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... more
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.
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With the idea of an eventual classification of 3-bridge links,\ we define a very nice class of 3-balls (called butterflies) with faces identified by pairs, such that the identification space is $S^{3},$ and the image of a prefered set of... more
With the idea of an eventual classification of 3-bridge links,\ we define a very nice class of 3-balls (called butterflies) with faces identified by pairs, such that the identification space is $S^{3},$ and the image of a prefered set of edges is a link. Several examples are given. We prove that every link can be represented in this way (butterfly representation). We define the butterfly number of a link, and we show that the butterfly number and the bridge number of a link coincide. This is done by defining a move on the butterfly diagram. We give an example of two different butterflies with minimal butterfly number representing the knot $8_{20}.$ This raises the problem of finding a set of moves on a butterfly diagram connecting diagrams representing the same link. This is left as an open problem.
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ABSTRACT Given a pseudo-periodic map \( f :{\Sigma_g}\rightarrow{\Sigma_g} \) Nielsen constructed a special homeomorphism which is homotopic to f and plays the role of a “standard form” in the mapping class of f , [53, Sect. 14]. In this... more
ABSTRACT Given a pseudo-periodic map \( f :{\Sigma_g}\rightarrow{\Sigma_g} \) Nielsen constructed a special homeomorphism which is homotopic to f and plays the role of a “standard form” in the mapping class of f , [53, Sect. 14]. In this Chapter, we will construct a similar standard form, slightly different from Nielsen’s, and will show its essential uniqueness. He wanted to avoid fixed points which might appear in annular neighborhoods of cut curves, while we do not care about such fixed points (Compare [22, Theorem 13.3]).
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ABSTRACT In this Chapter, we will review some basic results from Nielsen [51, 53].
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ABSTRACT The notation will be the same as in the previous chapter. The following fact is wellknown. Suppose F 0 has only normal crossings.
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In this chapter, we will complete the proof of Theorem 7.2. We wish to show that the monodromy correspondence \(\rho :\mathfrak{j}_{g} \rightarrow\mathfrak{{p}^-}_{g}\) is bijective for g ≥ 2.
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In this chapter we will prove the following theorem which was essential to the arguments in Chaps. 3 and 4.
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A periodic map f on a surface ∑ defines a quotient space ∑/f . In the case of a pseudo-periodic map f:S</font >®</font > S</font >, f:{\Sigma}\rightarrow {\Sigma}, however, the quotient space ∑/f would not be any... more
A periodic map f on a surface ∑ defines a quotient space ∑/f . In the case of a pseudo-periodic map f:S</font >®</font > S</font >, f:{\Sigma}\rightarrow {\Sigma}, however, the quotient space ∑/f would not be any reasonable space, if the term “quotient space” is taken in the usual sense, i.e., the orbit space under the action of f . To adjust this, we introduce the following definition.
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ABSTRACT A homeomorphism H:S® S¢ H:S\rightarrow S\prime between chorizo spaces is numerical if it preserves the orientation and the multiplicity on each part.
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ABSTRACT Let f:Sg®Sg f:{\Sigma_g}\rightarrow{\Sigma_g} be a pseudo-periodic map of negative twist. According to Corollary 4.5, the isomorphism class of the minimal quotient p:Sg® S[f]\pi:{\Sigma_g}\rightarrow {S[f]} and, in particular,... more
ABSTRACT Let f:Sg®Sg f:{\Sigma_g}\rightarrow{\Sigma_g} be a pseudo-periodic map of negative twist. According to Corollary 4.5, the isomorphism class of the minimal quotient p:Sg® S[f]\pi:{\Sigma_g}\rightarrow {S[f]} and, in particular, the numerical homeomorphism type of S[f] are conjugacy invariants of [f]eMg [f]\epsilon {\mathcal{M}_g} . However, the converse is not true: the minimal quotient does not necessarily determine the conjugacy class of [f]. (Nielsen [50, Sect. 15] incorrectly claims that this converse is true (compare Theorem 13.4 of [22] where this same claim is repeated)).
