An simple intermediate step of the proof that $partial^2 = 0$ in the case of singular homology












0












$begingroup$


Let $Delta_q$ be the standard $q$-simplex. The function $F_q^s : Delta_{q-1} to Delta_q$ is the $s$-th face of $Delta_q$, defined as the correstriction to $Delta_q$ of $(e_0,dots,e_{s-1},hat{e_s},e_{s+1},dots,e_q)$, the map $Delta_{q-1} to mathbb{R}^q$ that sends $e_i$ to itself for $i < s$ and $e_i$ to $e_{i+1}$ if $i geq s$.



I'm stuck on trying to prove the following elementary result,




Lemma. Let $0 leq j < i leq q$. Then $F_q^{i}F_{q-1}^{j} = F_q^{j}F_{q-1}^{i-1}$.




I apologize for the simplicity of the question, but I cannot seem to pinpoint where am I making the mistake, so any help is greatly appreciated.



By a direct calculation,



$$
begin{align}
F_q^{i}F_{q-1}^{j}(e_l) &= cases{F_q^{i}(e_l) text{ if $l < j$}\ F_q^{i}(e_{l+1}) text{ if $l geq j$}} \&=
cases{
e_l quad text{ if $l < j, l< i$}\
e_{l+1} text{ if $l < j, l geq i$}\
e_{l+1} text{ if $l geq j, l < i$}\
e_{l+2} text{ if $l geq j, l geq i$}} \&=
cases{
e_l quad text{ if $l < j$}\
e_{l+1} text{ if $l geq j, l < i$}\
e_{l+2} text{ if $l geq i$}}
end{align}
$$



and



$$
begin{align}
F_q^{j}F_{q-1}^{i-1}(e_l) &= cases{F_q^{j}(e_l) text{ if $l < i-1$}\ F_q^{j}(e_{l+1}) text{ if $l geq i-1$}} \ &=
cases{
e_l quad text{ if $l < i-1, l< j$}\
e_{l+1} text{ if $l < i-1, l geq j$}\
e_{l+1} text{ if $l geq i-1, l < j$}\
e_{l+2} text{ if $l geq i-1, l geq j$}} \ &=
cases{
e_l quad text{ if $l< j$}\
e_{l+1} text{ if $l < i-1, l geq j$}\
e_{l+2} text{ if $l geq i-1$}}
end{align}
$$



These don't seem to coincide. Where have I gone wrong?






algebraic-topology proof-explanation







share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    Many homology theories are described by a collection of maps called face and degeneracy maps, satisfying a set of axioms called simplicial identities (you may read more about them at the nLab). The ones you posted are precisely the ones which guarantee that their alternating sum is a differential.
    $endgroup$
    – F M
    Dec 17 '18 at 15:58
















0












$begingroup$


Let $Delta_q$ be the standard $q$-simplex. The function $F_q^s : Delta_{q-1} to Delta_q$ is the $s$-th face of $Delta_q$, defined as the correstriction to $Delta_q$ of $(e_0,dots,e_{s-1},hat{e_s},e_{s+1},dots,e_q)$, the map $Delta_{q-1} to mathbb{R}^q$ that sends $e_i$ to itself for $i < s$ and $e_i$ to $e_{i+1}$ if $i geq s$.



I'm stuck on trying to prove the following elementary result,




Lemma. Let $0 leq j < i leq q$. Then $F_q^{i}F_{q-1}^{j} = F_q^{j}F_{q-1}^{i-1}$.




I apologize for the simplicity of the question, but I cannot seem to pinpoint where am I making the mistake, so any help is greatly appreciated.



