Abstract For a free group F r F_{r} of finite rank ≥ 2 r\geq 2 and non-trivial element w ∈ w\in , the primitivity π ⁢ ( stretchy="false">) \pi(w) is small...

Two knots are homology concordant if they smoothly in a cobordism. The group C ? Z (respectively, ) was previously defined as the set of spheres that bounds balls S 3 ), modulo concordance. We prove / contains ? subgroup. construct our family examples by applying filtered mapping cone formula to L-space knots, and linear independence with help connected knot complex.