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## The cohomology of the n-torus

The goal of this post is to compute the cohomology of the $n$-torus $X = (S^1)^n \cong \mathbb{R}^n/\mathbb{Z}^n$ in as many ways as I can think of. Below, if no coefficient ring is specified then the coefficient ring is $\mathbb{Z}$ by default. At the end we will interpret this computation in terms of cohomology operations.

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## Decisions, decisions

Newcomb’s paradox is the name usually given to the following problem. You are playing a game against another player, often called Omega, who claims to be omniscient; in particular, Omega claims to be able to predict how you will play in the game. Assume that Omega has convinced you in some way that it is, if not omniscient, at least remarkably accurate: for example, perhaps it has accurately predicted your behavior many times in the past.

Omega places before you two opaque boxes. Box A, it informs you, contains \$1,000. Box B, it informs you, contains either \$1,000,000 or nothing. You must decide whether to take only Box B or to take both Box A and Box B, with the following caveat: Omega filled Box B with \$1,000,000 if and only if it predicted that you would take only Box B.

What do you do?

(If you haven’t heard this problem before, please take a minute to decide on an option before continuing.)

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