By a direct calculation,



$$
begin{align}
F_q^{i}F_{q-1}^{j}(e_l) &= cases{F_q^{i}(e_l) text{ if $l < j$}\ F_q^{i}(e_{l+1}) text{ if $l geq j$}} \&=
cases{
e_l quad text{ if $l < j, l< i$}\
e_{l+1} text{ if $l < j, l geq i$}\
e_{l+1} text{ if $l geq j, l < i$}\
e_{l+2} text{ if $l geq j, l geq i$}} \&=
cases{
e_l quad text{ if $l < j$}\
e_{l+1} text{ if $l geq j, l < i$}\
e_{l+2} text{ if $l geq i$}}
end{align}
$$



and



$$
begin{align}
F_q^{j}F_{q-1}^{i-1}(e_l) &= cases{F_q^{j}(e_l) text{ if $l < i-1$}\ F_q^{j}(e_{l+1}) text{ if $l geq i-1$}} \ &=
cases{
e_l quad text{ if $l < i-1, l< j$}\
e_{l+1} text{ if $l < i-1, l geq j$}\
e_{l+1} text{ if $l geq i-1, l < j$}\
e_{l+2} text{ if $l geq i-1, l geq j$}} \ &=
cases{
e_l quad text{ if $l< j$}\
e_{l+1} text{ if $l < i-1, l geq j$}\
e_{l+2} text{ if $l geq i-1$}}
end{align}
$$



These don't seem to coincide. Where have I gone wrong?






algebraic-topology proof-explanation







share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    Many homology theories are described by a collection of maps called face and degeneracy maps, satisfying a set of axioms called simplicial identities (you may read more about them at the nLab). The ones you posted are precisely the ones which guarantee that their alternating sum is a differential.
    $endgroup$
    – F M
    Dec 17 '18 at 15:58














0












0








0





$begingroup$


Let $Delta_q$ be the standard $q$-simplex. The function $F_q^s : Delta_{q-1} to Delta_q$ is the $s$-th face of $Delta_q$, defined as the correstriction to $Delta_q$ of $(e_0,dots,e_{s-1},hat{e_s},e_{s+1},dots,e_q)$, the map $Delta_{q-1} to mathbb{R}^q$ that sends $e_i$ to itself for $i < s$ and $e_i$ to $e_{i+1}$ if $i geq s$.



I'm stuck on trying to prove the following elementary result,




Lemma. Let $0 leq j < i leq q$. Then $F_q^{i}F_{q-1}^{j} = F_q^{j}F_{q-1}^{i-1}$.




I apologize for the simplicity of the question, but I cannot seem to pinpoint where am I making the mistake, so any help is greatly appreciated.



By a direct calculation,



$$
begin{align}
F_q^{i}F_{q-1}^{j}(e_l) &= cases{F_q^{i}(e_l) text{ if $l < j$}\ F_q^{i}(e_{l+1}) text{ if $l geq j$}} \&=
cases{
e_l quad text{ if $l < j, l< i$}\
e_{l+1} text{ if $l < j, l geq i$}\
e_{l+1} text{ if $l geq j, l < i$}\
e_{l+2} text{ if $l geq j, l geq i$}} \&=
cases{
e_l quad text{ if $l < j$}\
e_{l+1} text{ if $l geq j, l < i$}\
e_{l+2} text{ if $l geq i$}}
end{align}
$$



and



$$
begin{align}
F_q^{j}F_{q-1}^{i-1}(e_l) &= cases{F_q^{j}(e_l) text{ if $l < i-1$}\ F_q^{j}(e_{l+1}) text{ if $l geq i-1$}} \ &=
cases{
e_l quad text{ if $l < i-1, l< j$}\
e_{l+1} text{ if $l < i-1, l geq j$}\
e_{l+1} text{ if $l geq i-1, l < j$}\
e_{l+2} text{ if $l geq i-1, l geq j$}} \ &=
cases{
e_l quad text{ if $l< j$}\
e_{l+1} text{ if $l < i-1, l geq j$}\
e_{l+2} text{ if $l geq i-1$}}
end{align}
$$



These don't seem to coincide. Where have I gone wrong?






algebraic-topology proof-explanation







share|cite|improve this question











$endgroup$




Let $Delta_q$ be the standard $q$-simplex. The function $F_q^s : Delta_{q-1} to Delta_q$ is the $s$-th face of $Delta_q$, defined as the correstriction to $Delta_q$ of $(e_0,dots,e_{s-1},hat{e_s},e_{s+1},dots,e_q)$, the map $Delta_{q-1} to mathbb{R}^q$ that sends $e_i$ to itself for $i < s$ and $e_i$ to $e_{i+1}$ if $i geq s$.



