Classification of finite-dimensional compact topological algebras, preliminary report

A topological space $A$ is said to be compatible with a set $\Sigma$ of equations (involving operation symbols $F_t$) iff there are continuous operations $\overline F_t$ identically satisfying $\Sigma$ on $A$. The paper's main focus is on the compatibility relation for $A$ a finite simplicial complex. We review and extend the known compatibilities and incompatibilities in this context. Many such spaces are compatible with no non-trivial $\Sigma$. The paper concludes with open questions. For example, if $A$ is compatible with $\Sigma$, can this fact be deduced from the compatibility of $A$ with some $\Gamma$ that involves only ternary operations? If $A$ is compatible with $\Sigma$, can the operations $\overline F_t$ be chosen as piecewise multilinear? Is the compatibility relation (between finite $\Sigma$ and finite complexes $A$) algorithmic? For $A$ a one-simplex (i.e., a closed real interval), is there some simple characterization of the set of all $\Sigma$ compatible with $A$?