Fdtxjr

I seem to dance, I am not a dancer. Who am I?

Why is indicated airspeed rather than ground speed used during the takeoff roll?

Hausdorff dimension of the boundary of fibres of Lipschitz maps

In the 1924 version of The Thief of Bagdad, no character is named, right?

Existence of a celestial body big enough for early civilization to be thought of as a second moon

Recruiter wants very extensive technical details about all of my previous work

How to define limit operations in general topological spaces? Are nets able to do this?

When to use snap-off blade knife and when to use trapezoid blade knife?

Does the attack bonus from a Masterwork weapon stack with the attack bonus from Masterwork ammunition?

Do I need to be arrogant to get ahead?

Using Past-Perfect interchangeably with the Past Continuous

How could an airship be repaired midflight?

Probably overheated black color SMD pads

What favor did Moody owe Dumbledore?

Can a wizard cast a spell during their first turn of combat if they initiated combat by releasing a readied spell?

Could Sinn Fein swing any Brexit vote in Parliament?

PTIJ What is the inyan of the Konami code in Uncle Moishy's song?

Fewest number of steps to reach 200 using special calculator

What does Deadpool mean by "left the house in that shirt"?

Unfrosted light bulb

Print last inputted byte

Describing a chess game in a novel

A Ri-diddley-iley Riddle

Why are there no stars visible in cislunar space?

Lemma for Hurewicz Theorem (Bredon)

Homologous cycleHomotopies of edge paths in simplicial complexesHomology with twisted coefficients of the circleHomotopy and chain homotopy determine each otherWhy is the subdivision an extension?Why does the Hurewicz homomorphism factor through the abelianization of the fundamental group?Why is are the simplicial 1-chains $[A,B] neq -[B,A]$?Homology groups of a PokeballPrelude to Rotman's proof of the Hurewicz TheoremProof in Lee's Introduction to Topological manifoldsLifting homotopic paths

0

$begingroup$

I am trying to understand the following lemma:
If $f,g:Irightarrow X$ are paths s.t. $f(1)=g(0)$ then the 1-chain $f*g-f-g$ is a boudary.

Proof: On the standard 2-complex (should it say simplex?) $Delta_2$ put $f$ on the edge $(e_0,e_1)$ and $g$ on the edge $(e_1,e_2)$. Then define a singular 2-simplex $sigma:Delta_2rightarrow X$ to be constant on the perpendicular lines to the edge $(e_0,e_2)$ (HOW?). This results in the path $f*g$ being on the edge $(e_0,e_2)$ (WHY?). Therefore $partial sigma=g-(f*g)+f$

I know that $partial sigma=partial^0 sigma-partial^1 sigma-partial^2 sigma$
and that $partial^0 sigma(x,y,z)=sigma(0,y,z)$
$partial^1 sigma(x,y,z)=sigma(x,0,z)$
$partial^2 sigma(x,y,z)=sigma(x,y,0)$

I also drew a picture and still dont get it.

This is one of three previous lemma to the Hurewicz theorem, any help would be appreciated.

algebraic-topology homology-cohomology simplex simplicial-stuff

share|cite|improve this question

edited Mar 12 at 19:34

Thomas Andrews

130k12147298

asked Mar 12 at 19:32

AlfdavAlfdav

1027

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Take an equilateral triangle, and project onto one of its edges. This will produce a single edge, and follow this by the map $fg$, which will produce for you a particular $2$-chain. Restring this map to the boundary tells you that as a sum it is $f+g-fg$.
  $endgroup$
  – Andres Mejia
  Mar 12 at 19:45
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  See my answer here : math.stackexchange.com/questions/2838755/homologous-cycle/… (I don't know if you want to understand the lemma or understand a specific proof, that's why I don't know if it's a duplicate)
  $endgroup$
  – Max
  Mar 12 at 21:45
  
  

  
  
  
  

add a comment | 

0

$begingroup$

I am trying to understand the following lemma:
If $f,g:Irightarrow X$ are paths s.t. $f(1)=g(0)$ then the 1-chain $f*g-f-g$ is a boudary.