I'm stuck on trying to prove the following elementary result,




Lemma. Let $0 leq j < i leq q$. Then $F_q^{i}F_{q-1}^{j} = F_q^{j}F_{q-1}^{i-1}$.




I apologize for the simplicity of the question, but I cannot seem to pinpoint where am I making the mistake, so any help is greatly appreciated.



By a direct calculation,



$$
begin{align}
F_q^{i}F_{q-1}^{j}(e_l) &= cases{F_q^{i}(e_l) text{ if $l < j$}\ F_q^{i}(e_{l+1}) text{ if $l geq j$}} \&=
cases{
e_l quad text{ if $l < j, l< i$}\
e_{l+1} text{ if $l < j, l geq i$}\
e_{l+1} text{ if $l geq j, l < i$}\
e_{l+2} text{ if $l geq j, l geq i$}} \&=
cases{
e_l quad text{ if $l < j$}\
e_{l+1} text{ if $l geq j, l < i$}\
e_{l+2} text{ if $l geq i$}}
end{align}
$$



and



$$
begin{align}
F_q^{j}F_{q-1}^{i-1}(e_l) &= cases{F_q^{j}(e_l) text{ if $l < i-1$}\ F_q^{j}(e_{l+1}) text{ if $l geq i-1$}} \ &=
cases{
e_l quad text{ if $l < i-1, l< j$}\
e_{l+1} text{ if $l < i-1, l geq j$}\
e_{l+1} text{ if $l geq i-1, l < j$}\
e_{l+2} text{ if $l geq i-1, l geq j$}} \ &=
cases{
e_l quad text{ if $l< j$}\
e_{l+1} text{ if $l < i-1, l geq j$}\
e_{l+2} text{ if $l geq i-1$}}
end{align}
$$



These don't seem to coincide. Where have I gone wrong?






algebraic-topology proof-explanation




algebraic-topology proof-explanation






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 16 '18 at 6:39







Guido A.

















asked Dec 16 '18 at 6:33









Guido A.Guido A.

7,8701730




7,8701730








  • 1




    $begingroup$
    Many homology theories are described by a collection of maps called face and degeneracy maps, satisfying a set of axioms called simplicial identities (you may read more about them at the nLab). The ones you posted are precisely the ones which guarantee that their alternating sum is a differential.
    $endgroup$
    – F M
    Dec 17 '18 at 15:58














  • 1




    $begingroup$
    Many homology theories are described by a collection of maps called face and degeneracy maps, satisfying a set of axioms called simplicial identities (you may read more about them at the nLab). The ones you posted are precisely the ones which guarantee that their alternating sum is a differential.
    $endgroup$
    – F M
    Dec 17 '18 at 15:58








1




1




$begingroup$
Many homology theories are described by a collection of maps called face and degeneracy maps, satisfying a set of axioms called simplicial identities (you may read more about them at the nLab). The ones you posted are precisely the ones which guarantee that their alternating sum is a differential.
$endgroup$
– F M
Dec 17 '18 at 15:58




$begingroup$
Many homology theories are described by a collection of maps called face and degeneracy maps, satisfying a set of axioms called simplicial identities (you may read more about them at the nLab). The ones you posted are precisely the ones which guarantee that their alternating sum is a differential.
$endgroup$
– F M
Dec 17 '18 at 15:58










1 Answer
1






active

oldest

votes


















1












$begingroup$

You did not correctly calculate $F^s_q(e_{l+1})$. You have
$$
begin{align}
F_q^{i}F_{q-1}^{j}(e_l) &= cases{F_q^{i}(e_l) text{ if $l < j$}\ F_q^{i}(e_{l+1}) text{ if $l geq j$}} \&=
cases{
e_l quad text{ if $l < j, l< i$}\
e_{l+1} text{ if $l < j, l geq i$}\
e_{l+1} text{ if $l geq j, l+1 < i$}\
e_{l+2} text{ if $l geq j, l+1 geq i$}} \&=
cases{
e_l quad text{ if $l < j$}\
e_{l+1} text{ if $l geq j, l < i-1$}\
e_{l+2} text{ if $l geq i-1$}}
end{align}
$$



