Let be a natural number. Give an example of a compact -manifold such that .

# Author: dantegrevino

# Rank isn’t like dimension

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!)

# Your degree is a chance for expansion

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 .

# This’ a glue problem

Every -complex is homotopy equivalent to (the realization of) a simplicial complex.

# Too complex to be true but too symmetrical to be false

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

and .

Then and have the same distribution.

# What goes up must come down

Let such that . Prove that .

# The plane is too small 2

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.