Montesinos Amilibia, José María

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First Name
José María
Last Name
Montesinos Amilibia
Universidad Complutense de Madrid
Faculty / Institute
Ciencias Matemáticas
Geometría y Topología
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Now showing 1 - 10 of 112
  • Publication
    Closed oriented 3-manifolds as 3-fold branched coverings of S 3 of special type
    (Pacific Journal of Mathematics, 1976) Hilden, Hugh Michael; Montesinos Amilibia, José María; Thickstun, Thomas L.
    The first author [Amer. J. Math. 98 (1976), no. 4, 989–992] and the second author [Quart. J. Math. Oxford Ser. (2) 27 (1976), no. 105, 85–94] have shown that any closed orientable 3-manifold M is a 3-fold cover of S3 branched over a knot. In the present paper it is proved that matters may be arranged so that the curve in M which covers the branch set in S3 bounds a disc in M.
  • Publication
    4-manifolds, 3-fold covering spaces and ribbons.
    (American Mathematical Society, 1978-11) Montesinos Amilibia, José María
    It is shown that a PL, orientable 4-manifold with no 3- or 4-handles is a 3-fold irregular cover of the 4-ball, branched over a ribbon 2-manifold. The author also studies 2-fold branched cyclic covers and finds examples of surfaces in S4 whose 2-fold branched covers are again S4; this gives new examples of exotic involutions on S4 [cf. C. McA. Gordon, Proc. London Math. Soc. (3) 29 (1974), 98–110]. The conjecture that any closed, orientable 4-manifold is an irregular 4-fold branched cover of S4 is reduced to studying bordism classes of irregular 4-fold covers of S3 with covering space equal to a connected sum of copies of S1×S2.
  • Publication
    Harmonic manifolds and embedded surfaces arising from a super regular tesselation
    (World Scientific PublCo, 2017) Brumfiel, G.; Hilde, H.; Lozano, M. T.; Montesinos Amilibia, José María; Ramirez, E.; Short, H.; Tejada Cazorla, Juan Antonio; Toro, M.
    The main result of this paper is the construction of two Hyperbolic manifolds, M-1 and M-2, with several remarkable properties: (1) Every closed orientable 3-manifold is homeomorphic to the quotient space of the action of a group of order 16 on some covering space of M-1 or M-2. (2) M1 and M2 are tesselated by 16 dodecahedra such that the pentagonal faces of the dodecahedra fit together in a certain way. (3) There are 12 closed non-orientable hyperbolic surfaces of Euler characteristic -2 each of which is tesselated by regular right angled pentagons and embedded in M1 or M2. The union of the pentagonal faces of the tesselating dodecahedra equals the union of the 12 images of the embedded surfaces of Euler characteristic -2.
  • Publication
    Uncountably many wild knots whose cyclic branched covering are S3
    (Springer, 2003) Montesinos Amilibia, José María
    According to the confirmed Smith Conjecture [The Smith conjecture (New York, 1979), Academic Press, Orlando, FL, 1984;], a tame knot in the 3-sphere has a cyclic branched covering that is also the 3-sphere only if it is trivial. Here the author produces a nontrivial, wild knot whose n-fold cyclic branched cover is S3, for all n. In fact there are uncountably many inequivalent knots with this property, and the knots can be chosen to bound an embedded disk that is tame in its interior. One might conjecture that any wild knot whose nontrivial n-fold cyclic branched cover is S3 must bound such a disk that is tame in its interior.
  • Publication
    On hyperbolic 3-manifolds with an infinite number of fibrations over S1
    (Cambridge Univ Press, 2006-01) Hilden, Hugh Michael; Lozano Imízcoz, María Teresa; Montesinos Amilibia, José María
    W. P. Thurston [Mem. Amer. Math. Soc. 59 (1986), no. 339, i–vi and 99–130;] showed that if a hyperbolic 3-manifold with b1>1 fibers over S1, then it fibers in infinitely many different ways. In this paper, the authors consider a certain family of hyperbolic manifolds, obtained as branched covers of the 3-torus. They show explicitly that each manifold in this family has infinitely many different fibrations.
  • Publication
    Two questions on Heegaard diagrams of S3.
    (American Mathematical Society, 1988-02) Montesinos Amilibia, José María
    An important open question about 3-manifolds is whether or not there exists an algorithm for recognizing S3. The author poses two questions about Heegaard diagrams of S3, appropriate answers to either of which would give such an algorithm. If a Heegaard diagram contains either a wave or a cancelling pair, then one can find an equivalent diagram of smaller complexity (in the latter case, of smaller genus). Every nontrivial genus-2 diagram of S3 contains a wave [T. Homma, M. Ochiai and M. Takahashi Osaka J. Math 17 (1980), no. 3, 625–648; MR0591141 (82i:57013)], but this is false for higher genera. The author's first question is whether there are any Heegaard diagrams of S3 without waves and without cancelling pairs. {Reviewer's remark: An example of such a diagram is contained in an article of Ochiai [ibid. 22 (1985), no. 4, 871–873; MR0815455 (87a:57020)].} Given a Heegaard diagram, there is a reduction procedure which produces a so-called pseudominimal diagram. W. Haken [in Topology of manifolds (Athens, Ga., 1969), 140–152, Markham, Chicago, Ill., 1970; MR0273624 (42 #8501)] has suggested that perhaps the only pseudominimal diagrams of S3 are the trivial ones; no counterexamples are known. The author suggests a further reduction step which might be applied to a pseudominimal diagram, yielding several partial diagrams. If any of these has a cancelling pair, then the genus of the original diagram can be reduced. An example is given to show that, in general, for manifolds different from S3, even this enhanced procedure does not always detect the reducibility of a Heegaard splitting. The author's second question, however, is whether it does for splittings of S3. Thus the author is suggesting a possible algorithm for recognizing S3 which allows for the existence of nontrivial pseudominimal diagrams of S3.
