Category Archives: topology

this studies abstractions from geometry, algebra and analysis

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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made in México


maybe, for the presentation \langle a,b,c\mid a^2=1, b^2=1 , c^2=1\rangle, is this the its Cayley’s graph?Nsub3CayleyGcC

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Filed under algebra, group theory, low dimensional topology, math, topology, word algebra

double coset counting formula


the double coset counting formula is a relation inter double cosets HaK, where a\in G and H,K subgroups in G. This is:

\#(HaK)=\frac{|H||K|}{|H\cap aKa^{-1}|}

and

\#(G/K)=\sum_a[H;H\cap aKa^{-1}]

The proof is easy.

One is to be bounded to the study of the natural map H\times K\stackrel{\phi_a}\to HaK. And it uses the second abstraction lemma.

The formula allows you to see the kinds of subgroups of arbitrary H versus K a p-SS of G, p-SS for the set of the p– Sylow subgroups.

Or, you can see that through the action H\times G/K\to G/K via h\cdot aK=haK you can get:

  • {\rm Orb}_H(aK)=\{haK\} which comply the equi-partition
  • HaK=aK\sqcup haK\sqcup...\sqcup h_taK, so \#(HaK)=m|K|, for some m\in \mathbb{N}
  • {\rm St}_H(aK)=H\cap aKa^{-1}

then you can deduce:

|G|=\sum_a\frac{|H||K|}{|H\cap aKa^{-1}|}

Now, let us use those ideas to prove the next statement:

Let G be a finite group, with cardinal |G|=q_1^{n_1}q_2^{n_2}\cdots q_t^{n_t}, where each q_i are primes with q_1<q_2<...<q_t and n_i positive integers.

Let H be a subgroup of |G| of index [G:H]=q_1.

Then, H is normal.

Proof:

By employing K=H in the double coset partition, one get the decomposition:

G=HeH\sqcup Ha_1H\sqcup...\sqcup Ha_tH

So by the double coset counting formula you arrive to:

|G/H|=1+[H:H\cap a_1Ha_1^{-1}]+\cdots+[H:H\cap a_tHa_t^{-1}]

i.e.

q_1=1+\frac{|H|}{|H\cap a_1Ha_1^{-1}|}+\cdots+\frac{|H|}{|H\cap a_tHa_t^{-1}|}

From this, we get \frac{|H|}{|H\cap a_iHa_i^{-1}|}<q_1.

But |G|=q_1|H| as well |H|=|H\cap a_iHa_i^{-1}|[H:H\cap a_iHa_i^{-1}] so

|G|=q_1|H\cap a_iHa_i^{-1}|[H:H\cap a_iHa_i^{-1}], i.e.

[H:H\cap a_iHa_i^{-1}] divides |G|

Then [H:H\cap a_iHa_i^{-1}]=1. So |H|=|H\cap a_iHa_i^{-1}| for each a_i.

This implies H=H\cap a_iHa_i^{-1} and so H=a_iHa_i^{-1} for all the posible a_i, hence, H is normal.

QED.

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Filed under algebra, categoría, category theory, fiber bundle, group theory, math, math analysis, mathematics, maths, what is math, what is mathematics

las básicas


estas son las matemáticas antes llamadas “puras”

Image

o no? :D

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puntos críticos de una función suave en el círculo


En esta breve nota demostraremos que cada función f:S^1\to{\mathbb{R}}^1 que tenga un punto crítico aislado debe de tener otro.

Entonces supongamos que existe un punto p en S^1 talque {\rm grad}f(p)=\left[\frac{\partial f}{\partial x}|_p,\frac{\partial f}{\partial y}|_p\right]=\vec{0}, pero si elegimos la parametrización \phi:\ ]0,2\pi[\longrightarrow S^1 dada por t\longmapsto\left(\begin{array}{c}\cos(t)\\ \\ \sin(t)\end{array}\right), entonces tenemos una función g=f\circ\phi para la cual, la regla de la cadena implica que g'=f'(\phi)\phi' satisface

\frac{d g}{dt}|_{t_0}=\left[\frac{\partial f}{\partial x}|_p,\frac{\partial f}{\partial y}|_p\right]\left(\begin{array}{c}-\sin\\ \\ \cos\end{array}\right)_{|_{t_0}}

i.e.

\frac{d g(t_0)}{dt}=-\frac{\partial f(p)}{\partial x}\sin(t_0)+\frac{\partial f(p)}{\partial y}\cos(t_0)

entonces si {\rm grad}f(p)=\vec{0} tendremos \frac{d g(t_0)}{dt}=0, en otras palabras g tiene puntos críticos en t_0 y en t_0+2\pi.

Pero además g(t_0)=f\circ\phi(t_0)=f(p) tanto como

g(t_0+2\pi)=f\circ\phi(t_0+2\pi)=f\circ\phi(t_0)=f(p)

es decir g(t_0)=g(t_0+2\pi) y entonces –por el teorema de Rolle– existe t_1 en el intervalo abierto ]t_0,t_0+2\pi[ talque \frac{d g(t_1)}{dt}=0.

Pero si nos restringimos a S^1\setminus\{p\} entonces f=g\circ\phi^{-1},
y así (también por la regla de la cadena) tenemos {\rm grad}f=\frac{dg}{dt}\ {\rm grad}\ \phi^{-1} i.e.

\left[\frac{\partial f}{\partial x},\frac{\partial f}{\partial y}\right]=\frac{dg}{dt}\left[\frac{\partial\phi^{-1}}{\partial x},\frac{\partial\phi^{-1}}{\partial y}\right]

que evaluando en t_1 implica

\frac{\partial f(q)}{\partial x}=\frac{dg(t_1)}{dt}\frac{\partial\phi^{-1}(q)}{\partial x}=0

tanto como

\frac{\partial f(q)}{\partial y}=\frac{dg(t_1)}{dt}\frac{\partial\phi^{-1}(q)}{\partial y}=0

por lo tanto {\rm grad}f(q)=\vec{0}, donde q\neq p \Box

critical points of functions on the circle

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Filed under calculus on manifolds, low dimensional topology, math, math analysis

Levi-Civita tensor


to see

\varepsilon^i\wedge\varepsilon^j\wedge\varepsilon^k\wedge\varepsilon^l(e_s,e_t,e_u,e_v)={\varepsilon^{ijkl}}_{stuv}

since we are requiring “canonical” duality, i.e.  covectors, \varepsilon^k:V\to R, do

\varepsilon^k(e_l)={\delta^k}_l.

one uses

\varepsilon^i\!\wedge\!\varepsilon^j\!\wedge\!\varepsilon^k\!\wedge\!\varepsilon^l\!=\!\!\sum_{\sigma\in S_4}\!(\!-1\!)^{\sigma}\!\varepsilon^{\sigma(i)}\!\otimes\!\varepsilon^{\sigma(j)}\!\otimes\!\varepsilon^{\sigma(k)}\!\otimes\!\varepsilon^{\sigma(l)}

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Filed under differential geometry, fiber bundle, geometry, multilinear algebra