Let be a natural number. Give an example of a compact -manifold such that .
Let be the free group in the generators and . Consider the group morphism defined by . Prove that is a free group of infinite rank. (Hint: Think topologically!)
Let and be compact oriented smooth manifolds of the same dimension, with connected. Let be a smooth function. Take a regular value of and, for , define as if preserves orientation at and as if it reverses the orientation. We define the degree of as the number .
Prove the following extension theorem: Assume is a -dimensional sphere and is the boundary of a compact oriented -manifold . There exists a smooth extension of if and only if .
Every -complex is homotopy equivalent to (the realization of) a simplicial complex.
For , let be a complex domain and let . Let be a complex Brownian motion starting from . Set .
Suppose that there exists a conformal isomorphism such that . Set and define for
Then and have the same distribution.
Let such that . Prove that .
Let be the symmetric random walk in –, starting at the origin. So, and the transition probabilities are if and otherwise.
Let be the orthogonal projections of the random walk onto the diagonals and .
- Prove that are independent symmetric random walks in such that if and only if .
- Prove that , as , where and . Conclude that . This is equivalent to the recurrence of the random walk.