Operations via Morse Theory

Question

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. 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.

Answer 1

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 (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.

Answer 2

This paper by Cohen and Norbury discussed Steenrod operations, including Adem relations and Cartan formulae.

Here's the abstract:

In this paper we define and study the moduli space of metric-graph-flows in a manifold M.

This is a space of smooth maps from a finite graph to M, which, when restricted to each edge, is a gradient flow line of a smooth (and generically Morse) function on M. Using the model of Gromov-Witten theory, with this moduli space replacing the space of stable holomorphic curves in a symplectic manifold, we obtain invariants, which are (co)homology operations in M. The invariants obtained in this setting are classical cohomology operations such as cup product, Steenrod squares, and Stiefel-Whitney classes. We show that these operations satisfy invariance and gluing properties that fit together to give the structure of a topological quantum field theory. By considering equivariant operations with respect to the action of the automorphism group of the graph, the field theory has more structure. It is analogous to a homological conformal field theory. In particular we show that classical relations such as the Adem relations and Cartan formulae are consequences of these field theoretic properties.

These operations are defined and studied using two different methods. First, we use algebraic topological techniques to define appropriate virtual fundamental classes of these moduli spaces. This allows us to define the operations via the corresponding intersection numbers of the moduli space. Secondly, we use geometric and analytic techniques to study the smoothness and compactness properties of these moduli spaces. This will allow us to define these operations on the level of Morse-Smale chain complexes, by appropriately counting metric-graph-flows with particular boundary conditions.

Answer 3

Steenrod operations in Floer homology are constructed in the thesis of Matthias Schwarz: http://www.math.uni-leipzig.de/~schwarz/diss.pdf

Perhaps a similar construction is feasible for Morse homology?