Homology Q: Hatcher Top Page 125 - Isomorphism Explanation

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In summary, the first isomorphism in Hatcher's book on top of page 125 comes from the exact sequence of the pair (X\cup CA,CA) and the fact that CA is contractible, which results in every third group in the sequence vanishing and the remaining two groups being isomorphic.
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quasar987
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In Hatcher top of page 125, I do not understand where the first isomorphism comes from. He says it comes from the exact sequence of the pair, using the fact that CA is contractible, but what is the precise line of reasoning here?

Hatcher's book: http://www.math.cornell.edu/~hatcher/AT/AT.pdf

Thanks!
 
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Since CA is contractible [tex]\tilde{H}_n(CA)=0[/tex] for all n (there is a tilde over the H, but it's hard to see). Thus, if you write down the exact sequence of the pair [tex](X\cup CA,CA)[/tex] every third group in the sequence vanishes, which makes the two other groups isomorphic.
 
  • #3
I see, thanks!
 

FAQ: Homology Q: Hatcher Top Page 125 - Isomorphism Explanation

What is homology?

Homology is a concept in biology that refers to the similarity between different organisms or structures due to a shared evolutionary ancestry. It can also refer to the similarity between different genes or proteins.

How is homology determined?

Homology is determined through evolutionary comparisons, such as examining the anatomical, genetic, or developmental similarities between different organisms. Molecular techniques, such as DNA sequencing, can also be used to determine homology between genes or proteins.

What is an isomorphism?

In mathematics, an isomorphism is a mapping between two mathematical structures that preserves their structure and properties. In the context of homology, isomorphism refers to a one-to-one correspondence between the homology groups of two different structures.

How are isomorphisms used in homology?

Isomorphisms are used in homology to compare and classify different structures or organisms based on their shared evolutionary ancestry. They help to identify and understand the relationships between different organisms and their traits.

Can homology be used to predict evolutionary relationships?

Yes, homology can be used to predict evolutionary relationships between organisms. The more similar two organisms or structures are in terms of their homology, the more closely related they are likely to be. However, other factors, such as convergent evolution, must also be considered when predicting evolutionary relationships.

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