# Drawable 3-manifolds

For an embedded compact 3-manifold $M \subseteq \mathbb{R}^3$, $H_1(M)=0$ implies $\pi_1(M)=0$.

# Rock, paper, scissors

Explain why there is a “Rock, paper, scissors” game and a “Rock, paper, scissors, lizard, Spock” game, but there cannot be a “Rock, paper, scissor, lizard” game.

# Links of singular points in complex hypersurfaces

Consider the canonical embedding of a sphere of radius $\varepsilon$, say $S_\varepsilon^3$ inside $\mathbb C^2$ as the set of pairs $(z,w)$ with $|z|^2+|w|^2=\varepsilon^2$, and the curve $X= \{(z,w):z^k + w^l =0\}$. Calculate the homology of the intersection $X\cap S_\varepsilon^3$ for as many values of $(k,l)$ as possible where $k,l>1$, with $\varepsilon$ of your liking.

# Irreducible polynomial over finite field

Let $k$ be a finite field and $n$ a positive integer. Prove that there is an irreducible polynomial of degree $n$ in $k[x]$.

# Linear… algebra?

Let $A\in M_n(k)$ be an $n\times n$ matrix with coefficients in $k$ a field of characteristic $0$. Then $\mathrm{tr}(A)=0$ if and only if there exist matrices $X,Y\in M_n(k)$ such that $A = [X,Y]=XY-YX$.

# Euler characteristic of modules

Suppose $R$ is a ring that has the invariant basis number property, and assume $M$ is a module that has a finite free resolution (that is, a free resolution of finite length by finitely generated free modules). Define the Euler characteristic of $M$ to be the Euler characteristic of any given finite free resolution:

$0\to F_n\to\cdots \to F_0\to M\to 0$

has characteristic $\sum_{i=0}^n (-1)^i {\rm rank}\, F_i$.

1. Show the above is well defined, that is, it doesn’t depend on the given free resolution.
2. Show that the Euler characteristic of any module over a noetherian or over a commutative ring is always non-negative.
3. Exhibit a module over some ring that has negative Euler characteristic.

# Schanuel’s lemma fails for flats

Recall that Schanuel’s lemma asserts that if $0\to K_i \to P_i\to M\to 0$ are short exact sequences $i=1,2$ and $P_i$ is projective, then $P_1\oplus K_2\simeq P_2\oplus K_1$. Show this fails if the $P_i$ are only assumed to be flat.