Seminar on OpenAI's 722 Manuscripts

Let X be a (nice) topological space, and suppose that we wish to know if X is homotopy equivalent to a simplicial complex of dimension $\leq n$. An obvious necessary condition is that the cohomology of X (with arbitrary coefficients) must be concentrated in degrees $\leq n$. In the 1960's, Wall showed that this condition is also sufficient when $n \geq 3$, and asked if the same result holds in low dimensions. Last month, OpenAI gave a counterexample in the case $n=2$. In this talk, I'll review some of the mathematics connected to Wall's question and explain a simplified version of the counterexample (following even more recent work of Bartholdi-Bridson-Mikhailov-Nicholson).