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.)