An arXiv preprint presents a p-adic homotopy framework that turns finite realization into two algebraic checks for a specified class of spaces. For simply-connected, finite-type p-complete CW complexes, the model must descend to rational scalars, and mod-p cohomology must vanish in sufficiently high degrees. Within that class, the two conditions are equivalent to being the p-completion of a finite CW complex, a space assembled from finitely many basic cells. Here p-complete means completed at a chosen prime p.
The same preprint carries the construction into étale homotopy types, topological-style objects attached to algebraic varieties, and uses it to study Galois actions, p-adic representations and weight patterns. The document is an arXiv preprint, version 1, dated 26 Aug 2026.
Turning topology into algebra
At the core is a translation from topology into algebra. For p-complete spaces, the method starts with the singular-cochain E∞-algebra and replaces it with a quasi-isomorphic commutative differential graded algebra over Q_p. In ordinary language, the replacement is designed to preserve the relevant structure while giving the authors a more manageable algebraic object. The resulting minimal models are then used as compact records of the homotopy information.
For nilpotent p-complete finite-type spaces, a class covered by the paper's main recovery theorem, the minimal model captures the Q_p-homotopy groups and encodes Whitehead products, operations that describe how higher-dimensional features interact. The result is an algebraic route to information that would otherwise be expressed through the space itself. In the stated non-simply-connected setting, the Lie algebra dual to the 1-minimal model is the Lie algebra of the continuous Mal'cev Q_p-completion of the fundamental group. That gives the fundamental group an associated Q_p Lie-algebra description.
The finite-complex test
The finite-CW criterion is deliberately stronger than a check of cohomology alone. Its first requirement is rational scalar descent of the E∞-algebra or Q_p-minimal CDGA. Its second is eventual vanishing of mod-p cohomology. The theorem says both are needed, and together they are equivalent to finite-CW realization for the simply-connected, p-complete, finite-type class.
The paper reinforces that point with a counterexample. It constructs a simply-connected p-complete finite-type CW complex whose rational p-adic cohomology vanishes in high degrees but that is not the p-completion of any finite CW complex. Because this is an existence result, it does not say how common such examples are. It does show that high-degree cohomology vanishing by itself cannot guarantee a finite realization.
From spaces to varieties
Geometry enters through smooth proper varieties over separably closed fields. When the completed étale fundamental group is trivial, finite-CW realization of the variety's étale homotopy type is equivalent to descent of its Q_ℓ étale cohomology algebra from a graded Q-algebra. In effect, the question of whether the topological-style object has a finite model is reduced to whether its cohomology algebra has the required rational origin, under those assumptions.
A separate theorem says that the pro-ℓ étale homotopy type of every connected smooth proper variety over a separably closed field is Q_ℓ-formal. Formality here means that the homotopy type admits a simpler cohomological model. It is related to, but distinct from, finite-CW realization: the latter also requires the stated fundamental-group and descent conditions.
Arithmetic structure survives
The framework also records how arithmetic symmetries act on these models. For the stated smooth proper varieties, the Galois action on the homotopy automorphism group of the minimal model is continuous in the specified ℓ-adic topology. This is a structural continuity statement about the model and its symmetries.
For connected smooth proper p-adic adic spaces with a point, the relevant Mal'cev Lie quotients are successively de Rham. When the completed étale fundamental group is trivial, the higher étale homotopy representations are also successively de Rham. For connected smooth proper schemes over the ring of integers with good reduction, the corresponding Lie quotients are successively crystalline, and the higher representations share that property when the completed étale fundamental group is trivial. De Rham and crystalline are p-adic classifications of the representations, so the result specifies their arithmetic structure rather than supplying numerical estimates.
The paper also gives a weight range for higher étale homotopy representations. In homotopy degree n, for n at least 2, they are mixed of weights from n through 2n minus 1. Whitehead products are Galois-equivariant and additive in the weight grading. This links the algebraic operations in the minimal model to both the arithmetic action and the weight bookkeeping.
Relations become finite in special cases
At the level of the fundamental group, properness imposes a tight presentation. For proper varieties, the Mal'cev Lie algebra has defining relations generated in bracket length 2 and is determined by the quotient involving Γ3. The bracket length records how many nested Lie brackets are needed, while Γ3 marks the relevant stage of the lower-central filtration.
In the special case where ℓ is not p and the field is an algebraic closure of a finite field, the defining relations are generated in bracket lengths 2, 3 and 4, and the Lie algebra is determined by the quotient involving Γ5. This is a bounded presentation for that arithmetic setting, not a statement about every variety or every prime.
A framework with defined boundaries
These results concern particular classes and constraints. The finite-CW criterion is stated for simply-connected p-complete finite-type CW complexes, the étale criterion adds smoothness, properness, a separably closed field and a trivial completed étale fundamental group, and the representation results add conditions such as a point or good reduction. Those hypotheses define the scope of the conclusions.
The preprint therefore offers a common language for comparing homotopy, cohomology and arithmetic information. Its most concrete message is that finite realization can be characterized algebraically in the stated setting, but only when both the rational descent and high-degree mod-p vanishing requirements are checked. The counterexample shows why high-degree cohomology vanishing should not be treated as a universal shortcut.
Paper data and sources
Original title: $\mathbb{Q}_p$-Homotopy Types and Applications to Topology and Algebraic Geometry
Authors: Runjie Hu, Guozhen Wang
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text