Definition of boundary operator

[GatorPower]

Hi!

I am currently studying homology theory and am using Vicks book "Homology Theory, An introduction to algebraic topology". When I was reading I found a definition that troubles me, I simply cannot get my head around it.

Vick defines that: if PHI is a singular p-simplex we define d_i(PHI), a singular (p-1)-simplex in X by:

d_iPHI(t0, ..., t(p-1)) = PHI(t0,.., t(i-1), 0, ti,..., t(p-1))

As I see it, the PHI on the left has p arguments, and hence is a (p-1)-singular simplex while the one we end up with is a p-singular simplex since we add a zero. The "boundary operator" (i'th face operator) d_i is defined so we go down one dimension, but I cannot see how this works with the definition if we just fill in the rest of the arguments after we add a zero. Help please?

After some consideration I wonder if perhaps Vick should have t(i+1) on the right, but I have found other sources who uses t(i-1), 0, ti as well...

 

[lavinia]

Hi!

I am currently studying homology theory and am using Vicks book "Homology Theory, An introduction to algebraic topology". When I was reading I found a definition that troubles me, I simply cannot get my head around it.

Vick defines that: if PHI is a singular p-simplex we define d_i(PHI), a singular (p-1)-simplex in X by:

d_iPHI(t0, ..., t(p-1)) = PHI(t0,.., t(i-1), 0, ti,..., t(p-1))

As I see it, the PHI on the left has p arguments, and hence is a (p-1)-singular simplex while the one we end up with is a p-singular simplex since we add a zero. The "boundary operator" (i'th face operator) d_i is defined so we go down one dimension, but I cannot see how this works with the definition if we just fill in the rest of the arguments after we add a zero. Help please?

After some consideration I wonder if perhaps Vick should have t(i+1) on the right, but I have found other sources who uses t(i-1), 0, ti as well...

Don't know your book but the standard n simplex has singular n-1 simplices as faces. Each of these singular n-1 simplices is a linear map of the standard n-1 simplex onto the face that keeps the ordering of the vertices the same. Composing with PHI gives the singular n-1 simplex in your space.

 

[Jamma]

Your formula looks wrong- you need to add up the n+1 different faces (you PHI(t0,.., t(i-1), 0, ti,..., t(p-1)) are the faces- in a sense it is the face opposite the i'th vertex).

You will also see a term which looks like (-1)^i in there. This is just so that you get the orientation of the faces right.So, for example, a 2-simplex is a filled in triangle and the boundary map takes this simplex to the sum of 3 1-simplexes which are the 3 straight lines on the boundary of the triangle. They all point round in a circle from the orientation induced by your original ordering of the vertices (usually you would take it anti-clockwise- so the boundary map would take the simplex to the outside triangle with orientation going anticlockwise around the triangle).