Definition of boundary operator

In summary, the author Vick defines a boundary operator di that takes a singular p-simplex to a singular (p-1)-simplex in X by adding a zero and filling in the rest of the arguments in a specific order. This allows for the correct orientation of the faces to be maintained. However, there may be some discrepancies in notation between different sources.
  • #1
GatorPower
26
0
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 di(PHI), a singular (p-1)-simplex in X by:

diPHI(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) di 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...
 
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  • #2
GatorPower said:
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 di(PHI), a singular (p-1)-simplex in X by:

diPHI(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) di 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.
 
  • #3
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).
 

What is a boundary operator?

A boundary operator is a mathematical concept used in topology to define the boundary of a set or space. It is represented by the symbol ∂ and is used to determine the boundary points of a set.

How is a boundary operator used in topology?

In topology, a boundary operator is used to define the boundary of a set or space. It helps to determine which points are inside the set and which are on the boundary of the set. This information is important in understanding the properties and relationships between different sets or spaces.

What is the difference between an interior and a boundary point?

An interior point is a point that lies within a set or space, while a boundary point is a point that lies on the edge or boundary of a set or space. In other words, an interior point is surrounded by other points in the set, while a boundary point is not.

Can a set have both an empty interior and an empty boundary?

Yes, it is possible for a set to have both an empty interior and an empty boundary. This can occur in sets with disjoint components or in infinite sets with no clearly defined boundary points.

How is the boundary operator related to the concept of a boundary in real-world applications?

In real-world applications, the boundary operator is used to define the boundary of a physical space or object. For example, in geographical applications, the boundary operator can be used to determine the boundary between two countries or states. In engineering, it can be used to define the boundary of a physical structure or system.

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