Category Archives: 3-manifold

splitting into handlebodies


spleeti

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Filed under 3-manifold, 3-manifolds, fiber bundle, low dimensional topology

short exact sequence and center


Let us prove:

Let 1\to A\stackrel{f}\to B\stackrel{g}\to B/A\to 1 be a short exact sequence, if the center Z(B/A)=1  then Z(B)<A

Proof:  When x\in Z(B) then g(x)\in Z(B/A), so g(x)=1.

Therefore x\in\ker (g)={\rm im}(f)=A

\Box

 

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Filed under 3-manifold, algebra, group theory, word algebra

situation at some 3D-space


situation at some 3D-space

that is, a curve C,…

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Filed under 3-manifold, 3-manifolds, calculus on manifolds, differential geometry, fiber bundle, geometry, low dimensional topology, math, multilinear algebra, topology

umbrella auf Whitney


These are three level surfaces of the function f(x,y,z)=xy^2+z^2

they are at levels 1,0,-1.

This means that the orange points p on the surface \Sigma in the left graphic, that is, p\in\Sigma=f^{-1}(1) or f(p)=1.

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Filed under 3-manifold, calculus on manifolds, differential geometry, geometry, topology

elementary classes of 3-manifolds


Strangely, there are very few places that offer simple types or simple constructions of these topological beings. Sometimes innocent guessing make us to think that only \mathbb{R}^3 is the only space we need to appreciate all the available 3d-complexity.

Today there are: “obvious ones”

To learn more,  follow to the elementary classes of 3-manifolds lists.

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Filed under 3-manifold, low dimensional topology, topology