For an embedded compact 3-manifold , implies .
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.
Consider the canonical embedding of a sphere of radius , say inside as the set of pairs with , and the curve . Calculate the homology of the intersection for as many values of as possible where , with of your liking.
Let be a finite field and a positive integer. Prove that there is an irreducible polynomial of degree in .
Let be an matrix with coefficients in a field of characteristic . Then if and only if there exist matrices such that .
Suppose is a ring that has the invariant basis number property, and assume 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 to be the Euler characteristic of any given finite free resolution:
has characteristic .
- Show the above is well defined, that is, it doesn’t depend on the given free resolution.
- Show that the Euler characteristic of any module over a noetherian or over a commutative ring is always non-negative.
- Exhibit a module over some ring that has negative Euler characteristic.
Recall that Schanuel’s lemma asserts that if are short exact sequences and is projective, then . Show this fails if the are only assumed to be flat.