Visualizing a generator of $H^2(S^1 \times S^1)$ Does anyone know of a good visualization for the generator of $H^2(S^1 \times S^1)$? (In any coefficients and any cohomology theory). I gather it might help to think of it as the product of the two angle forms, and I'm trying to make that more transparent for myself.
My apologies if someone has asked this here before; I searched and found nothing.
 A: Here is a visualization for the regular singular homology theory with coefficients in $\mathbb{R}$. The generator for $H_2$ is going to be the cycle that is "the entire manifold" (this is called the fundamental class). Since our manifold is closed, it has no boundary, and it has no $3$-cells, so it generates $\mathbb{Z}$ (try contrasting this with $\mathbb{R}P^2$). Then cohomology is just the "dual," so we are assigning this fundaental class a number using our cohomology element. $H^2$ should be generated by the class that sends the fundamental class to $1$ in $\mathbb{R}$. We can interpret this generator of $H^2$ as the cup product of the $2$ generators in $H^1$. This is nicely explained here with a picture.
This will not work for other cohomology theories, and by the time we are seriously trying to do calculations with them, we will need to obtain a lot more tools and eventually realize that pretty much nothing is explicitly computable except for $\mathbb{C}P^N$, $\mathbb{R}P^N$, and maybe $S^N$ sometimes if we are lucky.
Addendum. By Poincaré duality, $H^2(S^1 \times S^1)$ is isomorphic to $H_0(S^1 \times S^1)$. The latter is infinite cyclic. Indeed, consdering the natural cellular decomposition of the torus, its $0$th cellular homology is generated by the unique $0$-cell.
Expanding on the above, for homology, $H_0$ is generated by any point, $H_1$ is generated by the two loops, $H_2$ is generated by the whole surface. For cohomology, $H^0$ is generated by the surface, $H^1$ is generated by the $2$ loops, $H^2$ is generated by a point. Now, the pairing is the "intersection pairing." In how many points do the corresponding cycles intersect? For instance, the generator of $H_2$ (the whole surface) intersects the generator of $H^2$ (a single point) in $1$ point. And each of the $1$-cycles has nontrivial pairing with the "opposite" cycle, but trivial pairing with itself. This is because it can be deformed off itself to have $0$ intersections.
Another way of seeing this. By the universal coefficient theorem, $H^2(S^1 \times S^1)$ is isomorphic to $\text{Hom}(H_2(S^1 \times S^1), \mathbb{Z})$. This is, more or less, generated by the fundamental class of the torus, i.e. more or less, the sum of all simplices for a simplicial decomposition of the torus.
Addendum 2. Another useful trick for understanding cohomology of manifolds is Alexander duality. If we embed $S%1 \times S^1$ into the $3$-sphere (which we can image as $\mathbb{R}^3$), then $H^2(S^1 \times S^1)$ is naturally isomorphic to the reduced $H_0$ of the complement. For the same reason, any surface that bounds a region in $S^3$ has nontrivial second cohomology.
A: It often helps to visualize cohomology classes as the Poincaré dual homology classes (or more specific their representants). Let $M$ be a closed oriented $n$-manifold. For $x\in H^nM \cong Hom(H_nM,\mathbb Z)$ we have a dual class $P.D.(x)\in H_0M$. Let $m\in M$ be any point representing this homology class (if $M$ is not connected choose one for each component). Now with the intersection pairing we get that $x$ represents the homomorphism which is given by transverse intersecting the representant of $P.D.(x)$ with submanifolds representing $n$-homology classes. Note that a representant of a generator of $H_nM$ is $M$ itself.
Also $kx$ is the homomorphism induced by intersecting with the representant of the Poincaré dual which is picking $k$ different oriented points per component.
A: Two things that come to mind are


*

*It's the dual (in the sense of universal coefficients) to the fundamental class, as Mike Miller says, and 

*It corresponds to any point in the torus by Poincaré duality, as Daniel Valenzuela says. 
The visualization via Poincaré duality might seem unsatisfying but in fact it works, suitably reinterpreted (augmented with Lefschetz duality and noncompact Poincaré duality), for spaces substantially more general than closed oriented manifolds. This idea is sometimes known as "geometric cohomology"; you can see a description of it, for example, in these notes (particularly Lectures 5 and 6), although I can make no guarantees as to their accuracy. 
