Reduced Homology: Modding Z(+)Z & Zeroth Homology

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In summary, the conversation discusses modding out by submodules of Z+Z and how all submodules are isomorphic to {0}, Z, or Z+Z. It also mentions different ways to embed Z and Z+Z into Z+Z and how the nature of the quotient is affected by the embedding. The conversation also clarifies that if there is an epimorphism from Z+Z to Z, then the kernel must be isomorphic to Z.
  • #1
homology
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Two questions:
(1) What can you mod Z(+)Z by and get Z?
Certainly modding out by Z works, but does anything else?


(2) I've just started reading about reduced homology (in particular the reduced zeroth homology (singular). So as a refresher: We define a homomorphism f (called the augmentation) between the zero chains C0 and the integers Z, and what f does is to add all the coefficients of the 0-chain together.

Its clear to me that all the zero boundaries are in the kernel of f, but why wouldn't it always be equal to it?

Another way of asking this is to say: why doesn't the reduced zeroth homology just have one less Z than the usual zeroth homology?


Thanks,
Kevin
 
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  • #2
the only thing you can mod out by woould be a submodule of Z+Z. now all submodules of Z+Z are isomorphic either to {0}, to Z, or to Z+Z.there are how ever many ways to embed Z and even Z+Z into Z+Z, and the embedding affects the nature of the quotient.any embedding of Z+Z will have rank 2 however and hence the quotient will be finite, so not isomorphic to Z.

hence only Z can be embedded so as to have quotient isomorphic to Z.

there are however many different embeddings of Z which have quotient Z, not just the obvious ones taking n to (0,n) or to (n,0).Any embedding is given by a 2 x 1 matrix of integers of rank 1.

then one can "diagonalize" this matrix by row and column operations, and afterwards, the quotient would appear to be isomorphic to Z if and only if the diagonalized matrix has a zero in it, so
 
  • #3
the only thing you can mod out by woould be a submodule of Z+Z. now all submodules of Z+Z are isomorphic either to {0}, to Z, or to Z+Z.



So does this mean if you have an epimorphism from Z+Z to Z that the kernel of the epimorphism must be isomorphic to Z? (Just want to clarify).
 
  • #4
yes. all submodules of a free finitely generated module over a pid are free of rank less than or equal to the rank of the top module.

and rank is additive over exact sequences, so the rank of a quotient equals the difference of the rank of the top and bottom modules.
 

1. What is reduced homology?

Reduced homology is a mathematical concept used in algebraic topology to measure the "holes" or "loops" in a space. It is a tool for understanding the topological structure of a set or space by assigning algebraic objects, known as homology groups, to different dimensions of the space.

2. How is reduced homology calculated?

To calculate reduced homology, we first construct a chain complex by assigning a group to each dimension of the space and a boundary map that describes the relationship between these groups. Then, we apply the process of modding, which involves taking the quotient of the group by its subgroup, to obtain the reduced homology groups.

3. What is the significance of modding Z(+)Z in reduced homology?

Modding Z(+)Z, or the direct sum of Z with itself, is a common step in calculating reduced homology. This is because Z(+)Z represents the group of 2-dimensional "loops" or "holes" in a space, which is an important component of the reduced homology groups.

4. What is the role of zeroth homology in reduced homology?

Zeroth homology, also known as the connected components of a space, is an essential part of reduced homology as it represents the 0-dimensional "loops" or "holes" in a space. It provides information about the number and structure of the different connected parts of a space.

5. How is reduced homology used in scientific research?

Reduced homology is a powerful tool used in various scientific fields, such as biology, physics, and computer science, to study the structure and properties of complex systems. It can be used to analyze data, classify shapes, and understand the behavior of complex networks, among other applications.

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