Pseudo-periodic homeomorphisms and degeneration of Riemann surfaces

We will announce two theorems. The first theorem will classify all topological types of degenerate fibers appearing in one-parameter families of Riemann surfaces, in terms of ``pseudoperiodic'' surface homeomorphisms. The second theorem will give a complete set of conjugacy invariants for the mapping classes of such homeomorphisms. This latter result implies that Nielsen's set of invariants [{\it Surface transformation classes of algebraically finite type}, Collected Papers 2, Birkh\"auser (1986)] is not complete.Comment: 6 page

Branched folded coverings and 3-manifolds

Under the framework of Fox spreads and its completions a theory thar generalizes coverings (folding covering theory) and a theroy that generalizes branched coverings (branched folding theory) is defined and some properties are proved. Two applications to 3-manifold theory are given.A problem is stated

A new classification of automorphisms in a vector space. (Spanish: Una nueva clasificación de automorfismos en un espacio vectorial)

The author gives a very interesting new equivalence for automorphisms u and u′ of a finite-dimensional vector space E over a field K. He defines u and u′ to be equivalent if there exists another automorphism H of E such that every u-invariant subspace is also H−1u′ H-invariant, and every u′-invariant subspace is also H−1u H-invariant. Suppose now that K contains the sets Q={q1,⋯,qr}, Q′={q1′,⋯,qs′} of roots of the minimal polynomials of u,u′, respectively. The author proves then that u and u′ are equivalent if and only if there exists a bijection λ:Q→Q′ such that the elementary divisor degrees of qi and λ(qi) are identical for i=1,⋯,r=s. For a given space E over an algebraically closed field K, the number of equivalence classes of automorphisms is finite. The author notes that this equivalence, less fine than similarity, preserves many of the geometric properties of automorphisms. {The reviewer notes that the definition of equivalence can be extended to arbitrary endomorphisms; the results quoted above seem to carry over to the more general case.

Heegaard diagrams for closed 4-manifolds

Let W4=H0∪λH1∪μH2∪γH3∪H4 be a handle decomposition of a closed, orientable PL 4-manifold. Let M4=H0∪λH1∪μH2 and let N4=N4(γ)=γH3∪H4=γ#(S1×B3). Then W4 is M4∪N4, identified along ∂M4=∂N4=γ#(S1×S2). The first observation in this paper is that W4 does not depend upon the method of attaching N4, as a consequence of a theorem of F. Laudenbach and V. Poénaru [Bull. Soc. Math. France 100 (1972), 337–344;], who showed (implicitly) that the homotopy group of ∂N4 is generated by maps which extend to N4. Dually, W4 does not depend upon the method of attaching H0∪λH1≅N4(λ). Hence W4 depends only on the cobordism C(λ,γ) from λ#(S1×S2) to γ#(S1×S2) defined by the 2-handles. The author calls (W4,C(λ,γ)) a Heegaard splitting of W4. The associated Heegaard diagram is a pair (λ#S1×S2,w) where w is a framed link in λ#S1×S2. It is noted that an arbitrary pair (λ#S1×S2,w) need not be a Heegaard diagram for a 4-manifold. Two diagrams are equivalent if there is a homeomorphism of pairs which preserves the framings. Moves are given which relate any two Heegaard diagrams for the same 4-manifold. The completeness of these moves is proved in Theorem 3 (and also Theorem 3′). A concept of a dual diagram is introduced. It is not known whether each Heegaard diagram is geometrically realizable as the diagram for some closed 4-manifold

On twins in the four-sphere. I.

E. C. Zeeman [Trans. Amer. Math. Soc. 115 (1965), 471–495; MR0195085 (33 #3290)] introduced the process of twist spinning a 1-knot to obtain a 2-knot (in S4), and proved that a twist-spun knot is fibered with finite cyclic structure group. R. A. Litherland [ibid. 250 (1979), 311–331; MR0530058 (80i:57015)] generalized twist-spinning by performing during the spinning process rolling operations and other motions of the knot in three-space. The first paper generalizes those results by introducing the concept of a twin. A twin W is a subset of S4 made up of two 2-knots R and S that intersect transversally in two points. The prototype of a twin is the n-twist spun of K (that is, the union of the n-twist spun knot of K and the boundary of the 3-ball in which the original knot lies). The exterior of a twin, X(W), is the closure of S4−N(W), where N(W) is a regular neighborhood of W in S4. The first paper considers properties of X(W), and uses these to characterize the automorphisms of a 2-torus standardly embedded in S4, which extend to S4, and also to prove that any homotopy sphere obtained by Dehn surgery on such a 2-torus is the real S4. The second paper is devoted to the fibration problem, i.e. given a twin in S4, try to understand what surgeries in W give a twin W′ which has a component that is a fibered knot (as in the Zeeman theorem). This approach yields alternative proofs of the twist-spinning theorem of Zeeman, and of the roll-twist spinning results of Litherland. New fibered 2-knots are produced through these methods

On continuous families of geometric Seifert conemanifold structures

In this paper, dedicated to Prof. Lou Kauffman, we determine the Thurston’s geometry possesed by any Seifert fibered conemanifold structure in a Seifert manifold with orbit space (Formula presented.) and no more than three exceptional fibers, whose singular set, composed by fibers, has at most three components which can include exceptional or general fibers (the total number of exceptional and singular fibers is less than or equal to three). We also give the method to obtain the holonomy of that structure. We apply these results to three families of Seifert manifolds, namely, spherical, Nil manifolds and manifolds obtained by Dehn surgery on a torus knot (Formula presented.). As a consequence we generalize to all torus knots the results obtained in [Geometric conemanifolds structures on (Formula presented.), the result of (Formula presented.) surgery in the left-handed trefoil knot (Formula presented.), J. Knot Theory Ramifications 24(12) (2015), Article ID: 1550057, 38pp., doi: 10.1142/S0218216515500571] for the case of the left handle trefoil knot. We associate a plot to each torus knot for the different geometries, in the spirit of Thurston