and



$$
begin{align}
F_q^{j}F_{q-1}^{i-1}(e_l) &= cases{F_q^{j}(e_l) text{ if $l < i-1$}\ F_q^{j}(e_{l+1}) text{ if $l geq i-1$}} \ &=
cases{
e_l quad text{ if $l < i-1, l< j$}\
e_{l+1} text{ if $l < i-1, l geq j$}\
e_{l+1} text{ if $l geq i-1, l+1< j$}\
e_{l+2} text{ if $l geq i-1, l+1 geq j$}} \ &=
cases{
e_l quad text{ if $l< j$}\
e_{l+1} text{ if $l < i-1, l geq j$}\
e_{l+2} text{ if $l geq i-1$}}
end{align}
$$






share|cite|improve this answer











$endgroup$













  • $begingroup$
    I see it now! I had rewritten this many times, but it seems that I had fixed my mistake already. Thanks a lot!
    $endgroup$
    – Guido A.
    Dec 17 '18 at 1:55











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1 Answer
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1 Answer
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active

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1












$begingroup$

You did not correctly calculate $F^s_q(e_{l+1})$. You have
$$
begin{align}
F_q^{i}F_{q-1}^{j}(e_l) &= cases{F_q^{i}(e_l) text{ if $l < j$}\ F_q^{i}(e_{l+1}) text{ if $l geq j$}} \&=
cases{
e_l quad text{ if $l < j, l< i$}\
e_{l+1} text{ if $l < j, l geq i$}\
e_{l+1} text{ if $l geq j, l+1 < i$}\
e_{l+2} text{ if $l geq j, l+1 geq i$}} \&=
cases{
e_l quad text{ if $l < j$}\
e_{l+1} text{ if $l geq j, l < i-1$}\
e_{l+2} text{ if $l geq i-1$}}
end{align}
$$



and



$$
begin{align}
F_q^{j}F_{q-1}^{i-1}(e_l) &= cases{F_q^{j}(e_l) text{ if $l < i-1$}\ F_q^{j}(e_{l+1}) text{ if $l geq i-1$}} \ &=
cases{
e_l quad text{ if $l < i-1, l< j$}\
e_{l+1} text{ if $l < i-1, l geq j$}\
e_{l+1} text{ if $l geq i-1, l+1< j$}\
e_{l+2} text{ if $l geq i-1, l+1 geq j$}} \ &=
cases{
e_l quad text{ if $l< j$}\
e_{l+1} text{ if $l < i-1, l geq j$}\
e_{l+2} text{ if $l geq i-1$}}
end{align}
$$






share|cite|improve this answer











$endgroup$













  • $begingroup$
    I see it now! I had rewritten this many times, but it seems that I had fixed my mistake already. Thanks a lot!
    $endgroup$
    – Guido A.
    Dec 17 '18 at 1:55
















1












$begingroup$

You did not correctly calculate $F^s_q(e_{l+1})$. You have
$$
begin{align}
F_q^{i}F_{q-1}^{j}(e_l) &= cases{F_q^{i}(e_l) text{ if $l < j$}\ F_q^{i}(e_{l+1}) text{ if $l geq j$}} \&=
cases{
e_l quad text{ if $l < j, l< i$}\
e_{l+1} text{ if $l < j, l geq i$}\
e_{l+1} text{ if $l geq j, l+1 < i$}\
e_{l+2} text{ if $l geq j, l+1 geq i$}} \&=
cases{
e_l quad text{ if $l < j$}\
e_{l+1} text{ if $l geq j, l < i-1$}\
e_{l+2} text{ if $l geq i-1$}}
end{align}
$$



