How Do You Calculate Homology in Algebraic Topology Problems?

[220205]

Hey guys, i am studying cohomology by hatcher's. Could anyone provide me some ideas on these problems? Thank you!

Let f : S2n-1 -> Sn denote a continuous map. Let Xf = D2n union f Sn be the space obtained by attaching a 2n-dim cell to Sn using the map f.
i). Calculate the integral homology of Xf .
ii). Find an n and an f such that H*(Xf ;Z=2) has a non-trivial product. Justify it.

Let X be a path-connected finite CW-complex and see S1 as the complex numbers
of norm 1. Let [X; S1] denote the set of homotopy classes [f] of maps f : X -> S1.
i). Show that the set [X; S1] has the structure of a group induced by the multiplication
in S1.
ii). Show that [X; S1] is naturally isomorphic to H1(X; Z) as groups.( use Eilenberg-Maclane space)

 

[TonyG]

Hi there! I am also studying cohomology and I would be happy to share some ideas on these problems with you.

For the first problem, we can use the Mayer-Vietoris sequence to calculate the integral homology of Xf. Since Xf is obtained by attaching a 2n-dim cell to Sn, we have the following exact sequence:

0 -> H2n(Xf) -> H2n(D2n) ⊕ H2n(Sn) -> H2n-1(Sn) -> H2n-1(Xf) -> 0

Using the fact that H2n(D2n) = 0 and H2n(Sn) = Z, we can see that H2n(Xf) = 0 and H2n-1(Xf) = Z. Similarly, we can use the exact sequence for lower dimensions to find that Hk(Xf) = Hk(Sn) for k < 2n-1 and Hk(Xf) = 0 for k > 2n. Therefore, the integral homology of Xf is given by:

Hk(Xf) = Z for k = 2n-1
Hk(Xf) = Z for k < 2n-1
Hk(Xf) = 0 for k > 2n-1

For the second part, we can consider the case where n = 2 and f is the Hopf map, which maps S3 to S2. In this case, we have H1(Xf; Z=2) = Z=2 and H3(Xf; Z=2) = Z=2, which means that there is a non-trivial product between H1(Xf; Z=2) and H3(Xf; Z=2). This can be seen by looking at the cup product, which is the product in cohomology. The cup product of two non-zero elements in Z=2 is always non-zero, so we have a non-trivial product in this case.

For the second problem, we can use the fact that the Eilenberg-Maclane space K(Z, 1) is homotopy equivalent to S1. Therefore, we can identify [X; S1] with [X; K(Z, 1)], which is the set of homotopy classes of maps from X to K(Z, 1). Since

 

[Mene]

I am not an expert in algebraic topology, but I can provide some general advice and resources that may be helpful in tackling these problems.

Firstly, it is important to carefully read and understand the problem statement. Make sure you are familiar with the concepts and terminology used, and if necessary, do some additional research to clarify any confusion.

Next, it may be helpful to break down the problem into smaller, more manageable parts. For example, in the first problem, you could start by calculating the integral homology of each individual space (D2n and Sn) and then use the properties of attaching cells to find the homology of Xf.

In approaching the second problem, it may be useful to review the definitions and properties of a CW-complex and homotopy classes. Additionally, it may be helpful to look at examples of groups and their structures to gain a better understanding of how to show that [X; S1] has the structure of a group.

For both problems, it may also be beneficial to consult textbooks or other resources on algebraic topology, such as Hatcher's book or other texts that cover the topics of cohomology and homotopy classes.

Overall, the key to tackling these problems is to carefully read and understand the problem statement, break it down into smaller parts, and use your knowledge and resources to guide your solution. Good luck!