Proof: On the standard 2-complex (should it say simplex?) $Delta_2$ put $f$ on the edge $(e_0,e_1)$ and $g$ on the edge $(e_1,e_2)$. Then define a singular 2-simplex $sigma:Delta_2rightarrow X$ to be constant on the perpendicular lines to the edge $(e_0,e_2)$ (HOW?). This results in the path $f*g$ being on the edge $(e_0,e_2)$ (WHY?). Therefore $partial sigma=g-(f*g)+f$

I know that $partial sigma=partial^0 sigma-partial^1 sigma-partial^2 sigma$
and that $partial^0 sigma(x,y,z)=sigma(0,y,z)$
$partial^1 sigma(x,y,z)=sigma(x,0,z)$
$partial^2 sigma(x,y,z)=sigma(x,y,0)$

I also drew a picture and still dont get it.

This is one of three previous lemma to the Hurewicz theorem, any help would be appreciated.

algebraic-topology homology-cohomology simplex simplicial-stuff

share|cite|improve this question

edited Mar 12 at 19:34

Thomas Andrews

130k12147298

asked Mar 12 at 19:32

AlfdavAlfdav

1027

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Take an equilateral triangle, and project onto one of its edges. This will produce a single edge, and follow this by the map $fg$, which will produce for you a particular $2$-chain. Restring this map to the boundary tells you that as a sum it is $f+g-fg$.
  $endgroup$
  – Andres Mejia
  Mar 12 at 19:45
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  See my answer here : math.stackexchange.com/questions/2838755/homologous-cycle/… (I don't know if you want to understand the lemma or understand a specific proof, that's why I don't know if it's a duplicate)
  $endgroup$
  – Max
  Mar 12 at 21:45
  
  

  
  
  
  

add a comment | 

$begingroup$

I am trying to understand the following lemma:
If $f,g:Irightarrow X$ are paths s.t. $f(1)=g(0)$ then the 1-chain $f*g-f-g$ is a boudary.

Proof: On the standard 2-complex (should it say simplex?) $Delta_2$ put $f$ on the edge $(e_0,e_1)$ and $g$ on the edge $(e_1,e_2)$. Then define a singular 2-simplex $sigma:Delta_2rightarrow X$ to be constant on the perpendicular lines to the edge $(e_0,e_2)$ (HOW?). This results in the path $f*g$ being on the edge $(e_0,e_2)$ (WHY?). Therefore $partial sigma=g-(f*g)+f$

I know that $partial sigma=partial^0 sigma-partial^1 sigma-partial^2 sigma$
and that $partial^0 sigma(x,y,z)=sigma(0,y,z)$
$partial^1 sigma(x,y,z)=sigma(x,0,z)$
$partial^2 sigma(x,y,z)=sigma(x,y,0)$

I also drew a picture and still dont get it.

This is one of three previous lemma to the Hurewicz theorem, any help would be appreciated.

algebraic-topology homology-cohomology simplex simplicial-stuff

share|cite|improve this question

edited Mar 12 at 19:34

Thomas Andrews

130k12147298

asked Mar 12 at 19:32

AlfdavAlfdav

1027

$endgroup$

share|cite|improve this question

edited Mar 12 at 19:34

Thomas Andrews

130k12147298

asked Mar 12 at 19:32

AlfdavAlfdav

1027

share|cite|improve this question

edited Mar 12 at 19:34

Thomas Andrews

130k12147298

asked Mar 12 at 19:32

AlfdavAlfdav

1027

edited Mar 12 at 19:34

Thomas Andrews

130k12147298

edited Mar 12 at 19:34

Thomas Andrews

130k12147298

asked Mar 12 at 19:32

AlfdavAlfdav

1027

asked Mar 12 at 19:32

AlfdavAlfdav

1027