Your degree is a chance for expansion

Let X and Y be compact oriented smooth manifolds of the same dimension, with Y connected. Let f : X \to Y be a smooth function. Take y \in Y a regular value of f and, for x \in f^{-1}(y), define o(x) as +1 if f preserves orientation at x and as -1 if it reverses the orientation. We define the degree of f as the number d(f) = \sum_{x\in f^{-1}(y)}o(x).

Prove the following extension theorem: Assume Y=S^k is a k-dimensional sphere and X=\partial W is the boundary of a compact oriented (k+1)-manifold W. There exists a smooth extension F:W\to Y of f if and only if d(f)=0.

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Your degree is a chance for expansion

The plane is too small 2

Let \{X_n|n\in\mathbb{N}_0\}\subset\mathbb{Z}^2 be the symmetric random walk in 2d, starting at the origin. So, X_0=(0,0) and the transition probabilities are p_{i,j}=1/4 if |i-j|=1 and 0 otherwise.

Let X_n^{+},X_n^{-} be the orthogonal projections of the random walk onto the diagonals \{y=x\} and \{y=-x\}.

  1. Prove that X_n^{+},X_n^{-} are independent symmetric random walks in \mathbb{Z}/2 such that X_n=0 if and only if X_n^{+}=0=X_n^{-}.
  2. Prove that p_{00}^{(2n)} = (\binom{2n}{n}2^{-2n})^2 \sim \frac{A}{n}, as n\to\infty, where A \in \mathbb{R}_{>0} and p_{00}^{(2n)}=P(X_{2n}=(0,0)|X_0=(0,0)). Conclude that \sum_{n=0}^{\infty} p_{00}^{(n)} = \infty. This is equivalent to the recurrence of the random walk.
The plane is too small 2