and



$$
begin{align}
F_q^{j}F_{q-1}^{i-1}(e_l) &= cases{F_q^{j}(e_l) text{ if $l < i-1$}\ F_q^{j}(e_{l+1}) text{ if $l geq i-1$}} \ &=
cases{
e_l quad text{ if $l < i-1, l< j$}\
e_{l+1} text{ if $l < i-1, l geq j$}\
e_{l+1} text{ if $l geq i-1, l+1< j$}\
e_{l+2} text{ if $l geq i-1, l+1 geq j$}} \ &=
cases{
e_l quad text{ if $l< j$}\
e_{l+1} text{ if $l < i-1, l geq j$}\
e_{l+2} text{ if $l geq i-1$}}
end{align}
$$






share|cite|improve this answer











$endgroup$













  • $begingroup$
    I see it now! I had rewritten this many times, but it seems that I had fixed my mistake already. Thanks a lot!
    $endgroup$
    – Guido A.
    Dec 17 '18 at 1:55














1












1








1





$begingroup$

You did not correctly calculate $F^s_q(e_{l+1})$. You have
$$
begin{align}
F_q^{i}F_{q-1}^{j}(e_l) &= cases{F_q^{i}(e_l) text{ if $l < j$}\ F_q^{i}(e_{l+1}) text{ if $l geq j$}} \&=
cases{
e_l quad text{ if $l < j, l< i$}\
e_{l+1} text{ if $l < j, l geq i$}\
e_{l+1} text{ if $l geq j, l+1 < i$}\
e_{l+2} text{ if $l geq j, l+1 geq i$}} \&=
cases{
e_l quad text{ if $l < j$}\
e_{l+1} text{ if $l geq j, l < i-1$}\
e_{l+2} text{ if $l geq i-1$}}
end{align}
$$



and



$$
begin{align}
F_q^{j}F_{q-1}^{i-1}(e_l) &= cases{F_q^{j}(e_l) text{ if $l < i-1$}\ F_q^{j}(e_{l+1}) text{ if $l geq i-1$}} \ &=
cases{
e_l quad text{ if $l < i-1, l< j$}\
e_{l+1} text{ if $l < i-1, l geq j$}\
e_{l+1} text{ if $l geq i-1, l+1< j$}\
e_{l+2} text{ if $l geq i-1, l+1 geq j$}} \ &=
cases{
e_l quad text{ if $l< j$}\
e_{l+1} text{ if $l < i-1, l geq j$}\
e_{l+2} text{ if $l geq i-1$}}
end{align}
$$






share|cite|improve this answer











$endgroup$



You did not correctly calculate $F^s_q(e_{l+1})$. You have
$$
begin{align}
F_q^{i}F_{q-1}^{j}(e_l) &= cases{F_q^{i}(e_l) text{ if $l < j$}\ F_q^{i}(e_{l+1}) text{ if $l geq j$}} \&=
cases{
e_l quad text{ if $l < j, l< i$}\
e_{l+1} text{ if $l < j, l geq i$}\
e_{l+1} text{ if $l geq j, l+1 < i$}\
e_{l+2} text{ if $l geq j, l+1 geq i$}} \&=
cases{
e_l quad text{ if $l < j$}\
e_{l+1} text{ if $l geq j, l < i-1$}\
e_{l+2} text{ if $l geq i-1$}}
end{align}
$$



and



$$
begin{align}
F_q^{j}F_{q-1}^{i-1}(e_l) &= cases{F_q^{j}(e_l) text{ if $l < i-1$}\ F_q^{j}(e_{l+1}) text{ if $l geq i-1$}} \ &=
cases{
e_l quad text{ if $l < i-1, l< j$}\
e_{l+1} text{ if $l < i-1, l geq j$}\
e_{l+1} text{ if $l geq i-1, l+1< j$}\
e_{l+2} text{ if $l geq i-1, l+1 geq j$}} \ &=
cases{
e_l quad text{ if $l< j$}\
e_{l+1} text{ if $l < i-1, l geq j$}\
e_{l+2} text{ if $l geq i-1$}}
end{align}
$$







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited Dec 16 '18 at 9:45

























answered Dec 16 '18 at 9:38









Paul FrostPaul Frost

11.6k3934




11.6k3934












  • $begingroup$
    I see it now! I had rewritten this many times, but it seems that I had fixed my mistake already. Thanks a lot!
    $endgroup$
    – Guido A.
    Dec 17 '18 at 1:55


















