Cup product in Morse cohomology Dualizing the Morse complex, we obtain the Morse cohomology, which is isomorphic to the usual singular cohomology and thus admits a cup product.
Does anybody know how this cup product would look like in terms of critical points of a Morse function? What is the geometric picture?
 A: It is given by counting Y-shaped flowlines, using Morse functions along each of the three edges. For more info:
Morse Homotopy, $A_\infty$-Category, and Floer Homologies (by Fukaya)
Googling around also helps for such questions!
A: I want to expand a little bit on Chris' answer. The cup product in singular homology arises from the composition
$$
H^*(M)\otimes H^*(M)\rightarrow H^*(M\times M)\rightarrow H^*(M)
$$
Here the first map is the cross product map, and the second map is the pullback $\Delta^*$ of the diagonal embedding $M\rightarrow M\times M$. This can also be interpreted Morse theoretically.
The cross product map can be computed in Morse homology as follows. Choose two critical points $x$ and $y$ of Morse functions $f_1$ and $f_2$ on $M$. Let $\eta_x$ be the map that sends $x$ to $1$ and all other critical points to zero (this is a basis of the dual complex). Then $\eta_x\otimes \eta_y$ is a generator of $C^*(M)\otimes C^*(M)$. Sending this to $\eta_{(x,y)}$ where $(x,y)$ is the critical point of $f(p,q)=f_1(p)+f_2(q)$ on $M\times M$ defines the cross product map.
In general a map $h:M\rightarrow N$ (equipped with Morse-Smale pairs $(f_M,g_M)$ and $(f_N,g_N)$ induces under the transversality assumption that $h|_{W^u(x;f_M,g_M)}\pitchfork W^s(y;f_N,g_N)$ a map $h^*:HM^*(N)\rightarrow HM^*(M)$ defined on the generators by
$$
h^*\eta_y=\sum_{y}\#(W^u(x;f_M,g_M)\cap h^{-1}(W^s(y;f_N,g_N))\eta_x.
$$
Applying this to the diagonal embedding $\Delta$, and composing this with the cross product we can compute the cup product map via
$$
\eta_x\cup\eta_y=\sum_z \#(W^s(x;f_1,g_1)\cap W^s(y;f_2,g_2)\cap W^u(z;f_3,g_3))\eta_z.
$$
Here the $(f_i,g_i)$ are Morse-Smale pairs such that the stable manifolds and unstable manifolds intersect mutually transversely. These are exactly the $Y$ shaped flowlines of Chris' answer. The transversality condition required for the diagonal map implies that the intersection above is transverse. It is not too hard to see that the expected dimension is what it should be.
A reference for point of view are the appendices to Abbondandolo Schwarz: Floer homology of cotangent bundles and the loop product.
