We classify isotopy classes of automorphisms (self-homeomorphisms) of 3manifolds satisfying the Thurston Geometrization Conjecture. The classification is similar to the classification of automorphisms of surfaces developed by Nielsen and Thurston, except an automorphism of a reducible manifold must first be written as a suitable composition of two automorphisms, each of which fits into our clas...

This is the third of three papers about the Compression Theorem: if Mm is embedded in Qq ×R with a normal vector field and if q−m ≥ 1, then the given vector field can be straightened (ie, made parallel to the given R direction) by an isotopy of M and normal field in Q× R. The theorem can be deduced from Gromov’s theorem on directed embeddings [4; 2.4.5 (C′)] and the first two parts gave proofs....

Let S be a closed orientable surface of genus g. The mapping class group Mod(S) of S is defined as the group of isotopy classes of orientationpreserving diffeomorphisms S → S. We will need also the extended mapping class group Mod(S) of S which is defined as the group of isotopy classes of all diffeomorphisms S → S. Let us fix an orientation of S. Then the algebraic intersection number provides...

It is certainly well-known that a suitable count of solutions to the Yang-Mills or SeibergWitten equations gives rise to invariants of a smooth 4-manifold. In recent years, these invariants have become reasonably computable, and have led to many advances in 4-manifold theory. In this paper, we use the tools of gauge theory to describe invariants of a diffeomorphism f which depend only on the is...

The space of n-sided polygons embedded in three-space consists of a smooth manifold in which points correspond to piecewise linear or “geometric” knots, while paths correspond to isotopies which preserve the geometric structure of these knots. The topology of these spaces for the case n = 6 and n = 7 is described. In both of these cases, each knot space consists of five components, but contains...

The pair (GH , ·) is called a special loop if (G, ·) is a loop with an arbitrary subloop (H, ·) called its special subloop. A special loop (GH , ·) is called a second Smarandache Bol loop (S2ndBL) if and only if it obeys the second Smarandache Bol identity (xs · z)s = x(sz · s) for all x, z in G and s in H. The popularly known and well studied class of loops called Bol loops fall into this clas...

It has long been known that a Milnor invariant with no repeated index is an invariant of link homotopy. We show that Milnor’s invariants with repeated indices are invariants not only of isotopy, but also of self Ck-moves. A self Ck-move is a natural generalization of link homotopy based on certain degree k clasper surgeries, which provides a filtration of link homotopy classes.