Homology Functor, Prod. Spaces, Chain Groups: Refs Needed

[WWGD]

Hi all,
Went to a seminar today, arrived a few minutes late; hope someone can tell me something about this topic and/or give a ref so that I can read on it . I know this is a lot of material; if you can refer me to at least some if, I would appreciate it :

1)Basically, understanding how/why the (co)homology functor can/does fail to agree with the product of spaces, when we do not work with coefficients in a field i.e.,

$H^k(X \times Y) \neq \bigoplus_{i+j=k} H^i (X) \otimes H^j(X)$

is not satisfied when we do not work with coefficients in a field; I am tooignorant at this point to understand why/how field coefficients matter ( I was considering the issue of torsion, but there was no mention that the fields had to be of characteristic zero). Also,

2)What kind of correction terms do we need when we consider (co)chain groups on product spaces, i.e., what is the relationship between$C_p(X \times Y)$ and $C_p(X), C_p(Y)$ and how the differential/boundary operator works on the (co)homology of products, i.e., if we have $\delta$ boundary operator on different (co)chain complexes, how/when can we define a boundary operator on the product $X \times X$. Does it make sense to consider a product of (co)chain complexes with different boundary operators?

Basically, we work with inclusion , diagonal and projection diagrams/operations, i.e., $x \rightarrow (x,x)$, etc.

Thanks.

 

[WWGD]

I guess I was being lazy; in case anyone is interested, the main topics are algebras, coalgebras, and their dualizations, the existence of coproducts in homology/cohomology, Hopf algebras , etc. Some of these issues and other related ones are dealt, e.g., in : http://mathoverflow.net/questions/415/does-homology-have-a-coproduct

Still, I would appreciate additional comments/refs.

 

[homeomorphic]

Have you read what Hatcher has to say about the chain complexes and homology/cohomology of products (especially for CW-complexes)?

http://www.math.cornell.edu/~hatcher/AT/ATpage.html

I tend to like to figure stuff out in the CW setting because it's more geometrical and when all is said and done, everything is weak-equivalent to a CW complex, so you don't lose anything as far as homology and cohomology goes.

I'm also being a bit lazy, but if I wanted to understand it more fully, I'd go back and read that. It's hard to see what you were missing because I don't know how you convinced yourself that the homology is that way if you have field coefficients. To me, I just know the chain complex works that way and it's not clear that when you take homology that it's going to break down the same way in terms of just the direct sum of tensor products thing that you have there. That only happens on the chain level.

 

[WWGD]

There was also the issue of the conditions , if any, under which the product in cohomology , i.e., the cup product can be dualized into a coproduct in homology, and general conditions under which homology and cohomology are duals of each other. It seems like finite-dimensionality and bgeing torsion-free are necessary, thou am not sure they are sufficient. But , yes, I do need to read some more; thanks for the link.

 

[blue_raver22]

Hi there,

Thank you for sharing your thoughts and questions about the homology functor, product spaces, and chain groups. These are all important concepts in algebraic topology and can be quite complex, so I understand your desire for more resources and references.

First, let's address your point about the (co)homology functor and its relationship with the product of spaces. This is a fundamental property of the functor that is often taken for granted when working with coefficients in a field. However, when we do not work with coefficients in a field, the (co)homology functor may no longer satisfy the product property you mentioned.

The reason for this is that the product of spaces is not a ring when we do not work with coefficients in a field. This means that the usual algebraic operations we use to define the (co)homology functor, such as addition and multiplication, may not hold in this context. This can lead to discrepancies between the (co)homology of the product space and the product of the (co)homology groups of the individual spaces.

As for your second point, the relationship between (co)chain groups on product spaces is also an important aspect to consider. In general, the (co)chain groups on the product space can be thought of as the tensor product of the (co)chain groups on the individual spaces. This means that the (co)chain groups on the product space will have elements that are products of elements from the (co)chain groups on the individual spaces.

The differential/boundary operator on the product space can then be defined as a combination of the differential/boundary operators on the individual spaces. This is where the inclusion, diagonal, and projection diagrams/operations come into play. By using these operations, we can define the differential/boundary operator on the product space in terms of the differential/boundary operators on the individual spaces.

In conclusion, the homology functor, product spaces, and chain groups are all interconnected and understanding their relationships is crucial in algebraic topology. As for references, I would recommend looking into textbooks on algebraic topology, such as "Algebraic Topology" by Allen Hatcher, for more in-depth explanations and examples.

I hope this helps and good luck with your research!