On the structures of the Johnson cokernels of the basis-conjugating automorphism groups of free groups

Abstract: In this paper, we study the Johnson homomorphisms of basis-conjugating automorphism groups of free groups. We construct obstructions for the surjectivity of the Johnson homomorphisms. By using it, we determine its cokernels of degree up to four, and give further observations for degree greater than four. As applications, we give the affirmative answer for the Andreadakis problem for degree four. We show that the cup product map of the first cohomology groups of the basis-conjugating automorphism group of a free group into the second cohomology group is surjective. Finally, we calculate the twisted first cohomology groups of the braid-permutation automorphism groups of a free group.