We will discuss techniques from geometry and topology which have been applied to problems in the Grove program. (This talk will be independent of last week's talk.) We will briefly discuss how, using Morse theoretic results, one uses the assumptions of positive sectional curvature and symmetry to reduce geometric problems in this setting to (curvature-free) topological ones. We will then describe some of the tools from topology, including cohomology operations and results from Smith theory and group cohomology, which provide answers to some of these problems. We will also discuss a few of the topological roadblocks, as well as conjectured detours and immediate applications to the Grove program.