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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.

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