Topological methods to solve equations over groups

We present a large class of groups (no group known to be not in the class) that satisfy the Kervaire-Laudenbach Conjecture about solvability of non-singular equations over groups. We also show that certain singular equations with coefficients over groups in this class are always solvable. Our method is inspired by seminal work of Gerstenhaber-Rothaus, which was the key to prove the Kervaire-Laudenbach Conjecture for residually finite groups. Exploring the structure of the p-local homotopy type of the projective unitary group, we manage to show that many singular equations with coefficients in unitary groups can be solved in the unitary group. Joint work with Anton Klyachko.