I am interested in seeing if and how Morse Theory can "do everything". Some core things are handle decomposition, Bott periodicity, and Euler characteristic. But what do the normal (co)homology operations look like from Morse Homology?

**Poincare duality** $H_*(M)\cong H^{n-\ast}(M)$ is the symmetry $f\to -f$, i.e. the reversal of flowlines.

**Cup product** $H^i(M)\otimes H^j(M)\to H^{i+j}(M)$ is given by counting Y-shaped flowlines, using Morse functions along each of the three edges. The **cap product** is connected to the above two.

**Kunneth isomorphism** $H_\ast(M\times N)\cong H_\ast(M)\otimes H_\ast(N)$ is combining flowlines from $f_1:M\to\mathbb{R}$ and $f_2:N\to \mathbb{R}$ to get flowlines for $f_1+f_2:M\times N\to\mathbb{R}$.

**Leray-Serre spectral sequence**: pull back a Morse function on the base (flowlines of total space project onto flowlines of base space) and use a filtration by ordering the critical-point indices.

Does someone know what goes on for the following?

- Slant product
- Alexander duality
- Steenrod operations (in particular, the
*Cartan formula*) - Massey triple product

My guess for (4) is counting X-shaped flowlines, and then I get suspicious about its relation to $A_\infty$-structures from Lagrangian-Intersection Floer homology.

[[Edit]] There was a MathOverflow post for (2), here. **Alexander duality** $H_\ast(S^n-M)\cong H^{n-1-\ast}(M)$ arises by taking a height function on $S^n$ and perturbing it to become Morse on the subspace $M\subset S^n$, and then separating the critical points according to its tubular neighborhood and its complement.

[[Edit]] Cohen and Schwarz' paper "A Morse Theoretic Description of String Topology" provides the **relative cohomology** and the **Thom isomorphism**, as well as **homomorphisms** arising from proper embeddings of submanifolds.