Proposition

Assumption 1 ([Universal framed disk replacement]). Let M be a compact connected oriented smooth four-manifold with boundary. Suppose that a finite collection of proper generic immersions f_i(D^2,S^1)⟶(M,∂M), 1≤i≤k, has pairwise disjoint …

Assumption 1 ([Universal framed disk replacement]). Let M be a compact connected oriented smooth four-manifold with boundary. Suppose that a finite collection of proper generic immersions f_i(D²,S¹)⟶(M,∂M), 1≤i≤k, has pairwise disjoint embedded boundary circles, and admits generically immersed framed spheres g₁,…,g_k such that λ(f_i,g_j)=δ_ij , λ(g_i,g_j)=0, μ(g_i)=0 all i,j. Here intersections are counted in ℤ[π_1M] , with fixed whiskers, the diagonal equations are included, and μ(g_i)∈ ℤ[π_1M]/ ⟨h-h^-1 ,\,ℤ·1⟩ uses the additive quotient. Then the f_i can be replaced by pairwise disjoint locally flat embedded disks with the same boundary maps and the same induced normal framings on those boundary circles.