Algebra question in algebraic topology

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SUMMARY

The discussion centers on a specific algebraic topology concept presented in Hatcher's book, particularly on page 262, where the relationship between cokernels and homology groups is established. The key takeaway is that the cokernel in question is identified as H_n(C) ⊗ G, derived from tensoring the exact sequence B_n → Z_n → H_n → 0. This clarification resolves initial confusion regarding the connection between these algebraic structures.

PREREQUISITES
  • Understanding of algebraic topology concepts, particularly homology groups.
  • Familiarity with exact sequences in algebraic structures.
  • Knowledge of tensor products in the context of vector spaces or modules.
  • Ability to read and interpret mathematical texts, specifically Hatcher's "Algebraic Topology".
NEXT STEPS
  • Study the properties of exact sequences in algebraic topology.
  • Explore the concept of tensor products in detail, focusing on their applications in homology.
  • Review Lemma 3A.1 in Hatcher's "Algebraic Topology" for deeper insights.
  • Investigate the implications of cokernels in various algebraic contexts.
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Students and researchers in mathematics, particularly those specializing in algebraic topology, as well as educators seeking to clarify concepts related to homology and exact sequences.

quasar987
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In Hatcher, p. 262 (http://www.math.cornell.edu/~hatcher/AT/ATch3.4.pdf), he writes, just before Lemma 3A.1, "the next lemma shows that this cokernel is just H_n(C)\otimes G. I can't say that I see how this follows.

Thanks!
 
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Ok I see it now. It just comes from tensoring the exact sequence B_n-->Z_n-->H_n-->0.
 

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