# Is H'_n(X) Isomorphic to H_n(X, y) in Relative Singular Homology?

• eastside00_99
In summary, the reduced homology of a topological space is isomorphic to the relative homology of the one point space {y} around the point y.
eastside00_99
Let X be a topological space and let y be a point in X. Denote H'_n(X) be the reduced homology of X and Denote H_n(X,y) the relative homology of X and the one point space {y}.

Prove: H'_n(X) is isomorphic to H_n(X,y).

I have proven that it is true for n>0. But, for n=0, I get H_n(X) isomorphic to H_n(X,y) and since the reduced homology of X is H_n(X) mod integers, I have run into a contradiction. Wikipedia gives my result as a property of relative homology, but my teacher has something different. (Note H_n(X)=H'_n(X) for n>0).

Help?!??!

Hrm. When I worked it out, I got the exact sequence

$$0 \to \mathbb{Z} \to H_0(X) \to H_0(X, y) \to 0.$$

Incidentally, note talk page for that wikipedia article. I'm going to edit the wikipedia page to remove confusion.

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Yeah, I got that exact sequence. But, somehow I wealsed to get the wrong, wikipedia answer, so I probably need to re-check my work! Thanks.

ok got it: just use the first isomorphism theorem and that H'_0(X)=H_0(X) modulo integers.

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eastside00_99 said:
ok got it: just use the first isomorphism theorem and that H'_0(X)=H_0(X) modulo integers.
Just to make sure -- there are a lot of ways the integers can be mapped into an abelian group. You have to make sure you actually have the same subgroup for the kernel in both!

e.g. the kernel of Z --> Z/2 and Z --> Z/4 are both abelian groups isomorphic to the integers.

Hurkyl said:
Hrm. When I worked it out, I got the exact sequence

$$0 \to \mathbb{Z} \to H_0(X) \to H_0(X, y) \to 0.$$

Actually should it just be

$$H_1(X,y) \to \mathbb{Z} \to H_0(X) \to H_0(X, y) \to 0.$$

If not how did you get the zero?

eastside00_99 said:
Actually should it just be

$$H_1(X,y) \to \mathbb{Z} \to H_0(X) \to H_0(X, y) \to 0.$$

If not how did you get the zero?

$$0 \to H_1(X) \to H_1(X,y) \to \mathbb{Z} \to H_0(X) \to H_0(X, y) \to 0.$$

So, I am still stuck. No matter how you word it, it boils down to the map from H_1(X) to H_1(X,y) being surjective and the map from H_0(X) to H_0(X,y) being injective. No matter what you need those facts and so I think you have to go the chain complexes and look at how the relative homology and the differential of the chain C_n(X)/C_n(y) are defined. We do know that

C_m(y) iso C_n(y) iso Integers for all m,n>=0. This surely must tell you something about the differential but I have no idea what.

I was half-thinking cohomology and had some of the arrows reversed.

No matter how you word it, it boils down to the map from H_1(X) to H_1(X,y) being surjective and the map from H_0(X) to H_0(X,y) being injective.
Those can't both be true; otherwise the sequence wouldn't be exact at Z.

Anyways, there is a direct description of $H_0(\cdot)$; doesn't it allow you to conclude that $H_0(y) \to H_0(X)$ is monic?

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eastside00_99 said:
$$0 \to H_1(X) \to H_1(X,y) \to \mathbb{Z} \to H_0(X) \to H_0(X, y) \to 0.$$

So, I am still stuck. No matter how you word it, it boils down to the map from H_1(X) to H_1(X,y) being surjective and the map from H_0(X) to H_0(X,y) being injective. No matter what you need those facts and so I think you have to go the chain complexes and look at how the relative homology and the differential of the chain C_n(X)/C_n(y) are defined. We do know that

C_m(y) iso C_n(y) iso Integers for all m,n>=0. This surely must tell you something about the differential but I have no idea what.

Ok, I got it:

$$0 \to H_1(X) \to H_1(X,y) \to \mathbb{Z} \to H_0(X)=C_0(X) \to H_0(X, y)=C_0(X)/C_0(y)=C_0(X)/\mathbb{Z} \to 0.$$

So, right away we have H'_0(X)=H_0(X) module integers = H(X,y). Furthermore, we know that the kernal of the map from is C_0(y) = integers which implies the map from the integers to H_0(X) is injective which implies the image of the map from H_1(X,y) to the integers is 0 which implies we have the exact sequence

$$0 \to H_1(X) \to H_1(X,y) \to 0$$

i.e., H'_1(X) iso to H_1(X,y). the proof for n>1 is trivial.

