In the study of surfaces in 3-manifolds, the so-called ?cut-and-paste? of surfaces is frequently used. In this paper, we generalize this method, in a sense, to singular-surfaces, and as an application, we prove that two collections of singular-disks in the 3-space R3 which span the same trivial link are link-homotopic in the upper-half 4-space R3 [0,8) keeping the link fixed. Throughout the paper, we work in the piecewise linear category, consisting of simplicial complexes and piecewise linear maps....

We investigate the knots in the boundary of the punctured complex projective plane. Our result gives an affirmative answer to a question raised by Suzuki. As an application, we answer to a question by Mathieu.

Let Γ be a 4-chart with at most two crossings. We show that if the closure of the surface braid obtained from Γ is one 2-sphere, then the sphere is a ribbon surface.

For $K=K_1\bigsqcup ...\bigsqcup K_s$ and a link map $f:K\to {ℝ}^m$ let $K^{\sim }={\bigsqcup }_{i<j}{K}_i\times K_j$, define a map $f^{\sim }:K^{\sim }\to S^{m-1}$ by $f^{\sim }(x,y)=(fx-fy)/|fx-fy|$ and a (generalized) Massey-Rolfsen invariant $\alpha \left(f\right)\in {\pi }^{m-1}\left(K\right)$ to be the homotopy class of $f^{\sim }$. We prove that for a polyhedron K of dimension ≤ m - 2 under certain (weakened metastable) dimension restrictions, α is an onto or a 1 - 1 map from the set of link maps $f:K\to {ℝ}^m$ up to link concordance to ${\pi }^{m-1}\left(K^{\sim }\right)$. If $K_1,...,K_s$ are closed highly homologically connected manifolds of dimension $p_1,...,p_s$ (in particular, homology spheres), then ${\pi }^{m-1}\left(K^{\sim }\right)\cong {\oplus }_{i<j}{\pi }_{p_i+p_j-m+1}^S$.