ne n\).

Let \(M\) be a closed, orientable 3-manifold such that \[ H_1(M;\mathbb {Z})=\mathbb {Z}\oplus \mathbb {Z}_3. \] Calculate the remaining homology and cohomology groups for \(M\) with \(\mathbb {Z}\) coefficients.