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A triple (M,D,ψ) is called a degenerating family of Riemann surfaces of genus g (abbreviated as degenerating family of genus g) if M is a complex surface, \( D=\{\xi \in \mathbf{C}||\xi| \(\psi:M\rightarrow D\) is a surjective proper... more
A triple (M,D,ψ) is called a degenerating family of Riemann surfaces of genus g (abbreviated as degenerating family of genus g) if M is a complex surface, \( D=\{\xi \in \mathbf{C}||\xi| \(\psi:M\rightarrow D\) is a surjective proper holomorphic map, for each \(\psi:M\rightarrow D \) , the fiber \( {F _\xi}={\psi^{-1}\Psi}(\xi)\) is connected, and \(\psi{|_{M^*}}:{M^*}\rightarrow {D^*}\) is a smooth (i.e.C∞) fiber bundle whose fiber is a Riemann surface of genus g, where \({D^*}=D-\{0\}\Psi\) and \({M^*}-{\psi^{-1}}(0)\) .
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... Hugh M. Hilden * Maria Teresa Lozano ** Jos~ Maria Montesinos ** I. Introduction. ... I/~ = i/~(I-2R+R3), and D, E and F are: Ii°i Eli] Conjugating the matrices AI, BI, and C 1 by P is equivalent to choosing a different embedding of... more
... Hugh M. Hilden * Maria Teresa Lozano ** Jos~ Maria Montesinos ** I. Introduction. ... I/~ = i/~(I-2R+R3), and D, E and F are: Ii°i Eli] Conjugating the matrices AI, BI, and C 1 by P is equivalent to choosing a different embedding of our dodecahedron in the Poincar~- Page 6. ...
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The archaeological study of the evolution of a city throughout history, due to the accumulation of cultural layers, is particularly complex because the archaeological site is presented with an overlay of traces that do not make feasible a... more
The archaeological study of the evolution of a city throughout history, due to the accumulation of cultural layers, is particularly complex because the archaeological site is presented with an overlay of traces that do not make feasible a single reading of it. This paper describes a geographical information system architecture to support archaeological research, where we use this type of
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It is proved that if M n {M^n} is a branched covering of a sphere, branched over a manifold, so is M n × S m {M^n}\\, \\times \\,{S^m} , but the number of sheets is one more. In particular, the n-dimensional torus is an n-fold simple... more
It is proved that if M n {M^n} is a branched covering of a sphere, branched over a manifold, so is M n × S m {M^n}\\, \\times \\,{S^m} , but the number of sheets is one more. In particular, the n-dimensional torus is an n-fold simple covering of S n {S^n} branched over an orientable manifold. The proof involves the development of a new technique to perform equivariant handle addition. Other consequences of this technique are given.
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There are two main results in this paper. First, we show that every closed orientable 3 3 -manifold can be constructed by taking a pair of disjoint bounded orientable surfaces in S 3 {S^3} , call them F 1 {F_1} and F 2 {F_2} ; taking... more
There are two main results in this paper. First, we show that every closed orientable 3 3 -manifold can be constructed by taking a pair of disjoint bounded orientable surfaces in S 3 {S^3} , call them F 1 {F_1} and F 2 {F_2} ; taking three copies of S 3 {S^3} ; splitting the first along F 1 {F_1} , the second along F 1 {F_1} and F 2 {F_2} , and the third along F 2 {F_2} ; and then pasting in the natural way. Second, we show that given any closed orientable 3 3 -manifold M 3 {M^3} there is a 3 3 -fold irregular branched covering space, p : M 3 → S 3 p:{M^3} \to {S^3} , such that p : M 3 → S 3 p:{M^3} \to {S^3} is the pullback of the 3 3 -fold irregular branched covering space q : S 3 → S 3 q:{S^3} \to {S^3} branched over a pair of unknotted unlinked circles.
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1. Introduction A link or knot in S3 is universal if it serves as common branching set for all closed, oriented 3-manifolds. A knot is simple if its exterior space is simple, ie any incompressible torus or annulus is parallel to the... more
1. Introduction A link or knot in S3 is universal if it serves as common branching set for all closed, oriented 3-manifolds. A knot is simple if its exterior space is simple, ie any incompressible torus or annulus is parallel to the boundary. No iterated torus knot or link is universal, but ...