Geometry and Topology Seminar

How can you tell if some power of a word is an n-fold commutator? How do cochains "see" the fundamental group? We give explicit answers to these questions, in joint work with A. Ozbek, D. Sinha and B. Walter. Thinking of a topological space as a presentation of its fundamental groups, I will introduce a collection of invariants for words in groups, starting from the 1-st cohomology of a space, using linking numbers of letters in words. These turn out to define a universal "finite type" invariant for arbitrary groups, a result which has not been achieved for other explicit functionals on words, namely those arising from Magnus expansion and Fox calculus. Moreover, the theory works over any PID, in particular, it relates mod-p Massey products and the p-central series of groups, and presents the rational Harrison complex as a dual to the Malcev Lie algebra.