Definition of the boundary map for chain complexes

In summary, the conversation discusses the two definitions of the boundary operator in homology theory, one with a factor of (-1)i and one without, and the speaker's reasoning behind why the factor may not matter since there is some choice in constructing the chain. The conversation also touches on the choice of coefficients, with the expert clarifying that in homology, one starts with a free abelian group and then chooses other coefficients.
  • #1
Tac-Tics
816
7
I've been poking around, learning a little about homology theory. I had a question about the boundary operator. Namely, how it's defined.

There's two definitions I've seen floating around. The first is at:

http://en.wikipedia.org/wiki/Simplicial_homology

The second, at

http://www.math.wsu.edu/faculty/bkrishna/FilesMath574/S12/LecNotes/Lec16_Math574_03062012.pdf

The only difference seems to be the inclusion of a factor of (-1)i inside the sums.

My guess is that the extra factor doesn't matter, since there is some choice in how you construct chain. In other words, the fact that you're working with a FREE abelian group over the p-simplexes of your complex, flipping the signs results in an isomorphic group.

(If that's not the case, my other guess would be that the latter only works in Z/2Z, where sign doesn't matter anyway).

Is my reasoning sound? Or am I missing something?
 
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  • #2
With Z2 coefficients signs don't matter since minus 1 and one are the same. The Wikipedia definition of boundary is correct in general. You can check this with examples.
 
  • #3
Ah. Thank you.

Now that I think about it, you can' "choose" what group you want the coefficients to be in if your generating your groups freely anyway.

(I'm guessing that would be some quotient of the free group, determined by the type of coefficient you're interested in, but I'll worry about that later).
 
  • #4
Tac-Tics said:
Ah. Thank you.

Now that I think about it, you can' "choose" what group you want the coefficients to be in if your generating your groups freely anyway.

(I'm guessing that would be some quotient of the free group, determined by the type of coefficient you're interested in, but I'll worry about that later).

In homology I think you start with the free abelian group on simplices, define the boundary operator, then choose other coefficients than the integers by tensoring (over Z) each group with the coefficient group.You never start with a free group, always a free abelian group. It is a characterisitc of homology that the groups are always abelian, unlike the fundamental group which usually is not abelian.
 
  • #5
Yes. I meant "abelian", but omitted it to introduce some confusion :)
 

1. What is the purpose of the boundary map in a chain complex?

The boundary map, also known as the boundary operator, is used to define the boundaries of the chain complex. It maps each element of the chain complex to the next one, and its main purpose is to help calculate the homology groups of the complex.

2. How is the boundary map defined in a chain complex?

The boundary map is defined as a linear transformation between the elements of the chain complex. It takes an element and maps it to the sum of its boundary components, which are defined by the relative positions of the elements in the complex.

3. What is the difference between the boundary map and the differential in a chain complex?

The terms boundary map and differential are often used interchangeably, but there is a subtle difference between the two. The boundary map is a specific type of differential that is used in the context of chain complexes, while the term differential is more general and can refer to a variety of mathematical operations.

4. How does the boundary map relate to the concept of cycles and boundaries in a chain complex?

The boundary map is essential in understanding the concepts of cycles and boundaries in a chain complex. A cycle is an element that maps to zero under the boundary map, while a boundary is an element that is the image of another element under the boundary map. The boundary map helps identify and distinguish between these two types of elements in a chain complex.

5. Can the boundary map be visualized in geometric terms?

Yes, the boundary map can be visualized in geometric terms, particularly in the context of simplicial or cellular complexes. In these cases, the boundary map is defined by the relative positions of the simplices or cells in the complex, and it can be thought of as the boundary of a higher-dimensional object. This visualization can aid in understanding the behavior of the boundary map and its relationship to the elements of the chain complex.

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