Échelles et faisceaux sur les quotients de feuilletages
For smooth knottings of compact (not necessarily orientable) n-dimensional manifolds in (or ), 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.
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.
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.