  • $begingroup$
    I see it now! I had rewritten this many times, but it seems that I had fixed my mistake already. Thanks a lot!
    $endgroup$
    – Guido A.
    Dec 17 '18 at 1:55
















$begingroup$
I see it now! I had rewritten this many times, but it seems that I had fixed my mistake already. Thanks a lot!
$endgroup$
– Guido A.
Dec 17 '18 at 1:55




$begingroup$
I see it now! I had rewritten this many times, but it seems that I had fixed my mistake already. Thanks a lot!
$endgroup$
– Guido A.
Dec 17 '18 at 1:55


















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Plaza Victoria

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Plaza Victoria Plaza Victoria en 2017 Otros nombres Plaza de La Victoria Localización El Almendral, Valparaíso, Chile Vías adyacentes Chacabuco, Molina, Plaza Victoria y Edwards Creación 1841 Metro Bellavista Ubicación 33°02′46″S 71°37′11″O  /  -33.04624167, -71.61980833 Coordenadas: 33°02′46″S 71°37′11″O  /  -33.04624167, -71.61980833 [editar datos en Wikidata] La Plaza Victoria , también llamada Plaza de la Victoria , [ 1 ] ​ es una plaza ubicada en el sector El Almendral del plan de la ciudad de Valparaíso, Chile, de importancia patrimonial, que incluye valiosas estatuas y especies vegetales. [ 2 ] ​ La plaza se creó como tal en la primera mitad del siglo XIX, tomando previamente los nombres de Plaza Nueva y Plaza de Orrego . Si bien desde sus inicios tuvo relevancia histórica, no fue sino a partir de fines de los años 1860 cuando comenzó a cobrar mayor importancia y se convirtió en uno de los principales centros sociales de l...
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Puebla de Zaragoza

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Para otros usos de este término, véase Puebla (desambiguación). Heroica Puebla de Zaragoza Localidad De arriba a abajo de izquierda a derecha: Estrella de Puebla, Volcán Popocatépetl, Catedral de Puebla, Monumento a Ignacio Zaragoza, Estadio Cuauhtémoc, Angelópolis y Vista panorámica de la ciudad. Escudo Otros nombres: El relicario de América/ La ciudad de los Ángeles/ La Angelópolis Lema: Muy noble y muy leal ciudad de Puebla de los Ángeles Heroica Puebla de Zaragoza Localización de Heroica Puebla de Zaragoza en México Heroica Puebla de Zaragoza Localización de Heroica Puebla de Zaragoza en Puebla Mapa interactivo Coordenadas 19°03′05″N 98°13′04″O  /  19.051388888889, -98.217777777778 Coordenadas: 19°03′05″N 98°13′04″O  /  19.051388888889, -98.217777777778 Idioma oficial Español Entidad Localidad  • País   México  • Estado Puebla  • Municipio Puebla Presidente municipal Claudia Rivera Vivanc...
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Musa

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Para el género de plantas, véase Musa (planta). Apolo y la Nueve Musas En la mitología griega, las musas (en griego antiguo μοῦσαι « mousai» ) son, según los escritores más antiguos, las divinidades inspiradoras de las Artes: cada una de ellas está relacionada con ramas artísticas y del conocimiento. Son hijas de Zeus y de Mnemósine, compañeras del séquito de Apolo, dios olímpico de la música y patrón de las bellas artes, quien tuvo romances con cada una de ellas, dejando descendientes. Bajaban a la tierra a susurrar ideas e inspirar a aquellos mortales que las invocaran. En la época más arcaica eran las ninfas inspiradoras de las fuentes, en las cuales eran adoradas. Finalmente, alrededor de los siglos VIII-VII a. C. [ 1 ] ​ prevaleció en todo el territorio de la Hélade la adoración de las nueve Musas, que son Calíope, Clío, Erato, Euterpe, Melpómene, Polimnia, Talía, Terpsícore y Urania. El culto a las musas era originalmente de Tracia y Beocia, y fueron de vital impo...
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