  • Publication
    On twins in the four-sphere. I.
    (Oxford University Press, 1983) Montesinos Amilibia, José María
    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.
  • Publication
    Classical tessellations and three-manifolds
    (Springer-Verlag, 1987) Montesinos Amilibia, José María
    Symmetry exists in nature and art. The rotational symmetry of a simple daisy must have at one time or another stirred geometric thoughts in the least mathematical mind. On a higher scale, the florets of Helianthus maximus naturally arrange themselves into two sets of logarithmic spirals with opposite sense of coiling. Indeed it is possible to argue with some force that mathematics is the study of pattern—the common theme that links symmetry in nature with our rational attempts at understanding. It is entirely appropriate that the author of this beautiful book comes from the country which produced the intricate wall decorations of the Alhambra in Granada. (Incidentally, for the benefit of Hispanophiles, this book produces photographic evidence once and for all that all 17 plane symmetry patterns appear in the Alhambra.) So what do 3-manifolds have to do with tessellations? At first sight there does not seem to be much of a connection, but consider the following example: Let M be the set of tilings of R2 by square tiles of unit side. Then an element of M is determined by a point z in R2 as a vertex of a tile and the angle which an edge makes with the x-axis. But the point of R2 is only well defined modulo unit translations and the angle is only well defined modulo π/2. So M is the quotient torus of the action on T2 with monodromy (01−10). So M is a 3-manifold: but there is more! Consider the circle action on points of M (tilings) as follows: If M is a tiling and α is an angle let α⋅m denote the tiling obtained from M by rotating the plane about the origin 0 through an angle α. Since this action has no fixed points M is decomposed as a disjoint union of circle fibres corresponding to the orbits under the S1 action. The orbits return after a complete turn except when 0 lies at a vertex, at the centre of a tile or at the centre of an edge. So M has a Seifert fibre structure with three exceptional fibres of order 4, 4 and 2. The contents of the book given by chapter titles are (1) S1-bundles over surfaces. (2) Manifolds of tessellations in the Euclidean plane. (3) Manifolds of spherical tessellations. (4) Seifert manifolds. (5) Manifolds of hyperbolic tessellations. There are also two appendices: (A) on orbifolds and (B) on the hyperbolic plane. The book is written in a readable style with many examples and clear diagrams together with three pages of colour photographs—mineral crystals and tessellations from the Alhambra. The reviewer greatly enjoyed reading this book and has only a few criticisms—there are only a few misprints, Figure 16 on page 64 is not clear, at least to me, and Exercise 1, page 76, is loosely worded since S˜2222 has the presentation {z,a,b,c,d|a2=b2=c2=d2=z,abcd=z2}. For those mathematicians with interests in this area this book will make a valuable contribution to their library.
  • Publication
    A method of constructing 3-manifolds and its application to the computation of the μ-invariant
    (American Mathematical Society, 1978) Hilden, Hugh Michael; Montesinos Amilibia, José María; Milgram, James R.
    If F and G are disjoint compact surfaces with boundary in S3=∂D4, let F′ and G′ be the result of pushing F and G into the interior of D4, keeping ∂F and ∂G fixed. The authors give an explicit cut and paste description of an irregular 3-fold branched cover W4(F,G) of D4 branched along F∪G. If M3=∂W4(F,G), they say that (F,G) "represents M3 by bands''. Their main result is that any closed oriented 3-manifold can be so represented. In particular, any such 3-manifold bounds a simply connected W4 which is an irregular 3-fold branched cover of D4. Moreover, F and G can always be chosen in a rather special form which leads to a formula for the μ-invariant of M3 when M3 is a (Z/2)-homology sphere.
  • Publication
    The Chern-Simons invariants of hyperbolic manifolds via covering spaces
    (Oxford University Press, 1991) Hilden, Hugh Michael; Lozano Imízcoz, María Teresa; Montesinos Amilibia, José María
    The Chern-Simons invariant was extended to 3-dimensional geometric cone manifolds in [H. M. Hilden, M. T. Lozano and J. M. Montesinos-Amilibia, J. Math. Sci. Univ. Tokyo 3 (1996), no. 3, 723–744; MR1432115 (98h:57056)]. The present paper is about the behavior of this generalized invariant under change of orientation and with respect to virtually regular coverings. (A virtually regular cover is a cover with the property that the branching index is constant along the fiber over each point of the branching set.) As one might suspect, CS(−M)=−CS(M). However, unlike the volume, the Chern-Simons invariant is not multiplicative with respect to branched coverings. There is a correction term depending on the intersection number of longitudes of inverse images of the singular set with the inverse image of the longitude of the singular set. The paper concludes with applications of the main formula to specific examples.