Another algebra question in algebraic topology

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SUMMARY

The discussion centers on Proposition 3A.5 from Hatcher's "Algebraic Topology," specifically the induced map from Tor(A,B) to Tor(B,A). The user questions the validity of the induced map, noting that the connecting homomorphism from Tor(A,B) to A⊗F₁ and the inclusion from Tor(B,A) to F₁⊗A do not guarantee the existence of an element y in Tor(B,A) corresponding to x'' derived from Tor(A,B). The resolution confirms the correctness of the induced map as stated in Hatcher's proof.

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  • Familiarity with tensor products, specifically A⊗F₁ and F₁⊗A
  • Knowledge of long exact sequences in homology
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quasar987
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In the proof of Proposition 3A.5 in Hatcher p.265 (http://www.math.cornell.edu/~hatcher/AT/ATch3.4.pdf), at the bottow of the page, he writes,

"Since the squares commute, there is induced a map Tor(A,B) -->Tor(B,A), [...]"

How does this follow? The map Tor(A,B)-->A\otimes F_1 is the connecting homomorphism coming from the long exact sequence (see (6) and its proof) and Tor(B,A)-->F_1\otimes A is inclusion.

It one starts with an element x of Tor(A,B), then pushes it to A\otimes F_1 to an element x' and then to F_1\otimes A to an element x'', there is no guarantee as far as I can see that there will be a y in Tor(B,A) with y=x''...

Thanks for any help.
 
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