Again, I'd like to point out that "H_0(X) modulo integers" is not well-defined -- for most groups, there are lots of ways to map Z into it. For example, for appropriate choices of a map Z -> Z, each the following sequences are exact:

$$0 \to \mathbb{Z} \to \mathbb{Z} \to 0$$
$$0 \to \mathbb{Z} \to \mathbb{Z} \to \mathbb{Z} / 2 \to 0$$
$$0 \to \mathbb{Z} \to \mathbb{Z} \to \mathbb{Z} / 3 \to 0$$
...

and for most choices of the map Z -> Q, the following sequence is not exact:

$$0 \to \mathbb{Z} \to \mathbb{Q} \xrightarrow{\pi} \mathbb{Q} / \mathbb{Z} \to 0$$

(in fact, it's usually not even a complex!) ($\mathbb{Z}$ is an honest to goodness subgroup of $\mathbb{Q}$, so $\mathbb{Q} / \mathbb{Z}$ makes sense. The map $\mathbb{Q} \xrightarrow{\pi} \mathbb{Q} / \mathbb{Z}$ in the above sequence is the projection map. Other maps could fit there, though!)

So, the fact that you have two exact sequences

$$0 \to \mathbb{Z} \to H_0(X) \to H_0(X, y) \to 0$$

$$0 \to \mathbb{Z} \to H_0(X) \to H'_0(X) \to 0$$

does not imply that you have an isomorphism $H_0(X, y) \cong H'_0(X)$.

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I agree with you the fact that H_0(X,y) iso H'_0(X) is trivially done by observation--i.e.

H_0(X,y)=C(X)/Z and H_0(X)=C(X) and H'_0(X)=H_0(X)/Z.

eastside00_99 said:
I agree with you the fact that H_0(X,y) iso H'_0(X) is trivially done by observation--i.e.

H_0(X,y)=C(X)/Z and H_0(X)=C(X) and H'_0(X)=H_0(X)/Z.
I'm saying exactly the opposite.

I'm saying:
(1) You cannot conclude H_0(X,y) is isomorphic from H'_0(X) given just the information you presented.

(2) The expressions C(X)/Z and H_0(X)/Z aren't even well defined. It's like writing 0/0 in ordinary arithmetic!

Quotient groups G/H are defined for when H is a subgroup of G. Z is not a subgroup of H_0(X). Z is isomorphic to (many) subgroups of H_0(X); if you pick a specific map $f:\mathbb{Z} \to H_0(X)$, then you can talk about the group $H_0(X) / \im f$. But if you pick another map $g:\mathbb{Z} \to H_0(X)$, then it is possible that $H_0(X) / \mathrm{im\ } f \not\cong H_0(X) / \mathrm{im\ } g$.

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Yes, yes, all that is true. So, let's concentrate on (2) because that is were the confusion is.

Theorem: H_0(X) = Z + Z + ... +Z (n direct sums of copies of Z where n is the number of connected components of X), and H'_0(X) = H_0(X)/Z (i.e. n-1 direct sums of copies of Z where n is the number of connected components of X).

So, in this way, H_0(X)/Z is well-defined.

Now, consider a point y in X, C_n({y}) is the free abelian group generated by all continuous functions of the form f:delta^n --> {y} where delta^n is the standard n-simplex. In other words C_n({y}) is the free abelian group with one generator. In other words, C_n({y}) iso Z.

So, we have a canonical isomorphism

H_0(X,y)=C_0(X)/C_0(y) = H_0(X)/C_0(y) iso H'_0(X)

as H_0(X) iso n direct sums of Z and C_0(y) iso Z; n direct sums of Z modulo Z iso n-1 direct sums of Z iso H'_0(X).

eastside00_99 said:
So, in this way, H_0(X)/Z is well-defined.
Not really; you've not produced any sort of quotient group. If you chose a specific copy of Z, then you will have a well-defined quotient group H_0(X)/Z -- but that goes back to what I've been saying the whole time: you have to specify the map Z --> H_0(X) before H_0(X)/Z makes sense!

So, we have a canonical isomorphism

H_0(X,y)=C_0(X)/C_0(y) = H_0(X)/C_0(y) iso H'_0(X)
You have not shown that.

Let's assume you are right in saying that $H_0(X, y) \cong H_0(X) / \mathop{\mathrm{image}} \left(C_0(y) \to H_0(X))$, where $C_0(y) \to H_0(X)$ is the map induced by the chain map induced by the inclusion $y \to X$.

