# Category Archives: 3-manifold

## 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)

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

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

$\Box$

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

## situation at some 3D-space

that is, a curve $C$,…

## 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 $f(p)=1$.

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.