One of my professors this semester very graciously offered to give me some projects to work on over the summer, and I happily accepted his offer. The only (minor) drawback is that my interests lie in topology, while his lie in Sylow subgroups and subgroup lattices (among other group theoretic topics). Algebra is my weakest subject, so it will be great to work more in it. However, I was wondering if anyone knew of any connections between Sylow subgroups and techniques in algebraic topology so that I could have some extra motivation when working through the various problems he gives me this summer!
I truly appreciate any insight you may be able to offer!
You could take a look at Homotopy properties of the poset of nontrivial p-subgroups of a group by Daniel Quillen and the many later papers citing this article. Alternatively find a chapter in Stephen D. Smith's book Subgroup Complexes that interests you. Finally you could also look at the work of Dave Benson by either picking an article from his Groups, Representations and Cohomology Preprint Archive or a chapter from his books titled "Representations and Cohomology".