For every N > 0 there exists a group of deficiency less than −N that arises as the fundamental group of a smooth homology 4–sphere and also as the fundamental group of the complement of a compact contractible submanifold of the 4–sphere. A group is the fundamental group of the complement of a contractible submanifold of the n–sphere, n > 4, if and only if it is the fundamental group of a homolo...

The existence of contractible edges is a very useful tool in graph theory. For 3-connected graphs with at least six vertices, Ota and Saito (1988) prove that the set of contractible edges cannot be covered by two vertices. Saito (1990) prove that if a three-element vertex set S covers all contractible edges of a 3-connected graph G, then S is a vertex-cut of G provided that G has at least eight...

Non-contractible quality dimensions are at risk of degradation when the provision of public services is privatized. However, privatization may increase quality by fostering performance-improving innovation, particularly if combined with increased competition. We assemble a large data set on elderly care services in Sweden between 1990 and 2009 and estimate how opening to private provision affec...

This paper studies non-contractibility of a contract designer's actions in an agency model with costly monitoring. It shows that non-contractibility may lead to an explicit randomness, which is not optimal under full contractibility. The randomness mitigates non-contractibility. Its e ectiveness increases with the ex post deducibility of the non-contractible variable. Mitigation is perfect, if ...