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Elementary moves for higher dimensional knots

Dennis Roseman (2004)

Fundamenta Mathematicae

For smooth knottings of compact (not necessarily orientable) n-dimensional manifolds in n + 2 (or n + 2 ), we generalize the notion of knot moves to higher dimensions. This reproves and generalizes the Reidemeister moves of classical knot theory. We show that for any dimension there is a finite set of elementary isotopies, called moves, so that any isotopy is equivalent to a finite sequence of these moves.

Elliptic operators and higher signatures

Eric Leichtnam, Paolo Piazza (2004)

Annales de l’institut Fourier

Building on the theory of elliptic operators, we give a unified treatment of the following topics: - the problem of homotopy invariance of Novikov’s higher signatures on closed manifolds, - the problem of cut-and-paste invariance of Novikov’s higher signatures on closed manifolds, - the problem of defining higher signatures on manifolds with boundary and proving their homotopy invariance.

Embedded surfaces in the 3-torus

Allan L. Edmonds (2008)

Fundamenta Mathematicae

Those maps of a closed surface to the three-dimensional torus that are homotopic to embeddings are characterized. Particular attention is paid to the more involved case when the surface is nonorientable.

Embedding of Hilbert manifolds with smooth boundary into semispaces of Hilbert spaces

J. Margalef-Roig, Enrique Outerelo-Domínguez (1994)

Archivum Mathematicum

In this paper we prove the existence of a closed neat embedding of a Hausdorff paracompact Hilbert manifold with smooth boundary into H × [ 0 , + ) , where H is a Hilbert space, such that the normal space in each point of a certain neighbourhood of the boundary is contained in H × { 0 } . Then, we give a neccesary and sufficient condition that a Hausdorff paracompact topological space could admit a differentiable structure of class with smooth boundary.

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