+++ title = “Products and Massey Products” section = “features” +++
Products and Massey Products
htpy computes and displays the multiplicative structure of Ext, including Yoneda products, Massey products, and matric Massey products.
Yoneda products
The Yoneda product on $\operatorname{Ext}_A(M, \mathbb{F}_p)$ is computed by composing chain maps along the minimal resolution. For the sphere spectrum, the standard generators are:
| Generator | Bidegree $(n, s)$ | Detects |
|---|---|---|
| $h_0$ | $(0, 1)$ | $2 \in \pi_0$ |
| $h_1$ | $(1, 1)$ | $\eta \in \pi_1$ |
| $h_2$ | $(3, 1)$ | $\nu \in \pi_3$ |
| $h_3$ | $(7, 1)$ | $\sigma \in \pi_7$ |
| $h_4$ | $(15, 1)$ | (Kervaire) |
Products with these generators are displayed as structlines in the chart: lines connecting a class $x$ to $h_i \cdot x$.
Viewing products
In the toolbar, each product has a toggle button. Click to show/hide its structlines. The detail panel for any class shows all nontrivial products involving that class.
Product colors
Each product generator has a distinct color in the chart:
- $h_0$: vertical lines (filtration $+1$, stem $+0$)
- $h_1$: diagonal lines (filtration $+1$, stem $+1$)
- $h_2$: steeper diagonal (filtration $+1$, stem $+3$)
Massey products
When $\alpha\beta = 0$ and $\beta\gamma = 0$ in Ext, the Massey product $\langle \alpha, \beta, \gamma \rangle$ is a coset in Ext modulo the indeterminacy $\alpha \cdot \operatorname{Ext} + \operatorname{Ext} \cdot \gamma$.
htpy computes Massey products at the chain level:
- Find null homotopies $a$ and $b$ with $d(a) = \alpha\beta$ and $d(b) = \beta\gamma$.
- The representative is $a\gamma + \alpha b$.
- Indeterminacy generators are computed from the kernel of the null homotopy construction.
Viewing Massey products
Computed Massey products are displayed in the detail panel when you select a class. Each entry shows:
- The factors $\alpha, \beta, \gamma$ with their bidegrees
- The computed value (as a linear combination of Ext generators)
- The indeterminacy (as a list of generators spanning the coset)
4-fold Massey products
htpy also computes $\langle \alpha, \beta, \gamma, \delta \rangle$ when all consecutive products and 3-fold sub-brackets vanish. These require additional null homotopies and have larger indeterminacy.
Matric Massey products
Matric Massey products generalize ordinary Massey products to matrices of Ext elements. Given matrices $A, B, C$ with $AB = 0$ and $BC = 0$, the matric bracket $\langle A, B, C \rangle$ is computed. htpy discovers matric vanishing relations automatically by searching for pairs of Ext elements whose products satisfy the required identities.
Role in the deduction engine
Products and Massey products feed into the deduction engine in several ways:
- Leibniz propagation (Pass 2): Differentials propagate through products via the Leibniz rule.
- Permanent cycle propagation (Pass 1.5): Products of permanent cycles are permanent.
- Moss’ theorem (Pass 1.6): The Massey product structure constrains differentials via the higher Leibniz rule.
- $E_r$ Massey products: Brackets on higher pages detect permanent cycles via Moss’ theorem (Belmont–Kong, Theorem 3.13).