+++ title = “Cell Complexes and Cofibers” section = “features” +++
Cell Complexes and Cofibers
htpy supports building new spectra from existing ones by attaching cells, constructing cofiber sequences, and using the induced long exact sequence for deduction.
Taking cofibers
The primary workflow for building cell complexes:
- In the viewer, navigate to the $E_\infty$ page of a module.
- Select a class at filtration 1 (these correspond to maps of spectra, i.e., potential attaching maps).
- Click “Take Cofiber” in the detail panel.
- Choose between:
- Review: Opens the module builder with suggested Steenrod operations pre-checked. You can adjust before computing.
- Quick compute: Constructs the cofiber definition automatically and computes immediately.
The resulting cofiber module tracks its provenance — the parent module, the cell degree, and the attaching operations — enabling the deduction engine to compute naturality maps automatically.
Built-in cell complexes
htpy includes several standard cell complexes:
| Complex | Description | Attaching map |
|---|---|---|
| $S^0$ | Sphere spectrum | (base case) |
| $C(p)$ | Moore spectrum $M(p)$ | $p : S^0 \to S^0$ |
| $C(\eta)$ | Cofiber of $\eta$ | $\eta : S^1 \to S^0$ |
| $C(\nu)$ | Cofiber of $\nu$ | $\nu : S^3 \to S^0$ |
| $C(\sigma)$ | Cofiber of $\sigma$ | $\sigma : S^7 \to S^0$ |
| $C(2, \eta)$ | Iterated cofiber | $\eta$ on $C(2)$ |
These are available from the Cell Complexes tab on the compute page. Each complex specifies its cells, attaching maps, and expected generator names.
The long exact sequence
For a cofiber sequence $M \xrightarrow{f} C \to C/M$, the long exact sequence in Ext:
$$\cdots \to \operatorname{Ext}^s(C/M) \xrightarrow{\delta} \operatorname{Ext}^{s+1}(M) \xrightarrow{f^*} \operatorname{Ext}^{s+1}(C) \to \cdots$$
provides three maps that the deduction engine uses:
- Inclusion $i^*$: Transfers differentials from $C$ to $M$ (Pass 1.75).
- Projection $\pi^*$: Computed automatically; enables reverse naturality and LES subcomplex arguments.
- Connecting homomorphism $\delta$: Computed via the chain-level snake lemma.
Vanishing line propagation
Vanishing lines propagate through cofiber sequences. If the parent module $M$ and the root module have vanishing lines, htpy computes a vanishing line for the cofiber $C$ using the long exact sequence bound:
$$\operatorname{Ext}^s(C) = 0 \text{ when both } \operatorname{Ext}^s(M) = 0 \text{ and } \operatorname{Ext}^{s-1}(C/M) = 0$$
The propagated vanishing line is computed from the slopes and intercepts of the parent and root vanishing lines, adjusted by the cell connectivity.
The cell complex builder
The standalone Cell Complex Builder (accessible from the landing page) lets you define cell complexes from scratch:
- Specify cells (dimension and name).
- Define attaching maps as Steenrod algebra elements evaluated at a stem value.
- Preview the module definition JSON.
- Compute and view the result.
The builder includes a scalar evaluator that translates expressions like
Sq4 Sq2 Sq1 into the corresponding module action.