Mapping of the standard k-simplex in R^n to X

In summary, the book discusses how to denote a standard k-simplex of a topological space X using the symbol σ:Δk→X. It then explains the concept of a free abelian group generated by these σ's, which are known as k-chains on X. The concept of a chain for a map σ is not clear, but it is understood as a mapping of the standard simplex into X multiple times, connecting at their boundaries. The group action for chains is not function composition, but rather a purely symbolic group with no group action. The orientation for these chains comes from the standard orientation of the standard k-simplex in Euclidean space, and the boundary of a mapping is the alternating sum of its restriction to each
  • #1
cpsinkule
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My book denotes by σ:Δk→X for some suitable topological space X a standard k-simplex of X. It then describes the free abelian group generated by such σ's as the group of k-chains on X. It is not clear to me what is meant by a chain for a map σ. I understand a chain in Rn to be sums of integer linear combination of simplicies, but I cannot wrap my head around what is meant by chains of mappings of Δk. Is each σ mapping Δk to different neighborhoods on X? Is the group action for chains function composition of each σ? Currently, I am understanding a chain to be a mapping of Δk multiple times into X such that they connect at their boundaries (or perhaps overlap?), but I feel like this is not the correct view. Any insight would be greatly appreciated.

EDIT: I incorrectly put R^n in the title, it should be R^k to match the dimension of the simplex
 
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  • #2
As I already wrote the k chains are a purely symbolic group. There is no group action.
I am not sure why you think there is.

if you take simplices to be maps of the standard simplex in Euclidean space then just form the formal free abelian group on all of these maps.

In this case, the boundary of the simplex is the restriction of the mapping to the boundary of the standard simplex in R^k.

Before I thought you were talking about a simplicial complex. Sorry for the confusion.
But the story here is essentially the same.

A map of the standard simplex in Euclidean space can be thought of as the generator of a purely formal free abelian group. One jus associates a symbol for this map and defines a group of symbols as I described in you previous post. When you consider all possible maps - in de Rham theory they are smooth maps - then one can formally talk about the free abelian group that they generate. This is an infinite dimensional free abelian group.

Orientation comes from the standard orientation of the standard k-simplex in Euclidean space.
The boundary of the standard simplex is the formal alternating sum of its k-1 faces taken in sequential order just as I described in the other post. The boundary of the mapping is the alternating sum of its restriction to each k-1 face.

So for the standard 2 simplex σ = <0,(1,0),(0,1)>

∂σ = <(1,0),(0,1)> - <0,(0,1)> + <0,(1,0)>

If f is a smooth map of σ into a manifold, then ∂f = f(∂σ) = f(<(1,0),(0,1)>)- f(<0,(0,1)>) + f(<0,(1,0)>)
 
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1. What is the standard k-simplex in R^n?

The standard k-simplex in R^n is a geometric shape that represents the space of all possible combinations of k+1 non-negative numbers that add up to 1 in n-dimensional space. It is often denoted as Δ^k and can be visualized as an n-dimensional triangle with k+1 vertices.

2. How is the standard k-simplex mapped to X?

The standard k-simplex is mapped to X using a linear transformation. This transformation takes the k+1 vertices of the simplex and maps them to points in the target space X. The resulting image is a geometric representation of the simplex in X.

3. What is the purpose of mapping the standard k-simplex to X?

The purpose of mapping the standard k-simplex to X is to understand the relationship between the two spaces. This mapping allows us to study the properties and structure of the simplex in a different context, and can also be used to solve optimization problems or analyze data sets.

4. How is the mapping of the standard k-simplex to X calculated?

The mapping of the standard k-simplex to X is calculated using a linear transformation matrix. This matrix is constructed based on the coordinates of the k+1 vertices of the simplex and the desired coordinates in X. The resulting transformation is a composition of linear transformations that maps the simplex to X.

5. Can the mapping of the standard k-simplex to X be reversed?

Yes, the mapping of the standard k-simplex to X can be reversed by using the inverse of the linear transformation matrix. This means that the points in X can be mapped back to the original simplex in R^n, allowing us to retrieve the original data or solution from X.

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