DEPARTMENT OF MATHEMATICS University of Utah Ph.D. PRELIMINARY EXAMINATION IN GEOMETRY/TOPOLOGY January 2015

Let \(S^2=\{(x,y,z)\in \mathbb {R}^3\mid x^2+y^2+z^2=1\}\) be the two sphere in \(\mathbb {R}^3\). Define \(f:\mathbb {R}^3\to \mathbb {R}\) by \(f(x,y,z)=x(y-2)\).

1. Show that 0 is a critical value of \(f\) and find all critical points in \(\mathbb {R}^3\).
2. Show that 0 is a regular value of \(f\) restricted to \(S^2\) and conclude that \(M=f^{-1}(0)\cap S^2\) is manifold. What manifold is \(M\)?
3. Define \(g:\mathbb {R}^3\to \mathbb {R}\) by \(g(x,y,z)=x+z\). Calculate the critical points of \(g\) restricted to \(S^2\) and the critical points of \(g\) restricted to \(M\).

Define \(r:\mathbb {R}^2\backslash \{0\}\to \mathbb {R}^2\) by \[ r(x,y)=\left (\frac {x}{\sqrt {x^2+y^2}},\frac {y}{\sqrt {x^2+y^2}}\right ). \] Calculate \(r^*(-y\,dx+x\,dy)\) and \(r^*(x\,dx+y\,dy)\). Show that for any two form \(\omega \) on \(\mathbb {R}^2\), \(r^*\omega =0\) and conclude that for any 1-form \(\alpha \) on \(\mathbb {R}^2\), \(r^*\alpha \) is closed (even if \(\alpha \) is not).

For \(t\in (0,1)\) define a map \(f_t:S^2\to S^2\) by \[ f_t(x,y,z)=\left ( \frac {x}{\sqrt {x^2+y^2+(z+t)^2}}, \frac {y}{\sqrt {x^2+y^2+(z+t)^2}}, \frac {z+t}{\sqrt {x^2+y^2+(z+t)^2}} \right ). \] Show that \(f_t\) is a Lefschetz map with isolated fixed points at the two antipodal points with \(z=\pm 1\) and the Lefschetz number is \(L(f)=2\). Conclude that every diffeomorphism of \(S^2\) that is homotopic to the identity has a fixed point. (Hint: One can simply compute the derivative of \(f_t\) to see that on the tangent spaces of the two fixed points it is a multiple of the identity. However, if you use that \(f_t\) is a composition of the map \((x,y,z)\mapsto (x,y,z+t)\) and the 3-dimensional version of the map \(r\) from (2) then the computation is easier.)

Let \(f:M\to N\) be a differentiable map between \(n\)-manifolds and assume that \(M\) is compact and \(N\) is connected. If \(x\in N\) is a regular value of \(f\) show that \(f^{-1}(x)\) is finitely many points in \(M\). If every point in \(N\) is a regular value of \(f\) define a function \(\#:N\to \mathbb {Z}\) by letting \(\#(x)\) be the number of points in \(f^{-1}(x)\) and show that \(\#\) is constant.

Let \(f:M\to N\) be a differentiable map and \(V\) and \(W\) vector fields on \(M\) and \(N\), respectively, with \(f_*(x)V(x)=W(f(x))\) for all \(x\in M\). Assume that \(\phi _t\) and \(\psi _t\) are the respective flows of \(V\) and \(W\) and show that \(f\circ \phi _t=\psi _t\circ f\).

State Stokes’ theorem and use it to give an example (with proof) of a manifold with a form that is closed but not exact.

Regard the \(n\)-simplex \(\Delta ^n\) as a \(\Delta \)-complex in the natural way, with \((n+1)\) vertices and a single \(n\)-simplex. Find all subcomplexes \(X\subset \Delta ^n\) such that \(\operatorname {H}_{n-1}(X)\ne 0\) (with a proof that there are no others).

Let \(p,q\ge 2\) be integers and denote by \(X_{p,q}\) the join of discrete sets \(\{v_1,\ldots ,v_p\}\) and \(\{w_1,\ldots ,w_q\}\). Thus \(X_{p,q}\) is a graph with \(p+q\) vertices and \(pq\) edges.

1. What is the fundamental group of \(X_{p,q}\)?
2. Denote by \(Y_{p,q}\) the 2-complex obtained from \(X_{p,q}\) by attaching 2-cells with attaching maps \(S^1\to X_{p,q}\) homeomorphisms to all possible cycles of length 4. (Thus there are \(\binom {p}{2}\binom {q}{2}\) 2-cells.) Show that \(Y_{p,q}\) is simply connected.

State the Path Lifting Theorem for covering spaces and sketch a proof of it.

Let \(M,N\) be two closed connected oriented \(n\)-manifolds and \(f:M\to N\) a degree 1 map. (Recall that this means that \(f_*[M]=[N]\) where \([M],[N]\) are the fundamental classes of \(M,N\) respectively.)

1. Show that \(f_*:H_k(M)\to H_k(N)\) is onto for every \(k\). Hint: \(f_*(\alpha \cap f^*(z))=f_*(\alpha )\cap z\). Explain.
2. Deduce that if \(M=S_g\), \(N=S_h\) are surfaces of genus \(g\), \(h\) respectively, then \(g\ge h\).

Let \(X\) be a connected surface with nonempty boundary \(\partial X\) and \(p\in X\). Show that inclusion \[ X\backslash \{p\}\hookrightarrow X \] induces an isomorphism in \(H_1\) if and only if \(p\in \partial X\). You may use the fact that \(H_2(X)=0\) without proof.

Prove that the fundamental group of any Lie group is abelian.