# Bounding the projective plane 2

Let $n$ be a natural number. Give an example of a compact $2n$-manifold $M$ such that $\partial M = \mathbb{R}P^{2n-1}$.

# Rank isn’t like dimension

Let $F_2(a,b)$ be the free group in the generators $a$ and $b$. Consider the group morphism $f: F_2(a,b) \to \mathbb{Z}$ defined by $f(a)=f(b)=1$. Prove that $Ker(f)$ is a free group of infinite rank. (Hint: Think topologically!)

# 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 $d(f)=0$.

# This’ a glue problem

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

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

For $i\in\{1,2\}$, let $D_i$ be a complex domain and let $z_i \in D_i$. Let $B_i$ be a complex Brownian motion starting from $z_i$. Set $T_i = \inf \{t > 0 : B_i(t) \notin D_i\}$.

Suppose that there exists a conformal isomorphism $\varphi : D_1 \to D_2$ such that $\varphi (z_1) = z_2$ . Set $\tilde{T} = \int_0^T |\varphi' (B_1(t))|dt$ and define for $t< \tilde{T}$

$\tau(t)= \inf \{s>0 : \int_0^s|\varphi' (B_1(r))|dr=t\}$ and $\tilde{B}(t) = \varphi(B_1(\tau(t)))$.

Then $(\tilde{T}, (\tilde{B}(t))_{t<\tilde{T}})$ and $(T_2, (B_2(t))_{t have the same distribution.

# What goes up must come down

Let $f \in C^1(\mathbb{R})$ such that $f,f' \in L^1(\mathbb{R})$. Prove that $\int_\mathbb{R}f'=0$.

# The plane is too small 2

Let $\{X_n|n\in\mathbb{N}_0\}\subset\mathbb{Z}^2$ be the symmetric random walk in $2$$d$, 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.