UNIVERSITY OF UTAH DEPARTMENT OF MATHEMATICS Ph.D. Preliminary Examination in Geometry / Topology Jan. 4, 2019.

Let \(M\) and \(N\) be smooth manifolds and let \(M\) be compact. Let \(F:M\times \mathbb {R}\to N\) be a smooth function and for any \(t\in \mathbb {R}\), let \(F_t:M\to N\) be the function defined by \(F_t(x)=F(x,t)\). If \(F_0\) is a submersion, prove that there is some \(\epsilon >0\) such that \(F_t\) is an submersion for all \(t\in (-\epsilon ,\epsilon )\).

Find all connected Lie subgroups of \(\mathrm {SL}_2(\mathbb {R})\) that contains the subgroup \[ H= \left \{ \begin {pmatrix} 1 & x\\ 0 & 1 \end {pmatrix} : x\in \mathbb {R} \right \}. \]

Let \(\alpha =y\,dx-x\,dy+dz\) be a one form on \(\mathbb {R}^3\). Define a 2-plane field distribution at \(p\in \mathbb {R}^3\) by \(\Delta _p=\{V\in T_p\mathbb {R}^3:\alpha (V)=0\}\). Is the plane field \(\Delta _p\) integrable? Why?

Let \(a:S^2\to S^2\) be the antipodal map \(a(x)=-x\) and let \[ p:S^2\to \mathbb {R}\mathrm {P}^2=S^2/x\sim -x,\qquad p(x)=[x] \] be the projection to the projective plane. Let \(\omega \) be a two form on \(\mathbb {R}\mathrm {P}^2\).

1. Prove that \(\int _{S^2}p^*\omega =0\). [Hint: \(p^*\omega \) is \(a\)-invariant.]
2. Prove that there is a one form \(\eta \) on \(\mathbb {R}\mathrm {P}^2\) such that \(d\eta =\omega \). You are allowed to use the fact that if \(\beta \) is a two-form on \(S^2\) and \(\int _{S^2}\beta =0\) then \(\beta \) is exact, but you are not allowed to use the de Rham Theorem.
3. Prove that \(\mathbb {R}\mathrm {P}^2\) is not orientable.

Let \(M\) be a compact, smooth two dimensional manifold without boundary embedded in \(\mathbb {R}^3\). Show that for almost every two-plane \(L\in \mathbb {R}^3\) such that \(L\cap M\) is nonempty, the intersection \(L\cap M\) consists of finitely many closed curves.

Let \(X=S^1\vee S^1\), with basepoint \(*\) at the point where the two circles are glued, and with \(a,b:S^1\to X\) the generators of \(\pi _1(X,*)\) corresponding to the identity maps onto each circle in the wedge sum. Construct a covering space \(p:(E,e)\to (X,*)\) such that \(p_*\pi _1(E,e)\) is the subgroup of \(\pi _1(X,*)\) generated by \(a^2\), \(ab^{-1}\), \(ab\). Justify that your answer works in two ways:

1. using covering space theory; and
2. using the van Kampen theorem.

Let \(G\) be any group. Construct a connected topological space whose fundamental group is isomorphic to \(G\). Justify your answer.

1. Explain how the set of lines in \(\mathbb {C}^{n+1}\) is topologized to yield the space \(\mathbb {C}\mathrm {P}^n\). Describe how to equip \(\mathbb {C}\mathrm {P}^n\) with a CW structure with \(m\)-skeleton isomorphic to \(\mathbb {C}\mathrm {P}^m\) for all \(m\le n\).
2. Compute the singular homology groups \(H_*(\mathbb {C}\mathrm {P}^n;\mathbb {Z})\). Be sure you indicate which properties of homology you are using.

Fix integers \(n\ge 1\) and \(m\ge 1\).

1. Construct a connected space \(X\) such that \[ \widetilde {H}_r(X;\mathbb {Z})= \begin {cases} 0 & \text {if } r\ne n,\\ \mathbb {Z}/m & \text {if } r=n. \end {cases} \] (Here \(\widetilde {H}\) denotes reduced homology.)
2. For any connected space \(X\) satisfying the conclusion of part (a), compute the groups \(H_r(X;\mathbb {Z}/2)\) for all \(r\ge 0\).

Let \(X\) be a connected compact orientable manifold of dimension \(n\). Explain carefully how Poincaré duality implies the existence of a bilinear pairing \[ H^k(X;\mathbb {Z})\times H^{n-k}(X;\mathbb {Z})\to \mathbb {Z} \] that modulo torsion becomes a perfect (i.e., non-singular) pairing. Deduce from this the structure of the cohomology ring \(H^*(\mathbb {C}\mathrm {P}^n;\mathbb {Z})\).