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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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

11.5k3934




11.5k3934












  • $begingroup$
    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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Bundesstraße 106

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Dieser Artikel beschreibt die Bundesstraße 106 in Deutschland. Zur gleichnamigen Straße in Österreich siehe Mölltal Straße. Vorlage:Infobox hochrangige Straße/Wartung/DE-B Bundesstraße 106 in Deutschland Karte Basisdaten Betreiber: Deutschland  Bundesrepublik Deutschland Straßenbeginn: Wismar ( 53° 54′  N , 11° 24′  O 53.900536 11.396875 ) Straßenende: Schwerin ( 53° 36′  N , 11° 24′  O 53.592785 11.404892 ) Gesamtlänge: 40,7 km Bundesland: Mecklenburg-Vorpommern Ausbauzustand: zweistreifig [1] Bundesstraße 106 in Schwerin (Umgehungsstraße) Straßenverlauf Land Mecklenburg-Vorpommern Landkreis Nordwestmecklenburg Ortsumgehung Wismar Wallensteingraben über   Dorf Mecklenburg Wallensteingraben Groß Stieten Bad Kleinen OT Niendorf Bahnstrecke Lübeck–Bad Kleinen Zickhusen Lübstorf Klein Trebbow OT Kirch Stück
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Verónica Boquete

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Verónica Boquete im Trikot der Nationalmannschaft (2013) Personalia Name Verónica Boquete Giadans Geburtstag 9. April 1987 Geburtsort Santiago de Compostela, Spanien Größe 159 cm Position Mittelfeld / Sturm Juniorinnen Jahre Station 2001–2005 Xuventú Aguiño Frauen Jahre Station Spiele (Tore) 1 2005–2008 Prainsa Zaragoza 83 (40) 2008–2010 Espanyol Barcelona 53 (30) 2010 Buffalo Flash 9 0 (9) 2010 Chicago Red Stars 3 0 (1) 2010–2011 Espanyol Barcelona 26 (41) 2011 Philadelphia Independence 13 0 (5) 2011 Energija Woronesch 5 0 (1) 2012–2014 Tyresö FF 39 (14) 2014 Portland Thorns FC 15 0 (4) 2014–2015 1. FFC Frankfurt 23 0 (7) 2015–2016 FC Bayern München 9 0 (0) 2016–2018 Paris Saint-Germain 31 0 (8) 2018 Peking BG Phoenix 2019– Utah Royals FC 0 0 (0) Nationalmannschaft Jahre Auswahl
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Ida-Boy-Ed-Garten

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Die Lage des Ida-Boy-Ed-Gartens, rot markiert Der Ida-Boy-Ed-Garten Der Ida-Boy-Ed-Garten, Blick von der Wakenitzmauer durch das Tor in der Stadtmauer Der Ida-Boy-Ed-Garten ist eine Straße der Lübecker Altstadt. Inhaltsverzeichnis 1 Lage 2 Geschichte 3 Literatur 4 Weblinks Lage | Der etwa 90 Meter lange Ida-Boy-Ed-Garten befindet sich an der nördlichen Spitze der Altstadtinsel, vor dem Burgtor. Die in einem Bogen durch eine Grünanlage hangaufwärts führende Straße verbindet die Wakenitzmauer mit der Burgtorbrücke. Geschichte | Der heutige Ida-Boy-Ed-Garten wurde ursprünglich als Verlängerung der Wakenitzmauer angelegt. 1903 wurde in den erhaltenen Teil der mittelalterlichen Stadtmauer, an dem die Wakenitzmauer bei der Einmündung der Kaiserstraße begann, ein Tor mit einem großen Bogen für die Durchführung der Straße und zwei kleineren seitlichen Öffnungen für Fußgänger gebrochen, um für die neu errichteten Wohnhäuser in diesem Bereich
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