errata and more: “On the trigenus of Surface Bundles over S^1”


coming now…

… details about the errors, flaws, lies, successes, hopes, dreams, desire, aims, goals, joy…  about the paper:

“ON THE TRIGENUS OF SURFACE BUNDLES OVER THE S^1

now available at tri-genus of surface bundles

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  • in the abstract i asserted that the seven periodic N_3 bundles over S^1 have tri-genera (1,1,1),… recent investigations indicate that some of them have (1,1,3), this happens for the hyper-elliptic involution (t_at_b)^3=\left(\begin{array}{cc}-1 & 0\\ 0 & -1\end{array}\right) Zumbeispiel.

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what aren’t errata, are:

  • an intensive and exhaustive use of the abstract kernel C\to {\rm Out}A of an exact sequence of groups 1\to A\to E\to C\to 1, this, give the possible extensions E of A by C
  • A complete topological description of the four (nonorientable 3-manifolds) N_2-bundles over the one-sphere:

fundamental group,

holomogy,

Heegaard genus,

Seifert fibration invariants,

Bockstein’s morphism of their w_1\in H^1(E_{\varphi}:\mathbb{Z}_2),

tri-genus…

  • a list of four group automorphisms of GL_2(\mathbb{Z}) in terms of Dehn’s twists, at … Schematical remarks on the presentations of GL_2(\mathbb{Z})

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Suppose that E=N_3\times _fS^1  for some autohomeomorphism f:M\to M and E=W\cup_T(M\ddot{o}\times S^1) for an orientable 3-manifold W with a 2-torus boundary: at a section of the Mayer-Vietoris sequence, we have:

0\to H_3(T)\to H_3(W)\oplus H_3(M\ddot{o}\times S^1)\to H_3E\to  H_2(T)

\to H_2(W)\oplus H_2(M\ddot{o}\times S^1)\to H_2(E)\to H_1(T)\to

H_1(W)\oplus H_1(M\ddot{o}\times S^1)\to H_1(E)\to 0

which reduces to

0\to 0\to H_3(W)\oplus 0\to 0\to \mathbb{Z}\to H_2(W)\oplus(\mathbb{Z}\oplus\mathbb{Z}_2)

\to \mathbb{Z}\oplus\mathbb{Z}_2\to \mathbb{Z}^2\to H_1(W)\oplus(\mathbb{Z}^2)\to \mathbb{Z}^2\oplus{\mathbb{Z}_2}^2\to0

(cuz H_3(\quad)=0 for each non orientable 3-manifold), then H_3(W)=0 which contradicts W is orientable… What?

One response to “errata and more: “On the trigenus of Surface Bundles over S^1”

  1. в конце приводится четыре разных копредставления группы GL(2,Z), со ссылками на первоисточники, и даются явные изоморфизмы между ними.

    (at the end provides four different presentation of a group GL (2, Z), with links to primary sources, and give explicit isomorphisms between them.)

    read at the bottom of http://mathreader.livejournal.com/
    or direct http://mathreader.livejournal.com/26482.html

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