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?


1. Slant product
2. Alexander duality
3. Steenrod operations (in particular, the *Cartan formula*)
4. 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](https://mathoverflow.net/questions/53619/alexander-duality-in-terms-of-morse-function). **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.



Operations via Morse Theory
Massey products are discussed in Section 1.3 of 



> 
> Fukaya, Kenji. Morse homotopy,
>  $A\_{\infty}$-category, and Floer
>  homologies. Proceedings of GARC
>  Workshop on Geometry and Topology '93
>  (Seoul, 1993), 1--102,
> 
> 
> 


available [here](http://www.math.kyoto-u.ac.jp/~fukaya/mfikki.pdf) (pdf). The Massey products are obtained by counting gradient flow graphs with four external edges and one (finite-, possibly zero-, length) internal edge. Fukaya sketches a construction of an $A\_{\infty}$ category whose objects are Morse functions $f$ and with morphisms from $f$ to $g$ given by the Morse chain complex of $f-g$. In particular the Massey products can then be seen as arising from the $A\_{\infty}$ structure in a standard formal way.


There is a relation to Lagrangian Floer theory: a Morse function $f:M\to \mathbb{R}$ corresponds to a Lagrangian submanifold $graph(df)$ of $T^\*M$ and intersections between $graph(df)$ and $graph(dg)$ are in obvious bijection with critical points of $f-g$. There are results, pioneered by 



> 
> Fukaya, Kenji; Oh, Yong-Geun.
>  Zero-loop open strings in the
>  cotangent bundle and Morse homotopy.
>  Asian J. Math. 1 (1997), no. 1,
>  96--180,
> 
> 
> 


which relate the gradient flow graphs appearing in the Morse $A\_{\infty}$ operations to the holomorphic curves appearing in the Lagrangian Floer $A\_{\infty}$ operations.


