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On singular cut-and-pastes in the 3-space with applications to link theory.

Fujitsugu Hosokawa, Shin'ichi Suzuki (1995)

Revista Matemática de la Universidad Complutense de Madrid

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....

On the connectivity of finite subset spaces

Jacob Mostovoy, Rustam Sadykov (2012)

Fundamenta Mathematicae

We prove that the space e x p k S m + 1 of nonempty subsets of cardinality at most k in a bouquet of m+1-dimensional spheres is (m+k-2)-connected. This, as shown by Tuffley, implies that the space e x p k X is (m+k-2)-connected for any m-connected cell complex X.

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