Another algebra question in algebraic topology

In summary, Hatcher is proving Proposition 3A.5, and at the bottom of page 265, he explains that since the squares commute, there is a map Tor(A,B)-->Tor(B,A) induced. This follows from the connecting homomorphism Tor(A,B)-->A\otimes F_1 and Tor(B,A)-->F_1\otimes A, as shown in (6) and its proof. However, there is no guarantee that an element x of Tor(A,B) will have a corresponding y in Tor(B,A) after being pushed to A\otimes F_1 and then to F_1\otimes A.
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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)-->[itex]A\otimes F_1[/itex] is the connecting homomorphism coming from the long exact sequence (see (6) and its proof) and Tor(B,A)-->[itex]F_1\otimes A[/itex] is inclusion.

It one starts with an element x of Tor(A,B), then pushes it to [itex]A\otimes F_1[/itex] to an element x' and then to [itex]F_1\otimes A[/itex] 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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Related to Another algebra question in algebraic topology

1. What is algebraic topology?

Algebraic topology is a branch of mathematics that studies topological spaces by assigning algebraic structures to them. It uses algebraic tools to analyze and understand geometric shapes and spaces.

2. What is the difference between algebraic topology and other branches of topology?

Algebraic topology focuses on studying topological spaces using algebraic structures, while other branches of topology, such as point-set topology, focus on the properties of spaces without using algebraic techniques.

3. What is the purpose of using algebraic techniques in topology?

The use of algebraic techniques in topology allows for a more systematic and general approach to studying topological spaces. It also provides a way to classify and compare different spaces based on their algebraic properties.

4. How is algebraic topology applied in real-world situations?

Algebraic topology has many applications in diverse fields such as physics, engineering, computer science, and biology. It can be used to analyze and model complex systems, study networks and data structures, and understand the properties of physical spaces.

5. What are some common objects and structures studied in algebraic topology?

Algebraic topology studies a wide range of objects and structures, including manifolds, graphs, knots, and homotopy groups. It also deals with fundamental concepts such as topological spaces, continuous maps, and homotopies.

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