Fundamentals of Tor Functor for an Intro Algebraic Topology Course I'm learning about the universal coefficient theorem in my first-semester algebraic topology course, and to state and prove the theorem we needed to introduce the Tor functor. Here the class ran into the unfortunate problem that the definition of the functor is very abstract and category theoretical (the class made a brief diversion into the notion of derived functors, but nothing near a comprehensive treatment [we didn't even get close to defining an abelian category]). I understand that to appreciate how the Tor functor really works would take a long digression into homological algebra, but for my purposes (the universal coefficient theorem and the Künneth formula) all that is overkill.
I'm looking for a direct minimal treatment that can provide a working definition of the Tor functor in the case of tensor products with abelian groups (all I'm interested in is the first derived functor, not the others), from which all the necessary properties of the functor can be deduced. I believe the following are all the properties I should care about: (this is taken from Massey's Singular Homology Theory p. 121):

*

*$\operatorname{Tor}(A,B)$ and $\operatorname{Tor}(B,A)$ are naturally isomorphic.

*If either $A$ or $B$ is torsion-free, then $\operatorname{Tor}(A,B) = 0$.

*Let $0 \to F_1 \overset{h}{\to} F_0 \overset{k}{\to} A \to 0$ be a short exact sequence with $F_0$ a free abelian group; it follows that $F_1$ is also free. Then there is an exact sequence as follows: $$0 \to \operatorname{Tor}(A,B) \to F_1 \otimes B \overset{h \otimes 1}{\to} F_0 \otimes B \overset{k \otimes 1}{\to} A \otimes B \to 0.$$

*For any abelian group $G$, $\operatorname{Tor}(\mathbb{Z}_n,G)$ is isomorphic to the subgroup of $G$ consisting of all $x \in G$ such that $nx = 0$.

*$\operatorname{Tor}$ is an additive functor in each variable.

*Let $0 \to A' \overset{h}{\to} A \overset{k}{\to} A'' \to 0$ be a short exact sequence of abelian groups; then we have the following long exact sequence: $$0 \to \operatorname{Tor}(A',B) \overset{\operatorname{Tor}(h,1)}{\to} \operatorname{Tor}(A,B) \overset{\operatorname{Tor}(k,1)}{\to} \operatorname{Tor}(A'',B) \to A' \otimes B \overset{h \otimes 1}{\to} A \otimes B \overset{k \otimes 1}{\to} A'' \otimes B \to 0.$$
The book claims that property 3 can be used to define the Tor functor since any abelian group is the homeomorphic image of a free abelian group. I don't quite see how this would go. Do we just define it as the kernel of $h \otimes 1$? How is this independent of the free group we choose to map into $A$? Also, what are the maps $\operatorname{Tor}(h,1)$ and $\operatorname{Tor}(k,1)$ in the last exact sequence?
I don't need a full explanation of everything here, but if anyone could point me to a reference that provides just enough information about $\operatorname{Tor}$ for my purposes it would be much appreciated. The sources suggested in Massey (Cartan and Eilenberg, Hilton and Stammbach, and MacLane) are all full-on homological algebra books that don't define Tor until about 100 pages in.
 A: It's part of the machinery of homological algebra that the group you get from defining Tor using property 3 is independent of the choice of free resolution. This is indeed not obvious from "bare hands"! If you're willing to take this on faith property 3 is a pretty hands-on definition (by which I mean you just define $\text{Tor}(A, B)$ to be $\text{ker}(h \otimes 1)$) and determines $\text{Tor}$ for abelian groups. Here is a sequence of exercises you can try:

*

*Prove that property 3 implies property 2 in the special case that either $A$ or $B$ is $\mathbb{Z}^n$.

*Prove that property 3 implies property 4.

*Prove that property 3 implies property 5. Without assuming property 1 this will require two different proofs for additivity in $A$ and additivity in $B$. Alternatively you can prove additivity in $B$ and assume property 1 to deduce it in $A$.

*Use property 3 to compute $\text{Tor}(A, B)$ if $A$ is finitely generated, in terms of $B$. In particular, if $A$ and $B$ are both finitely generated, verify a weak form of property 1 that $\text{Tor}(A, B) \cong \text{Tor}(B, A)$ (not necessarily naturally).

*Prove that property 3 implies that $\text{Tor}(A, -)$ preserves filtered colimits. Using the fact that an abelian group is torsion-free iff it it's a filtered colimit of copies of $\mathbb{Z}^n$, and assuming property 1, prove property 2.

Once you believe that Tor preserves filtered colimits, using the fact that every abelian group is a filtered colimit of its finitely generated subgroups, the computation of Tor for finitely generated abelian groups actually determines it in general. For example it implies that $\text{Tor}(\mathbb{Q}/\mathbb{Z}, B)$ is exactly the subgroup of all torsion elements in $B$. This can also be proven using property 6 and the short exact sequence $0 \to \mathbb{Z} \to \mathbb{Q} \to \mathbb{Q}/\mathbb{Z} \to 0$, which is another nice exercise.
