DEPARTMENT OF MATHEMATICS University of Utah Ph.D. PRELIMINARY EXAMINATION IN GEOMETRY/TOPOLOGY August 2014

Identify \(\mathcal {M}(2)\), the set of two-by-two matrices, with \(\mathbb {R}^4\). Let \(\mathrm {SL}(2,\mathbb {R})\subset \mathcal {M}(2)\) be the set of matrices with determinant one. Show that \(\mathrm {SL}(2,\mathbb {R})\) is a smooth submanifold and calculate the tangent space \(T_{id}\mathrm {SL}(2,\mathbb {R})\) as a subspace of \(T_{id}\mathcal {M}(2)=\mathbb {R}^4\). Is the line \[ \begin {pmatrix}t&0\\0&1\end {pmatrix},\quad t\in \mathbb {R} \] transverse to \(\mathrm {SL}(2,\mathbb {R})\)? Justify your answer.

Define \(\phi :\mathbb {R}^2\to \mathbb {R}^2\backslash \{0\}\) by \(\phi (x,y)=(e^x\cos y,e^x\sin y)\). Let \(\omega \) be a 1-form on \(\mathbb {R}^2\backslash \{0\}\) such that \(dy=\phi ^*\omega \). Show that \(\omega \) is closed but not exact on \(\mathbb {R}^2\backslash \{0\}\).

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 \(U\subset \mathbb {R}^n\) be a connected open set with \(p,q\in U\). Show that there exists a diffeomorphism \(\phi \) of \(\mathbb {R}^n\) such that \(\phi (p)=q\) and \(\phi \) is the identity outside of \(U\). (Hint: First assume that \(U\) is convex with compact closure and find a vector field with support in \(U\) whose flow takes \(p\) to \(q\).)

Let \(M\) be a differentiable manifold. Prove that its tangent bundle \(TM\) and and its cotangent bundle are isomorphic as smooth vector bundles.

Give two definitions of a differentiable manifold being orientable; one via charts and the other via forms. Show that the two definitions are equivalent.

Let \(X\) be a path connected, locally path connected, and semi-locally simply connected space. (Recall that \(X\) is semi-locally simply connected if every point in \(X\) has a neighborhood \(U\) such that every closed loop in \(U\) can be contracted to a point in \(X\).) Outline the construction of the universal cover \(p:\widetilde {X}\to X\). Specifically,

1. Define \(\widetilde {X}\) as a set and the projection \(p:\widetilde {X}\to X\),
2. Define the topology on \(\widetilde {X}\) (you don’t have to check that this is a topology),
3. Give an argument that \(\widetilde {X}\) is simply connected.

Indicate where each of the three assumptions on \(X\) is used.

Describe a cell structure on the real projective space \(\mathbb {R}P^n\). Explicitly describe the attaching maps of all cells. Use this cell structure to compute \(H_i(\mathbb {R}P^n;\mathbb {Z}/2)\) for \(i\ge 0\).

Let \(X\) be the space obtained from the circle \(S^1\) by attaching two 2-cells, one with degree 3 attaching map and the other with degree 5 attaching map.

1. Using van Kampen’s theorem show that \(X\) is simply connected.
2. Show that \(X\) is homotopy equivalent to the 2-sphere. Hint: Use repeateadly the fact that the spaces \(Y\cup _f e^2\) and \(Y\cup _g e^2\) obtained from \(Y\) by attaching a single 2-cell with homotopic attaching maps \(f,g\) are homotopy equivalent.

Suppose a space \(X\) can be written as the union \(\bigcup _{i=1}^n U_i\) of open subsets \(U_i\). Also assume that every \(U_i\), as well as every nonempty intersection \(U_{i_1}\cap U_{i_2}\cap \cdots \cap U_{i_k}\), is contractible. Prove that \(\widetilde {H}_i(X;\mathbb {Z})=0\) for \(i\ge n-1\). (Hint: For \(i=1,\ldots ,n-1\) let \(V_i=U_n\cap U_i\) and apply induction.)

Show that the degree of every map \(\mathbb {C}P^2\to \mathbb {C}P^2\) of the complex projective plane to itself is nonnegative.