DEPARTMENT OF MATHEMATICS, UNIVERSITY OF UTAH Ph.D. PRELIMINARY EXAMINATION IN GEOMETRY/TOPOLOGY August 12, 2002, 9:00--12:00

Prove that no compact differentiable manifold (without boundary), except for the point, is contractible.

Consider a compact differentiable \(m\)-manifold \(M\) without boundary and let \(N=M\times M\). Let \(\Delta \subset N\) be the diagonal. Prove that there is no compact differentiable \(m+1\)-dimensional submanifold with boundary \(B\subset N\), so that the boundary of \(B\) is \(\Delta \).

State and prove Brouwer’s fixed-point theorem.

Prove that the Moebius band is not contractible.

Using methods of differential topology prove that for each compact orientable differentiable manifold (without boundary) \(M\) of dimension \(m\), \(H^m(M,\mathbb {R})\ne 0\), where \(H^m(M,\mathbb {R})\) is the de Rham cohomology group.

True or false (provide justification): Let \(M\subset \mathbb {R}^n\) be a smooth submanifold of dimension \(m\). Then there are \(n-m\) smooth functions \(f_1,\ldots ,f_m:\mathbb {R}^n\to \mathbb {R}\) such that:

1. \(M=\{x\in \mathbb {R}^n:f_1(x)=\cdots =f_{n-m}(x)=0\}\).
2. For each \(x\in M\) the gradients \(\nabla (f_1)(x),\ldots ,\nabla (f_{n-m})(x)\) are linearly independent.

Give definition of a (differentiable) orientable manifold. Prove that product of two orientable manifolds is again orientable.

Give definition of the tangent bundle of a smooth \(n\)-dimensional manifold. Prove that the tangent bundle is also a smooth manifold.

Compute the pull-back of the differential form \[ \omega =x\,dx\wedge dy+z\,dy\wedge dz \] in \(\mathbb {R}^3\) under the map \(f:\mathbb {R}^2\to \mathbb {R}^3\) given by \[ f(u,v)=(u^2,u+v,\sin (u+v)). \]

Define the index of a vector field on a smooth manifold. Construct an explicit vector field on the 2-sphere \(S^2\) and compute the Euler characteristic of \(S^2\) by computing the index of this vector field.

Express the Euler characteristic \(\chi (M\#N)\) in terms of \(\chi (M)\) and \(\chi (N)\).

Express the fundamental group \(\pi _1(M\#N)\) in terms of \(\pi _1(M)\) and \(\pi _1(N)\) if \(n>1\).

Prove that \(M\#N\) is orientable if and only if both \(M\) and \(N\) are orientable.

Prove that the Euler characteristic of any closed odd-dimensional manifold is zero.

The 4-dimensional cube \([0,1]\times [0,1]\times [0,1]\times [0,1]\) has 16 vertices, 32 edges and 24 2-dimensional faces. How many 3-dimensional faces does it have?

Find a degree 2 map from \(S^1\times S^1\) to \(S^2\).

Let \(\Sigma \) be a simply-connected closed 3-manifold. Prove that \(H_k(\Sigma )=H_k(S^3)\) for any integer \(k\).

Describe all possible ways in which the homology of a space can change after attaching an \(n\)-cell.

Let \(X\) be a simply-connected closed 4-manifold. Prove that \(H_k(X)\) is torsion-free for any integer \(k\).

Let \(X\) be a topological space such that \(H_0(X)=\mathbb {Z}\), \(H_1(X)=0\), \(H_2(X)=\mathbb {Z}_2\), \(H_3(X)=\mathbb {Z}_3\), \(H_4(X)=\mathbb {Z}\), \(H_k(X)=0\), \(k>4\). Compute \(H_*(X;\mathbb {Z}_2)\) and \(H^*(X;\mathbb {Z})\).