DEPARTMENT OF MATHEMATICS UNIVERSITY OF UTAH\ 1.2 ] Ph.D. Preliminary Exam in Geometry-Topology Fall 2000

Let \(\Sigma \) be a smooth, compact, orientable surface (2 dimensional manifold) of genus \(g\). Prove that \(\Sigma \) admits a smooth non-vanishing vectorfield if and only if \(g=1\).

Let \(M\) be a smooth, orientable \(n\)-dimensional manifold. Prove that the \(n\) dimensional DeRham cohomology with compact supports, \(H_c^n(M)\) is nonzero.

Without resorting to the DeRham Theorem, prove that \(H^2(\mathbb {R}^2)=0\) (where \(H^2(\mathbb {R}^2)\) denotes the 2 dimesional DeRham cohomology).

A. Let \[ S=\{(z_1,\ldots ,z_n)\in \mathbb {C}^n:z_1^2+\cdots +z_n^2=1\}. \] Prove that \(S\) is a smooth (real) submanifold of \(\mathbb {C}^n=\mathbb {R}^{2n}\), and find it’s dimension.

B. Prove that the orthogonal group \(O(n)\) is a smooth submanifold of the space of \(n\times n\) matrices \(M(n)=\mathbb {R}^{n^2}\), and find it’s dimension.

A. Let \(f:S^n\to T^n\) be continuous, \(n>1\), where \(S^n\) and \(T^n\) denote the \(n\)-sphere, and the \(n\) dimensional torus respectively. Prove that the degree of \(f\) is zero. (If you prefer, you may assume that \(f\) is smooth).

B. Let \(f:X\to Y\) be a covering map between two connected manifolds. Prove that if \(f\) is homotopic to a constant map then \(X\) is contractible.

A. Find an explicit CW complex structure for the complex projective space \(\mathbb {C}P^3\), and compute the singular homology groups (over \(\mathbb {Z}\)) using cellular homology.

B. Find an explicit CW complex structure for \(S^2\times S^4\), and compute the singular homology groups using cellular homology.

A. Prove the Brouwer Fixed Point Theorem using homology theory. That is, if \(f:B^n\to B^n\) is continuous, where \(B^n\) denotes the closed unit ball in \(\mathbb {R}^n\), then \(f\) has a fixed point.

B. Prove the Fundamental Theorem of Algebra using degree theory.

A. Compute the DeRham cohomology (or if you prefer, the singular homology) of \((S^2\times S^2)\#\mathbb {C}P^2\), where, \(M\#N\) denotes the connected sum of the manifolds \(M\) and \(N\).

B. Let \(p\) be a positive integer. Compute the singular homology with coefficients in \(\mathbb {Z}_p\) of real projective 3-space (\(\mathbb {Z}_p\) being the cyclic group of order \(p\)). That is, find \(H_k(\mathbb {R}P^3,\mathbb {Z}_p)\) for \(k\ge 0\).

A. Let \(F_g\) be a closed orientable surface of genus \(g\). Prove that \(\pi _1(F_g)\) is infinite, and non-abelian, provided \(g>1\).

B. Let \(G\) be the one point union of two circles (a figure eight). Draw a picture of an explicit irregular covering map \(\widetilde {G}\to G\).