DEPARTMENT OF MATHEMATICS University of Utah Ph.D. PRELIMINARY EXAMINATION IN GEOMETRY/TOPOLOGY August 21, 2006, 9:00--12:00

Show that the set \(M_1\) of real \(2\times 2\) matrices of rank 1 is a 3-dimensional submanifold of the space \(M(2,2)\cong \mathbb {R}^4\) of all real \(2\times 2\) matrices.

For which values of \(a>0\) does the hyperboloid \[ x^2+y^2-z^2=1 \] intersect the sphere \(x^2+y^2+z^2=a\) transversally (in \(\mathbb {R}^3\))?

Let \(X,Y\subset \mathbb {R}^3\) be two 1-dimensional submanifolds. Show that there is \(v\in \mathbb {R}^3\) such that \(X\) is disjoint from \(Y+v:=\{y+v\mid y\in Y\}\).

Define the Lefschetz index of an isolated fixed point of a smooth map \(f:M\to M\). Compute the Lefschetz index at 0 of the map \(f:\mathbb {C}\to \mathbb {C}\) given by \(f(z)=z+z^m\), \(m>0\).

Compute the Gaussian curvature of the hyperboloid \[ x^2+y^2-z^2=1 \] at the point \((1,0,0)\).

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.

Give an explicit example of a closed 1-form on \(\mathbb {R}^2-\{0\}\) which is not exact, and prove both properties.

Verify the formula \[ d\omega (X,Y)=X(\omega (Y))-Y(\omega (X))-\omega ([X,Y]) \] for any 1-form \(\omega \) on \(\mathbb {R}^n\) and any two vector fields \(X,Y\) on \(\mathbb {R}^n\). This is using Spivak’s normalization conventions \[ dx\wedge dy\left (\frac {\partial }{\partial x},\frac {\partial }{\partial y}\right )=1; \] using Guillemin-Pollack’s one should multiply the left-hand side by 2.

Let \(X\) be the quotient space of \(S^2\) obtained by identifying the north and south poles to a single point. Carefully define a cell complex structure on \(X\) (including a description of attaching maps) and use this to compute \(\pi _1(X)\).

For a connected CW complex \(X\), call a connected covering space \(\widetilde {X}\to X\) abelian if it is normal and has abelian deck transformation group. Show that \(X\) has an abelian covering space that is a covering space of every other abelian covering space of \(X\), and that such ‘universal’ abelian covering space is unique up to isomorphism. Describe this covering space explicitly for \(X=S^1\vee S^1\). Carefully state theorems you are using.

Define a \(\Delta \)-complex structure on the Klein bottle \(K\) and use it to compute \(H_*(K;\mathbb {Z})\), \(H_*(K;\mathbb {Z}_2)\), and \(H_*(K;\mathbb {Q})\).

Let \(X\) be the quotient space of the 2-sphere \(S^2\) under the identifications \(x\sim -x\) for \(x\) in the equator \(S^1\). Compute the homology groups \(H_i(X)\) using any method you like.

Apply the Lefschetz fixed point theorem to show that every map \(f:\mathbb {C}P^n\to \mathbb {C}P^n\) has a fixed point when \(n\) is even (state the Lefschetz fixed point theorem and any fact about the ring structure of \(H^*(\mathbb {C}P^n)\) you are using). Construct a fixed point free map \(f:\mathbb {C}P^n\to \mathbb {C}P^n\) when \(n\) is odd.

Show that a \(p\)-sheeted covering map \(M\to N\) between closed connected oriented smooth manifolds has degree \(\pm p\).

Show that if a closed orientable manifold \(M\) of dimension \(2k\) has \(H_{k-1}(M;\mathbb {Z})\) torsionfree, then \(H_k(M;\mathbb {Z})\) is also torsionfree.

Show that if the closed orientable surface \(M_g\) of genus \(g\) retracts onto a graph \(X\subset M_g\), then \(H_1(X)\) has rank at most \(g\). You may use the following algebraic fact: a nonsingular skew-symmetric bilinear pairing over the rationals \(\mathbb {Q}\), of the form \(\mathbb {Q}^n\times \mathbb {Q}^n\to \mathbb {Q}\), cannot be identically 0 when restricted to \(V\times V\) for any \(\mathbb {Q}\)-linear subspace \(V\subset \mathbb {Q}^n\) of dimension \(>n/2\).