The preprint constructs an inclusion chain of two-complexes, K(X) ⊂ K(Y) ⊂ K(Z), in which the map induced on π2 is zero at both steps. Here π2 is the second homotopy group the paper follows through the chain. K(X) is non-aspherical; K(Y) is non-aspherical and Cockcroft; K(Z) is not Cockcroft.
That is the paper’s answer to a question about how long an inclusion chain can be when it starts with a non-aspherical complex and every inclusion induces the zero map on π2. Terms such as non-aspherical and Cockcroft are formal labels for properties being tested in these presentations. The result is tied to the displayed complexes, not presented as a general statement about every possible presentation.
The middle stage carries the key property
On the group side, X starts with eight generators and eight relators, or defining relations. Y adds two two-cells along τ and σ. Z then adds four two-cells along a, b, c and d. No generators are added at these stages.
The group construction is based directly on a short balanced presentation of the binary icosahedral group. The nested complexes therefore share the same generator set while the presentation is extended through additional two-cells.
To analyze the inclusions, the paper separates ordinary homology from π2. Ordinary homology is computed from the exponent-sum cellular boundary matrix, whereas π2 is analyzed with the full Fox boundary matrix over the group ring. The two calculations provide the paper’s cellular and group-ring views of the presentations.
At the middle stage, the ordinary second-homology calculation gives H2(K(Y)) = Z^6, a free abelian group of rank six. The paper also establishes that K(Y) is non-aspherical and Cockcroft.
The first inclusion, from K(X) into K(Y), induces the zero map from π2(K(X)) to π2(K(Y)). The pair (K(Y), K(X)) also has the identity property: the projection of π2(K(Y)) onto the coordinates of the two new cells vanishes.
The second inclusion, from K(Y) into K(Z), also induces the zero map on π2. At the terminal stage, K(Z) is simply connected, has an Euler characteristic of χ(K(Z)) = 7, and has π2(K(Z)) isomorphic to H2(K(Z)), with both groups isomorphic to Z^6. The paper classifies K(Z) as not Cockcroft.
A separate result in Lie rings
A second result changes the setting from group-presentation complexes to Lie-ring presentations. It studies an inclusion L1 ⊂ L2, contrasting a smaller presentation with a larger one. The corresponding quantity in this part of the paper is written Π2.
The contrast is exact: Π2(L1) is nonzero, whereas Π2(L2) is zero. The smaller presentation contains a displayed identity among its defining relations, ω = y er1 + x er2, and the proof shows that this expression is nonzero.
That difference remains after base change to every field k: the smaller presentation remains non-aspherical, while the larger one is aspherical. The authors interpret this as a negative answer to the Lie-algebra version of Whitehead’s asphericity question over every field.
This Lie-ring result is not a statistical comparison. It is a formal construction supported by identities, presentation calculations and the stated base-change result, with no participants, measured outcomes or population estimates.
What the construction leaves open
The length of the group construction has a specific boundary. The chain reaches K(Z), but K(Z) is not Cockcroft, so the displayed example does not supply a further Cockcroft stage.
The conclusions also concern the particular group complexes and Lie-ring presentations constructed in the paper. They do not establish that the same asphericity behavior holds for presentations beyond those examples.
An open question is whether a longer inclusion chain can begin with a non-aspherical complex while every inclusion-induced π2 map remains zero. The authors leave that extension unresolved, so the preprint does not settle the maximum possible length.
The document is marked arXiv:2608.20270v1 [math.GT] and dated 20 Aug 2026. It also discloses assistance from ChatGPT 5.6 Sol, while noting that the specific contribution is difficult to isolate.
Paper data and sources
Original title: Two results on asphericity
Authors: Roman Mikhailov
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text