  This paper develops a recent line of economic theory seeking to understand
public goods economies using methods of topological analysis. Our first main
result is a very clean characterization of the economy's core (the standard
solution concept in public goods). Specifically, we prove that a point is in
the core iff it is Pareto efficient, individually rational, and the set of
points it dominates is path connected.
  While this structural theorem has a few interesting implications in economic
theory, the main focus of the second part of this paper is on a particular
algorithmic application that demonstrates its utility. Since the 1960s,
economists have looked for an efficient computational process that decides
whether or not a given point is in the core. All known algorithms so far run in
exponential time (except in some artificially restricted settings). By heavily
exploiting our new structure, we propose a new algorithm for testing core
membership whose computational bottleneck is the solution of $O(n)$ convex
optimization problems on the utility function governing the economy. It is
fairly natural to assume that convex optimization should be feasible, as it is
needed even for very basic economic computational tasks such as testing Pareto
efficiency. Nevertheless, even without this assumption, our work implies for
the first time that core membership can be efficiently tested on (e.g.) utility
functions that admit "nice" analytic expressions, or that appropriately defined
$\varepsilon$-approximate versions of the problem are tractable (by using
modern black-box $\varepsilon$-approximate convex optimization algorithms).
