# Constructing Mono/Epi Functions for Algebraic Topology

• Tchakra
In summary, the conversation discusses exact sequences in algebraic topology and how to determine if a sequence is exact. The process involves examining the maps between groups and determining if they are injective or surjective. The conversation also mentions the use of Abelian groups and provides an example of an exact sequence with non-Abelian groups. The conversation concludes with a challenge to determine all possible exact sequences of certain forms and to prove the exactness of a specific sequence.
Tchakra
Hey, can anyone help me with this please. I am doing algebraic topology and am particularly stuck on exact sequences. I "understand" the idea of the definition for example:

$$0\rightarrow A\stackrel{\alpha}{\rightarrow}B\stackrel{\beta}{\rightarrow}C\rightarrow 0$$

in this short exact sequence, alpha has to be mono or injective and beta has to be epi or surjective. However, what i don't get is: given a sequence of some groups how does one test whether it is exact. In other words how does one construct function between groups which are either epi or mono? for example from Z2 to Z4 or others.

(this may seem a strange question for someone doing "algebraic ..." but i have not done any algebra beyond an introduction few years ago)

thank you

Tchakra said:
Hey, can anyone help me with this please. I am doing algebraic topology and am particularly stuck on exact sequences. I "understand" the idea of the definition for example:

$$0\rightarrow A\stackrel{\alpha}{\rightarrow}B\stackrel{\beta}{\rightarrow}C\rightarrow 0$$

in this short exact sequence, alpha has to be mono or injective and beta has to be epi or surjective. However, what i don't get is: given a sequence of some groups how does one test whether it is exact. In other words how does one construct function between groups which are either epi or mono? for example from Z2 to Z4 or others.

(this may seem a strange question for someone doing "algebraic ..." but i have not done any algebra beyond an introduction few years ago)

thank you

In algebraic topology exact sequences are almost always given to you. You do not construct them. They arise naturally in comparing chain complexes.

In the the case of Z2 and Z4 there is only one possible exact sequence,

0 -> Z2 -> Z4 -> Z2 -> 0

It is easy to construct.

The generator of Z2 must be mapped to the unique element of order 2 in Z4. The generators of Z4 are mapped onto the generator of Z2.

I would be happy to help you with the exact sequences used algebraic topology.

Last edited:
I am looking at hatcher's book and one excercise asks to determine if there exist a short exact sequence: $$0 \rightarrow \mathbb{Z}_{4}\rightarrow \mathbb{Z}_{8} \oplus\mathbb{Z}_{2} \rightarrow \mathbb{Z}_{4} \rightarrow 0$$

Now when i look at that i have absolutely no idea where to start. So what i am hoping for is a general procedure to determine. How did you determine that there is a unique exact sequence for
0->Z2->Z4->Z2->0

Could you send my way, if you know any book or website that dwells in details into this.

Tchakra said:
I am looking at hatcher's book and one excercise asks to determine if there exist a short exact sequence: $$0 \rightarrow \mathbb{Z}_{4}\rightarrow \mathbb{Z}_{8} \oplus\mathbb{Z}_{2} \rightarrow \mathbb{Z}_{4} \rightarrow 0$$

Now when i look at that i have absolutely no idea where to start. So what i am hoping for is a general procedure to determine. How did you determine that there is a unique exact sequence for
0->Z2->Z4->Z2->0

Could you send my way, if you know any book or website that dwells in details into this.

With Abelian groups the logic is simple. You do not need a general procedure in my opinion.
This is how to do 0->Z2->Z4->Z2->0. If Z2 -> Z4 is injective then its generator must map to an element of order 2 in Z4. There is only one element of order 2 in Z4 so there is only one injective map. The quotient is clearly isomorphic to Z2 so you are done.

For Z4 -> Z8 + Z2 the generator must map to an element of order 4. These are the elements (z^2,0) (z^2,1) (Z^6,0) and (z^6,1) where z is any generator of Z_8. Just check it out to see if any quotient is isomorphic to Z_4. Maybe you can generalize from this example.

Tchakra said:
I am looking at hatcher's book and one excercise asks to determine if there exist a short exact sequence: $$0 \rightarrow \mathbb{Z}_{4}\rightarrow \mathbb{Z}_{8} \oplus\mathbb{Z}_{2} \rightarrow \mathbb{Z}_{4} \rightarrow 0$$

Now when i look at that i have absolutely no idea where to start. So what i am hoping for is a general procedure to determine. How did you determine that there is a unique exact sequence for
0->Z2->Z4->Z2->0

Could you send my way, if you know any book or website that dwells in details into this.

Try this problem. How many exact sequences are there of the form 0 -> Z2 -> Z2 + Z2 ->Z2 ->0? Construct all of them. How about 0 -> Z2 -> Z4 + Z4 -> G - > 0. What are the possible groups,G?

Here is an exact sequence of groups that are not all Abelian. 0 -> L2 -> G -> Z2 -> 0
G is not Abelian. L2 is the standard lattice in the plane i.e. all points in the plane whose co-ordinates are integers. It is a free Abelian group on two generators. G is the group generated by this lattice and one other transformation of the plane. This other transformation is multiply the y coordinate by -1 then translate by 1/2 in the x direction i.e.
(x,y) -> (x+1/2,-y). G is the fundamental group of the Klein bottle. Show that g/L2 is Z2 and prove that this is an exact sequence.

## 1. What is the purpose of constructing mono/epi functions in algebraic topology?

The purpose of constructing mono/epi functions in algebraic topology is to study and classify topological spaces by considering their algebraic properties. Mono/epi functions play a crucial role in this process as they allow us to identify and distinguish different topological spaces based on their algebraic characteristics.

## 2. How are mono/epi functions defined in the context of algebraic topology?

In algebraic topology, a function is considered mono if it preserves injectivity, meaning that distinct points in the domain map to distinct points in the codomain. On the other hand, a function is considered epi if it preserves surjectivity, meaning that every point in the codomain has at least one corresponding point in the domain that maps to it.

## 3. What are some properties of mono/epi functions in algebraic topology?

One important property of mono/epi functions is that they are preserved under composition. This means that if f and g are mono/epi functions, then so is their composition g ◦ f. Additionally, mono/epi functions are also preserved under pullbacks and pushouts, which are important constructions in algebraic topology.

## 4. How do mono/epi functions relate to other concepts in algebraic topology?

Mono/epi functions are closely related to other important concepts in algebraic topology, such as homotopy and homology. In fact, mono/epi functions can be used to define these concepts and are often used in the proofs of important theorems in algebraic topology.

## 5. What are some applications of mono/epi functions in algebraic topology?

Mono/epi functions have a wide range of applications in algebraic topology. They are used in the study of fundamental groups, homology groups, and homotopy groups, which are important algebraic invariants of topological spaces. Mono/epi functions also play a crucial role in the construction of important algebraic structures such as the Universal Coefficient Theorem and the Mayer-Vietoris Sequence.

• Science and Math Textbooks
Replies
10
Views
5K
• Topology and Analysis
Replies
2
Views
3K
• Differential Geometry
Replies
5
Views
3K
• Calculus
Replies
17
Views
1K
• Calculus and Beyond Homework Help
Replies
8
Views
703
• Set Theory, Logic, Probability, Statistics
Replies
6
Views
2K
Replies
2
Views
1K
• Math Proof Training and Practice
Replies
69
Views
4K
• Linear and Abstract Algebra
Replies
12
Views
2K
• Math Proof Training and Practice
Replies
25
Views
2K