GeomTopF2016

Prove that \(S^2\) is not a Lie group.

Give an example of a foliation on a compact manifold \(M\) such that there are infinitely many leaves, and each leaf is dense \(M\).

For any \(p\in \mathbb {R}^3\), let \(K_p\) be the kernel of the form \(2dz-3y\,dx\) taken at the point \(p\). Then \(K_p\) defines a plane field on the manifold \(\mathbb {R}^3\). Is the plane field integrable? Why?

Show that the Lie algebra of \(\mathrm {SL}_3(\mathbb {R})\) is the space of \(3\times 3\) matrices whose trace equals 0.

Suppose \(f:M\to N\) is a smooth function of smooth manifolds. Suppose \(Q\) is an embedded submanifold of \(N\), and that \(f\) is transverse to \(Q\). Prove that if \(p\in f^{-1}(Q)\), then \[ T_p(f^{-1}(Q))=(D_pf)^{-1}[T_{f(p)}Q]. \] (The above line says that the tangent space for \(f^{-1}(Q)\) at \(p\) is the inverse image of the tangent space of \(Q\) at \(f(p)\) under the differential of \(f\) at \(p\).)

Let \(G\) be a Lie group with Lie algebra \(\mathfrak {g}\). If \(\mathfrak {h}\) is a subalgebra of \(\mathfrak {g}\), then prove there is a Lie subgroup \(H\le G\) whose Lie algebra is \(\mathfrak {h}\).

Give an example of an irregular (i.e. not normal) covering space of the Klein bottle (with a proof).

Let \(X\) be a compact topological space and \(\widetilde {X}\) its universal cover. Show that \(\pi _1(X)\) is finite if and only if \(\widetilde {X}\) is compact.

Prove that \(H_i(S^2\vee S^4,\mathbb {Z})\) and \(H_i(\mathbb {C}P^2,\mathbb {Z})\) are isomorphic for all \(i\), but that \(S^2\vee S^4\) is not homotopy equivalent to \(\mathbb {C}P^2\).

Let \(\Sigma \) be a closed oriented surface of genus \(g\) and suppose that \(X\subset \Sigma \) is a graph that is a retract of \(\Sigma \). Prove that: \[ \operatorname {rank}(H_1(X))\le g. \]

Let \(M\) be a closed, orientable manifold of dimension \(2k\) and assume that \(H_{k-1}(M,\mathbb {Z})\) is torsion-free. Show that \(H_k(M,\mathbb {Z})\) is torsion free.

Let \(G\) be a group of homeomorphisms acting freely on the \(n\)-sphere \(S^n\) with \(n\) even. Prove that \(G\) has at most 2 elements.