# Tag Archives: math

## not all is watching soccer

Filed under group theory, math

## producto semi-directo diagram chasing the wreath

2014/05/03 · 14:50

Filed under math

## Vektorraum und seines Dualraum sind wie Hasen

Zwei natürlich vektorräume $V$ und seines Dualraum $V^*$ entstanden wie Hasen, ein Überfluß von Vektoren räume sind ist, wenn wir sein tensorprodukt bedenken. Zum Beispiel, der zwei-Rang sind  Räume $V\otimes V$, $V\otimes V^*$ und $V^*\otimes V^*$

Aber drei-Rang  sind $V\otimes V\otimes V,V\otimes V\otimes V^*,V\otimes V^*\otimes V^*$ und $V^*\otimes V^*\otimes V^*$ und so weiter…

Eine gute Übung soll feststellen, welches ihre Basis und zu ist, bestimmen, wie die Bauteile für jedes Element in einem besonderen räume ändern, wenn wir uns die Basis ändern

Filed under algebra, differential geometry, multilinear algebra

## Circle-bundles over surfaces are more known than surface-bundles over the circle

Three dimensional $S^1$-bundles which must be based on a two-manifolds were studied since 1930’s by H. Seifert. Commonly dubbed Seifer fiber spaces, they only include bundles by using singular fibers above cone points in their Zerlegungfläschen, that is  2D-cone-orbifolds

A most general point of view is considered when we ask for 3D-manifolds foliated by circles, these must be called Seifert-Scott fiber spaces and it is needed to consider reflector lines or circles in the orbit-surface

Recently,  it was unveiled a $N_3$-bundle over $S ^1$ by using the monodromy $-\mathbb{I}$, yes, the minus-identity auto-homeomorphism, of the genus-three-non-orientable-surface $N_3$.

Its Orlik-Raymond presentation is $\{0;(n_2,2,0,1);(1,0)\}$, remember, $n_2$ corresponds to the class $No$ in Seifert symbols

So, in this compact 3-manifold we can see a double structure, as a surface bundle $N_3\subset \{0;(n_2,2,0,1);(1,0)\}\to S^1$

or as a circle bundle $S^1\subset \{0;(n_2,2,0,1);(1,0)\}\to \omega$

where $\omega$ is a 2-orbifold with three cone-points and a reflector circle