What Are Homology and Graded Groups in Mathematics?

[homology]

Hello,

I'm studying homology and graded groups have come up. I don't completely understand what they are. Wikipedia didn't have an entry on graded groups, but on graded algebras and other graded stuff and the definitions there seemed different than the way graded groups have been used in my book (Homology by Vick). Could someone flesh out some of the principles?

I understand my request is kind of vaque, I suppose I just want to chat with someone about the object.

kevin

 

[matt grime]

Let X be *any* mathematical object, then a grading is (usually) a way of writing X as the union of subsets X(i) with i in an indexing set, usually the integers or the naturals, sometimes Z/2Z. If there is more structure to the underlying object we require that structure to 'compatible' with the grading as the following example illustrates:

A graded group, (graded by N, say) is a group of the form:

$$G= \coprod G_i$$

with i in N, ie the union and with the following rule: that if x is in G_p and y is in G_q then the product xy is in G_{p+q}

the number i in G_i is often said to be the degree. Thus in any graded gruop the identity element must lie in the degree 0 part. Obviously polynomials are the canonical example of a graded object, thinking of the degree of the poly as the degree in the grading and multiplication as the operation that the grading is compatible with,

Lots of things, particulalry homological objects naturally come with a grading, usually from a chain complex or something.

 

[homology]

Thanks for your reply matt grime, naturally I have some more questions.

(1) So if (given your example) x is in G_p and y is in G_q then what operation is the "product" between x and y, yielding xy in G_{p+q}? How would this be defined?

(2) In particular, now I'm confused as to how graded groups show up in homology. Given a chain complex I can't imagine how I would combine an p-chain and a q-chain to get a {p+q}-chain.

(3) With further regards to question (1), I don't understand the "degree." Certainly in the context of polynomials this term makes sense, but what would it mean in other contexts. That is, for x in G_p, what would it mean for x to have degree p?

(4) Expanding on question (3) could you give me an example in homology? (is degree just the dimension of the chains? i.e. S_n is the space of n-chains, so the the n-chains have degree n?

(5) What's that funny upside down product symbol thingy?

That'll do it for now, thanks again,

Kevin

 

[matt grime]

(1) the group composition

(2) tensor them together

(3) the term degree is used *by analogy*, thus we declare something to have degree p if that degree fits in with the group operation.

(4) probably better off reading the book.

(5) coproduct or direct sum.

i have probably confusingly mixed together graded rings and graded groups. But the idea is the same in either case.

A grading is just a way of splitting things into different parts that behaves well with the composition operation. It isn't very interesting in itself.

The integers themselves come with a grading

define the degree of n in Z to be n. Then adding things in degree n and m porduces something in degree n+m.

 

[homology]

(1) the group composition

Hmm, what is group composition...?

 

[matt grime]

the "binary operation" that allows you to compose two elements in a group.

 

[a-lbi]

According to Spanier's book.

Graded group is simply a sequence of abelian groups $$\{C_q\}$$ indexed by integers.

Homomorphism of degree d is abelian groups homomorphism $$C_q \rightarrow C_{q+d}$$. Of course composition of homomorphisms with degrees d1 and d2 is homomorphism of degree d1+d2.

 

[lavinia]

Hmm, what is group composition...?

I don't think that there is a binary operation between arbitrary elements of a graded group, only generally between elements of the same grade.

For some graded groups there are multiplications but they are not usually commutative in the usual sense.

In cohomology there are products and these turn the graded group of cohomology groups into a graded algebra.

 

[Bacle]

Just to add a few things to what has been said, maybe a slightly-different angle:



Grading in homology allows you put together(homology) groups of different

dimension that belong to the same

space. This organizes the homology of the space into a single group that contains the

total information about the total homology. If you have a chain complex in homology

( yes, named after you! ), which organizes and puts together the chain groups of

different dimension into a single group. The boundary operator allows you to go from

a given grading "k" , to grading "k-1" . In cohomology, cochain complexes go in the

opposite direction. As Lavinia said, you can (formally) add objects of the same grade--

chains of the same dimension--but it is not meaningful in this context to (formally)

add chains of different dimension. The grading expresses the fact that the space is

formed by a collection of groups of elements that are related to others of the same

grade (dimension), but that are otherwise not related; the information about the

k-th and j-th homology groups of a space.