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.

7,3601730




7,3601730








  • 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

10.6k3933




10.6k3933












  • $begingroup$
    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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Ellipse (mathématiques)

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Pour les articles homonymes, voir Ellipse. L'ellipse est le lieu des points dont la somme des distances à deux points fixes, dits foyers, est constante. Section du cône ou projection du cercle. En géométrie, une ellipse est une courbe plane fermée obtenue par l’intersection d’un cône de révolution avec un plan, à condition que celui-ci coupe l'axe de rotation du cône ou du cylindre : c'est une conique d'excentricité strictement comprise entre 0 et 1. On peut également la définir comme le lieu des points dont la somme des distances à deux points fixes, dits foyers, est constante (sa construction par la méthode du jardinier est très simple). Dans la vie courante, l’ellipse est la forme qu'on perçoit en regardant un cercle en perspective, ou la figure formée par l’ombre d'un disque sur une surface plane. On retrouve aussi, en première approximation [ 1 ] , des ellipses dans les trajectoires des corps célestes (planètes, comètes ou satellites arti...
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Quarter-circle Tiles

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up vote 0 down vote favorite I have $99$ identical square tiles, each with a quarter-circle drawn on it like this: [asy] size(1.5cm); draw(Arc((2,0),1,90,180),red+1); draw((0,0)--(2,0)--(2,2)--(0,2)--(0,0)); [/asy] When I arrange the tiles in a $9times 11$ rectangular grid, each with a random orientation, what is the expected value of the number of full circles I form? I think this problem has to do with finding the chance any given 2x2 square has a circle, but I can't find it. expected-value share | cite | improve this question asked Nov 20 at 15:03 6minecraftninja 1 2 ...
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Mont Emei

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Mont Emei.mw-parser-output .entete.map{background-image:url("//upload.wikimedia.org/wikipedia/commons/7/7a/Picto_infobox_map.png")} Vue du sommet Wanfo du mont Emei Géographie Altitude 3 099  m , Wanfo Coordonnées 29° 31′ 11″ nord, 103° 19′ 57″ est Administration Pays   Chine Province Sichuan Ville-préfecture Leshan Géolocalisation sur la carte : Sichuan Mont Emei Géolocalisation sur la carte : Chine Mont Emei modifier   Paysage panoramique du mont Emei, incluant le paysage panoramique du grand Bouddha de Leshan *   Patrimoine mondial de l'UNESCO Passerelle en bois sur le versant ouest Pays   Chine Subdivision Sichuan Type Mixte Critères (iv) (vi) (x) Superficie 15 400 ha Numéro d’identification 779 Zone géographique Asie et Pacifique  ** Année d’inscription 1996 (20 e session) * Descriptif officiel UNESCO ** Classification géog...
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