You have not shown just what that image is. Suppose that X is path connected, so that $H_0(X) \cong \mathbb{Z}$. What if the map $C_0(y) \to H_0(X)$ is isomorphic to the "multiplication by 2" map $\mathbb{Z} \xrightarrow{\cdot 2} \mathbb{Z}$? In that case, we'd have $H_0(X, y) \cong \mathbb{Z} / 2$. What if it's the zero map? Then we'd have $H_0(X, y) \cong \mathbb{Z}$. Neither of those are isomorphic to $H'_0(X)$.

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You're right!

I don't think you can prove it this way unless I am missing something. I think you have to show there is a chain homotopy equivalence from the augmented chain complex of C(X) to the chain complexC(X)/C(y). This will induce a isomorphism in homology. I haven't tried this technique yet.

Your basic idea is sound; in order to apply it, you simply need to figure out exactly what the map C_0(y) --> H_0(X) is. (and that shouldn't be too difficult)

Hi:

Hope not to distract too much from 00_99's question. Just a general question
on relative homology. Unfortunately, I haven't done much more than simplicial
homology for a while.

I am reading Griffiths and Harris' book:

Given complexes (K*,d) and (J*,d) (with std. meaning: a complex is

a seq. of Abelian groups with a differential d such that d^2=0 ), such

that (K*,d) is a subcomplex of (J*,d) , i.e, for all non-neg. integer p,

all groups J^p<K^p , for all Abelian groups K^p in (K*,d).

Then G&H define the quotient complex :

K*/J* , with "the obvious differential" . What is this "obvious

differential"?.

Now, the only actual case I can think of is that of simplicial

homology, and relative homology, with A<B , where A is a simplicial

subcomplex of B . Then a relative cycle c_p is given

by

C_p(A/B)=C_p(A)/C_p(B)= (by Abelianness)

C_p(A)-C_p(B) (A/B is just meant to stand for A relative to B,

not a topological quotient.)

[c_p]=c_p(+)C_p-1(B) , i.e. just the coset rep. of c_p in the quotient

where C_p-1 is the group of (p-1)-chain maps in B.

And the relative differential d_r s:

d_r([c_p])=d(c_p)(+)C_(p-1)(B)

with d the differential of the original complex.

Is this the generalized formula for relative homology groups, differentials, etc. ?

Thanks.

A map of complexes, by definition, commutes with the differential. So if you have a map $f : A^\cdot \to B^\cdot$, then you have commutative squares

$$\begin{array}{ccc} A^n & \xrightarrow{d} & A^{n-1} \\ {}^{{}_f}\!\!\!\downarrow & & {}^{{}_f}\!\!\!\downarrow \\ B^n & \xrightarrow{d} & B^{n-1} \\ \end{array}$$

i.e. for any element $a \in A^n$, f(da) = df(a)

In particular, if A is a subcomplex of B, then this applies to the inclusion $A \to B$ and also to the projection $B \to B/A$. (i.e. if denotes the equivalence class of b, then $d = [db]$)

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## 1. What is relative singular homology?

Relative singular homology is a mathematical tool used in algebraic topology to study the topological properties of spaces. It is a generalization of singular homology, which is a way of assigning algebraic objects (known as homology groups) to spaces in order to measure their "holes" or "higher-dimensional voids". Relative singular homology takes into account not only the space itself, but also a subset of that space, known as the "relative space". This allows for a more nuanced understanding of the topological features of a given space.

## 2. How is relative singular homology calculated?

The calculation of relative singular homology involves several steps. First, the space and its relative space are decomposed into pieces known as simplices. These simplices are then used to construct a chain complex, which is a sequence of algebraic groups connected by homomorphisms. The homology groups of this chain complex are then computed using techniques from abstract algebra, resulting in the relative singular homology groups.

## 3. What is the purpose of relative singular homology?

The purpose of relative singular homology is to provide a way to measure the topological features of a space, taking into account not only the space itself, but also a specific subset of that space. This allows for a more precise understanding of the topological properties of a given space, and can be useful in various fields such as geometry, physics, and computer science.

## 4. What are some real-world applications of relative singular homology?

Relative singular homology has various applications in different fields. In geometry, it can be used to classify and compare different shapes and spaces. In physics, it can be used to study the behavior of physical systems and understand the topological properties of the universe. In computer science, it can be used in data analysis and pattern recognition. Additionally, relative singular homology has been used in image and signal processing, as well as in the study of biological structures such as DNA and proteins.

## 5. Are there any limitations to using relative singular homology?

Like any mathematical tool, relative singular homology has its limitations. It may not be able to capture all the topological features of a space, as it is based on a particular decomposition of that space. Additionally, the complexity of its calculation can increase significantly for more complicated spaces. Some alternative methods, such as cohomology, may be more suitable for certain situations. It is important to carefully consider the specific problem at hand and choose the appropriate tool for the job.

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