Just as links may be algebraically described as certain morphisms in the category of tangles, compact surfaces smoothly embedded in R4 may be described as certain 2-morphisms in the 2-category of ‘2-tangles in 4 dimensions’. In this announcement we give a purely algebraic characterization of the 2-category of unframed unoriented 2-tangles in 4 dimensions as the ‘free semistrict braided monoidal...

Given any connected, open 3-manifold U having finitely many ends, a non-compact 3-manifold M is constructed having the following properties: the interior of M is homeomorphic to U ; the boundary of M is the disjoint union of finitely many planes; M is not almost compact; M is eventually end-irreducible; there are no proper, incompressible embeddings of S ×R in M ; every compact subset of M is c...

The concepts of tropical semiring and tropical hypersurface, are extended to the case of an arbitrary ordered group. Then, we define the tropicalization of a polynomial with coefficients in a Krull-valued field. After a close study of the properties of the operator “tropicalization” we conclude with an extension of Kapranov’s theorem to algebraically closed fields together with a valuation over...

We investigate categorical versions of algebraically closed (= pure) embeddings, existentially closed embeddings, and the like, in the context of locally presentable categories. The definitions of S. Fakir [Fa, 75], as well as some of his results, are revisited and extended. Related preservation theorems are obtained, and a new proof of the main result of Rosický, Adámek and Borceux ([RAB, 02])...

Introduction. Let A be a commutative semi-simple Banach algebra and let A(.4) be the set of nonzero multiplicative functionals on A. Denote by A' the strongly closed span of A(^4) and by A" the Banach space adjoint of A'. Modifying a construction of R. Arens [l] we introduce a multiplication in A" under which A" becomes a commutative Banach algebra. A is algebraically isomorphic to a subalgebra...

We consider geometric decompositions of aspherical 4manifolds which fibre over 2-orbifolds. We show first that no such manifold admits infinitely many fibrations over hyperbolic base orbifolds. If E is Seifert fibred over a hyperbolic surface B and either B has at most one cone point of order 2 or the monodromy has image in SL(2,Z) then E it has a decomposition induced from a decomposition of B...

In this paper, we introduce the concept of IVF weakly m∗-continuous mappings on between IVF minimal spaces and investigate some characterizations for such mappings. Also we study the relationships IVF weakly m∗-continuous mappings and IVF M -compactness. Key ords : IVF minimal space, IVF weakly m∗-continuous, IVF M -compact, almost IVF M -compact, nearly IVF M -compact